K-d tree
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at K-d tree. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
A k-d tree (short for k-dimensional tree) is a space-partitioning data structure for organizing points in a k-dimensional space. k-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key (e.g. range searches and nearest neighbor searches). k-d trees are a special case of binary space partitioning trees.
k-d trees are not suitable, however, for efficiently finding the nearest neighbor in high dimensional spaces. As a general rule, if the dimensionality is k, the number of points in the data, N, should be N ≫ 2k. Otherwise, when k-d trees are used with high-dimensional data, most of the points in the tree will be evaluated and the efficiency is no better than exhaustive search, and other methods such as approximate nearest-neighbor are used instead.
Task: Construct a k-d tree and perform a nearest neighbor search for two example data sets:
- The Wikipedia example data of [(2,3), (5,4), (9,6), (4,7), (8,1), (7,2)].
- 1000 3-d points uniformly distributed in a 3-d cube.
For the Wikipedia example, find the nearest neighbor to point (9, 2) For the random data, pick a random location and find the nearest neighbor.
In addition, instrument your code to count the number of nodes visited in the nearest neighbor search. Count a node as visited if any field of it is accessed.
Output should show the point searched for, the point found, the distance to the point, and the number of nodes visited.
There are variant algorithms for constructing the tree. You can use a simple median strategy or implement something more efficient. Variants of the nearest neighbor search include nearest N neighbors, approximate nearest neighbor, and range searches. You do not have to implement these. The requirement for this task is specifically the nearest single neighbor. Also there are algorithms for inserting, deleting, and balancing k-d trees. These are also not required for the task.
AArch64 Assembly
/* ARM assembly AARCH64 Raspberry PI 3B */
/* program kdtree64.s */
/* the coordinates of the points are integers numbers.
The displayed distance represents the square of the distance between 2 points */
/* This program draws heavily from the published C program. Thanks to its creator. */
/************************************/
/* Constantes */
/************************************/
.include "../includeConstantesARM64.inc"
.equ MAXI, 3
.equ NBPOINTSRAND, 2000
.equ MAXCOORD, 10000
/***********************************************/
/* structures */
/**********************************************/
/* Définition node */
.struct 0
nodeKD_val: // value array
.struct nodeKD_val + (8 * MAXI)
nodeKD_left: // left pointer
.struct nodeKD_left + 8
nodeKD_right: // right pointer
.struct nodeKD_right + 8
nodeKD_end:
/*********************************/
/* Initialized data */
/*********************************/
.data
szMessAddressTree: .asciz "Node address : "
szMessTreeValue: .asciz " Value : "
szMessLeft: .asciz " Left : "
szMessRight: .asciz " Right : "
szMessResult: .asciz "Nearest point = "
szMessRandCoor: .asciz "\n\nRandom point coordonnés = "
szMessVisited: .asciz "Node visited = "
szMessDistance: .asciz "square distance :"
szMessStart: .asciz "Program 64 bits start.\n"
szCarriageReturn: .asciz "\n"
.align 4
tabPoint1: .quad 2,3, 5,4, 9,6, 4,7, 8,1, 7,2
//tabPoint1: .quad 1,7, 3,4, 4,6, 6,2, 8,12, 10,9, 12,3, 14,1
//tabPoint1: .quad 3,5, 1,3, 2,8, 4,6, 5,4
.equ NBPOINTS, (. - tabPoint1) / 8
tabPoint2: .quad 9,2
//tabPoint2: .quad 3,7
.equ NBPOINTS2, (. - tabPoint2) / 8
/*********************************/
/* UnInitialized data */
/*********************************/
.bss
stNode1: .skip nodeKD_end
KDtree1: .skip nodeKD_end * NBPOINTS
KDtree2: .skip nodeKD_end * NBPOINTS2
KdtreeRand: .skip nodeKD_end * NBPOINTSRAND
tabPointsRand: .skip nodeKD_end
sZoneConv: .skip 24 // conversion buffer
sZoneConv1: .skip 24 // conversion buffer
sBufferConv16: .skip 24
iDistance: .skip 8 // best distance
stNearest: .skip nodeKD_end // best node
nbNodeVi: .skip 8 // visited node
/*********************************/
/* code section */
/*********************************/
.text
.global main
main: // entry of program
ldr x0,qAdrszMessStart
bl affichageMess
ldr x0,qAdrtabPoint1 // points array address
ldr x1,qAdrKDtree1 // KD tree address
mov x2,#NBPOINTS // array points size
mov x3,#2 // domension 2
bl initKDTree // init tree
ldr x0,qAdrKDtree1 // KD tree address
mov x1,#0 // start index
mov x2,#NBPOINTS / 2 // end = array points size
mov x3,#0 // level
mov x4,#2 // dimension 2
bl createTree
mov x8,x0 // save median
cmp x0,#-1
beq 100f
ldr x0,qAdrKDtree1 // KD tree address
mov x1,#2 // dimension 2
bl displayTree
ldr x0,qAdrtabPoint2 // points array address
ldr x1,qAdrKDtree2 // search KDtree address
mov x2,#NBPOINTS2 // search array points size
mov x3,#2 // dimension 2
bl initKDTree // init search tree
ldr x0,qAdrKDtree1 // KDtree address
mov x1,#nodeKD_end
madd x0,x8,x1,x0 // compute median address
ldr x1,qAdrKDtree2 // search KDtree address
mov x2,#0 // first index
mov x3,#2 // dimension 2
bl searchNearest // search nearest point
ldr x0,qAdrszMessResult // display résult
bl affichageMess
ldr x0,qAdrstNearest // nearest point address
ldr x0,[x0]
mov x1,#2
bl displayCoord // coordonnés display
ldr x0,qAdrnbNodeVi // visited nodes
ldr x0,[x0]
ldr x1,qAdrsZoneConv
bl conversion10
mov x0,#3
ldr x1,qAdrszMessVisited
ldr x2,qAdrsZoneConv
ldr x3,qAdrszCarriageReturn
bl displayStrings
ldr x0,qAdriDistance // best distance address
ldr x0,[x0]
ldr x1,qAdrsZoneConv
bl conversion10
mov x0,#3
ldr x1,qAdrszMessDistance
ldr x2,qAdrsZoneConv
ldr x3,qAdrszCarriageReturn
bl displayStrings
bl traitRandom
100: // standard end of the program
mov x0, #0 // return code
mov x8,EXIT
svc #0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrsZoneConv: .quad sZoneConv
qAdrszMessDistance: .quad szMessDistance
qAdrszMessResult: .quad szMessResult
qAdrszMessVisited: .quad szMessVisited
qAdrszMessStart: .quad szMessStart
qAdrtabPoint1: .quad tabPoint1
qAdrtabPoint2: .quad tabPoint2
qAdrKDtree1: .quad KDtree1
qAdrKDtree2: .quad KDtree2
/***************************************************/
/* traitement random points */
/***************************************************/
traitRandom:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
ldr x8,qAdrKdtreeRand // KD tree address
mov x7,#nodeKD_end
ldr x10,iNbPointsRand
mov x3,#0
1: // loop generate random coordonnes
mov x0,0
mov x1,#MAXCOORD
bl extRandom // X
mov x4,x0
mov x0,0
mov x1,#MAXCOORD
bl extRandom // Y
mov x5,x0
mov x0,0
mov x1,#MAXCOORD
bl extRandom // Z
mov x6,x0
madd x0,x3,x7,x8
add x1,x0,#nodeKD_val // node address
str x4,[x1] // store X,Y,Z
str x5,[x1,#8]
str x6,[x1,#16]
mov x4,#-1 // init pointer left and right
str x4,[x0,#nodeKD_left]
str x4,[x0,#nodeKD_right]
add x3,x3,#1
cmp x3,x10
blt 1b
mov x0,x8 // KD tree address
mov x1,#0 // start indice
mov x2,x10 // array points size
mov x3,#0 // level
mov x4,#3 // dimension 2
bl createTree
mov x9,x0 // save median
// create random search point
mov x0,0
mov x1,#MAXCOORD
bl extRandom
mov x4,x0
mov x0,0
mov x1,#MAXCOORD
bl extRandom
mov x5,x0
mov x0,0
mov x1,#MAXCOORD
bl extRandom
mov x6,x0
ldr x3,qAdrtabPointsRand
add x0,x3,#nodeKD_val
str x4,[x0]
str x5,[x0,#8]
str x6,[x0,#16]
ldr x0,qAdrszMessRandCoor
bl affichageMess
mov x0,x3
mov x1,#3
bl displayCoord // search
ldr x0,qAdriDistance // init best distance address
mov x1,#0
str x1,[x0]
madd x0,x9,x7,x8 // median KD tree address
mov x1,x3 // search point address
mov x2,#0 // index
mov x3,#3 // dimension 2
bl searchNearest
ldr x0,qAdrszMessResult
bl affichageMess
ldr x0,qAdrstNearest
ldr x0,[x0]
mov x1,#3
bl displayCoord
ldr x0,qAdrnbNodeVi
ldr x0,[x0]
ldr x1,qAdrsZoneConv
bl conversion10
mov x0,#3
ldr x1,qAdrszMessVisited
ldr x2,qAdrsZoneConv
ldr x3,qAdrszCarriageReturn
bl displayStrings
ldr x0,qAdriDistance // best distance address
ldr x0,[x0]
ldr x1,qAdrsZoneConv
bl conversion10
mov x0,#3
ldr x1,qAdrszMessDistance
ldr x2,qAdrsZoneConv
ldr x3,qAdrszCarriageReturn
bl displayStrings
100:
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
qAdrKdtreeRand: .quad KdtreeRand
qAdrtabPointsRand: .quad tabPointsRand
qAdrszMessRandCoor: .quad szMessRandCoor
iNbPointsRand: .quad NBPOINTSRAND
/***************************************************/
/* init KN tree */
/***************************************************/
/* x0 array points */
/* x1 array tree address */
/* x2 points number */
/* x3 dimension */
initKDTree:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
stp x6,x7,[sp,-16]! // save registers
stp x8,x9,[sp,-16]! // save registers
mov x8,#0 // points indice
mov x4,#0 // node tree indice
mov x6,#nodeKD_end // structure size
1:
madd x5,x4,x6,x1 // compute node address
mov x9,#0 // index value
2:
ldr x7,[x0,x8,lsl #3] // load one point coord
str x7,[x5,x9,lsl #3] // store in node value
add x8,x8,#1 // next point
add x9,x9,#1 // next node value
cmp x9,x3 // = dimension ?
blt 2b // no loop
mov x7,#-1 // init pointer left and right
str x7,[x5,#nodeKD_left]
str x7,[x5,#nodeKD_right]
add x4,x4,#1 // increment node tree indice
cmp x8,x2 // end points ?
blt 1b
100:
ldp x8,x9,[sp],16 // restaur registers
ldp x6,x7,[sp],16 // restaur registers
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* create KD tree */
/***************************************************/
/* x0 array tree address */
/* x1 start index */
/* x2 end index
/* x3 level */
/* x4 dimention */
createTree:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
stp x6,x7,[sp,-16]! // save registers
stp x8,x9,[sp,-16]! // save registers
stp x10,x11,[sp,-16]! // save registers
stp x12,x13,[sp,-16]! // save registers
cmp x1,x2 // if start = end -> return -1
mov x5,-1
csel x0,x5,x0,ge
//movge x0,#-1
bge 100f
add x5,x1,#1 // if start + 1 = end -> return start
cmp x5,x2
csel x0,x1,x0,eq
//moveq x0,x1
beq 100f
mov x8,x0 // save address
mov x7,x1 // save start index
mov x12,x2 // save end index
mov x5,x3 // save level
mov x10,x4 // save dimension
mov x9,#nodeKD_end // node structure size
mov x1,x7 // start
mov x2,x12 // end
bl findMedian
cmp x0,#0 //
mov x6,-1
csel x0,x6,x0,lt
//movlt x0,#-1
blt 100f
mov x6,x0 // save indice median
add x5,x5,#1 // compute new value index
cmp x5,x10 // => dimension ?
csel x5,xzr,x5,ge
//movge x5,#0
mov x0,x8 // tree address
mov x1,x7 // start
mov x2,x6 // end = median
mov x3,x5 // index
mov x4,x10 // dimension
bl createTree
madd x1,x6,x9,x8 // compute median address
cmp x0,#-1 // left address is ok ?
beq 1f
madd x0,x9,x0,x8 // yes compute address
1:
str x0,[x1,#nodeKD_left] // and store
mov x0,x8 // KDtree address
add x1,x6,#1 // start = indice median + 1
mov x2,x12 // end
mov x3,x5 // index
mov x4,x10 // dimension
bl createTree
madd x1,x6,x9,x8 // compute median address
cmp x0,#-1 // indice right ok ?
beq 2f
madd x0,x9,x0,x8 // yes compute address
2:
str x0,[x1,#nodeKD_right] // and store in pointer
mov x0,x6 // return indice median node
100:
ldp x12,x13,[sp],16 // restaur registers
ldp x10,x11,[sp],16 // restaur registers
ldp x8,x9,[sp],16 // restaur registers
ldp x6,x7,[sp],16 // restaur registers
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* find median and sort points */
/***************************************************/
/* x0 tree address */
/* x1 start tree indice
/* x2 end tree indice */
/* x3 indice */
/* x0 return median point index */
findMedian:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
stp x6,x7,[sp,-16]! // save registers
stp x8,x9,[sp,-16]! // save registers
stp x10,x11,[sp,-16]! // save registers
stp x12,x13,[sp,-16]! // save registers
cmp x2,x1
mov x7,-1
csel x0,x7,x0,le
//movle x0,#-1
ble 100f
mov x7,#nodeKD_end // node size
add x5,x1,#1 // next node address
cmp x2,x5 // equal to end ?
csel x0,x1,x0,eq
//moveq x0,x1
beq 100f // yes return
mov x8,x1 // save start
mov x12,x0 // save tree address
add x4,x2,x1 // end + start
lsr x9,x4,#1 // divide by 2 = median index
madd x10,x7,x9,x0 // mediam md address
lsl x5,x3,#2 // index * 4
1:
cmp x2,x8 // end <= start
csel x0,x2,x0,le // stop ?
//movle x0,x2 // stop ?
ble 100f
add x6,x10,#nodeKD_val
add x11,x6,x5 // add shift index
ldr x6,[x11] // load pivot value
mov x0,x10 // median address
sub x1,x2,#1 // compute address end - 1
mul x1,x7,x1
add x1,x1,x12
bl inversion // inversion median and end - 1
mov x11,x8 // store=indice start
mov x4,x8 // p =indice start
2:
cmp x4,x2 // compare p and end
bge 5f
madd x3,x4,x7,x12 // compute p address
add x3,x3,x5 // add shift index
ldr x0,[x3] // load value
cmp x0,x6 // compare to pivot
bge 4f
cmp x4,x11 // compare p et store
beq 3f
madd x0,x4,x7,x12 // compute p address
madd x1,x11,x7,x12 // compute store address
bl inversion // inversion p and store
3:
add x11,x11,#1 // increment store
4:
add x4,x4,#1 // increment p
b 2b
5:
csel x0,x9,x0,eq // equal return median index
//moveq x0,x9 // equal return median index
beq 100f
madd x0,x11,x7,x12 // store address
sub x1,x2,#1 // end - 1
madd x1,x7,x1,x12 // compute address
bl inversion // inversion store and end - 1
ldr x0,[x0,#nodeKD_val] // load store value
add x0,x0,x5 // add shift index
ldr x1,[x10,#nodeKD_val] // load median value
add x1,x1,x5 // add shift index
cmp x0,x1 // compare values
csel x0,x9,x0,eq // equal return median index
// moveq x0,x9 // equal return median index
beq 100f
cmp x11,x9 // compare index store and median
csel x2,x11,x2,gt // new end
csel x8,x11,x8,le // new start
//movgt x2,x11 // new end
//movle x8,x11 // new start
b 1b // and loop
100:
ldp x12,x13,[sp],16 // restaur registers
ldp x10,x11,[sp],16 // restaur registers
ldp x8,x9,[sp],16 // restaur registers
ldp x6,x7,[sp],16 // restaur registers
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* compute distanceo 2 */
/***************************************************/
/* x0 tree node address */
/* x1 search points address */
/* x2 dimension */
distance:
stp x3,lr,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
stp x6,x7,[sp,-16]! // save registers
add x0,x0,#nodeKD_val // root node values array address
add x1,x1,#nodeKD_val // search node values array addresse
mov x3,#0 // first value index
mov x7,#0 // init distance
1:
ldr x4,[x0,x3,lsl #3] // load value
ldr x5,[x1,x3,lsl #3] // load value
sub x6,x5,x4 // différence
add x3,x3,#1
madd x7,x6,x6,x7 // compute square TODO: overflow
cmp x3,x2 // end ?
blt 1b // no -> loop
mov x0,x7 // return distance
100:
ldp x6,x7,[sp],16 // restaur registers
ldp x4,x5,[sp],16 // restaur registers
ldp x3,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* search nearest point */
/***************************************************/
/* x0 tree address */
/* x1 search points address */
/* x2 index */
/* x3 dimension */
searchNearest:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
stp x6,x7,[sp,-16]! // save registers
stp x8,x9,[sp,-16]! // save registers
stp x10,x11,[sp,-16]! // save registers
stp x12,x13,[sp,-16]! // save registers
cmp x0,#-1
beq 100f
mov x7,x0 // start with median
mov x8,x1 // save serach point address
lsl x9,x2,#2 // shift index
mov x10,x3 // save dimension
mov x2,x3 // dimension
bl distance // compute distance
mov x11,x0
ldr x1,[x7,x9]
ldr x12,[x8,x9]
sub x12,x12,x1
mul x6,x12,x12 // distance axis
ldr x4,qAdriDistance
ldr x5,[x4]
cmp x5,#0 // first distance ?
csel x5,x11,x5,eq
csel x3,x7,x3,eq
//moveq x5,x11 // new best distance
//moveq x3,x7 // new best node
beq 1f
cmp x11,x5 // compare new distance and best distance
bgt 2f
mov x5,x11 // new best distance
mov x3,x7 // new best node
1:
str x5,[x4] // store new best distance
ldr x4,qAdrstNearest // and store best node address
str x3,[x4]
2:
ldr x1,qAdrnbNodeVi // increment visited node
ldr x3,[x1]
add x3,x3,#1
str x3,[x1]
cmp x5,#0 // distance = 0 -> end //
beq 100f
cmp x12,#0 // else search in childen tree
bge 3f
ldr x0,[x7,#nodeKD_left]
b 4f
3:
ldr x0,[x7,#nodeKD_right]
4:
mov x1,x8
lsr x2,x9,#2
add x2,x2,#1
cmp x2,x10
csel x2,xzr,x2,ge
//movge x2,#0
mov x3,x10
bl searchNearest
ldr x4,qAdriDistance // best distance
ldr x5,[x4]
cmp x6,x5 // compare distance
bge 100f
cmp x12,#0 // else search in childen tree
blt 5f
ldr x0,[x7,#nodeKD_left]
b 6f
5:
ldr x0,[x7,#nodeKD_right]
6:
bl searchNearest
100:
ldp x12,x13,[sp],16 // restaur registers
ldp x10,x11,[sp],16 // restaur registers
ldp x8,x9,[sp],16 // restaur registers
ldp x6,x7,[sp],16 // restaur registers
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
qAdrstNearest: .quad stNearest
qAdriDistance: .quad iDistance
qAdrnbNodeVi: .quad nbNodeVi
/***************************************************/
/* inversion */
/***************************************************/
/* x0 tree address */
/* x1 tree */
inversion:
stp x2,lr,[sp,-16]! // save registers
stp x3,x4,[sp,-16]! // save registers
add x0,x0,#nodeKD_val
add x1,x1,#nodeKD_val
mov x2,#0
1:
ldr x4,[x0,x2,lsl #3]
ldr x3,[x1,x2,lsl #3]
str x3,[x0,x2,lsl #3]
str x4,[x1,x2,lsl #3]
add x2,x2,#1
cmp x2,#MAXI
blt 1b
100:
ldp x3,x4,[sp],16 // restaur registers
ldp x2,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* display tree */
/***************************************************/
/* x0 tree address */
/* x1 dimension */
displayTree:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
stp x6,x7,[sp,-16]! // save registers
stp x8,x9,[sp,-16]! // save registers
stp x10,x11,[sp,-16]! // save registers
stp x12,x13,[sp,-16]! // save registers
mov x10,x0
mov x7,x1
mov x11,#0
mov x12,#nodeKD_end
1:
madd x9,x12,x11,x10
mov x0,x9
ldr x1,qAdrsBufferConv16
bl conversion16
mov x0,#2
ldr x1,qAdrszMessAddressTree
ldr x2,qAdrsBufferConv16
bl displayStrings
mov x8,#0
2:
add x4,x9,#nodeKD_val
ldr x0,[x4,x8,lsl #3]
ldr x1,qAdrsZoneConv
bl conversion10
mov x0,#2
ldr x1,qAdrszMessTreeValue
ldr x2,qAdrsZoneConv
bl displayStrings
add x8,x8,#1
cmp x8,x7
blt 2b
ldr x0,qAdrszCarriageReturn
bl affichageMess
add x0,x9,#nodeKD_left
ldr x0,[x0]
ldr x1,qAdrsZoneConv
bl conversion16
add x0,x9,#nodeKD_right
ldr x0,[x0]
ldr x1,qAdrsZoneConv1
bl conversion16
mov x0,#5
ldr x1,qAdrszMessLeft
ldr x2,qAdrsZoneConv
ldr x3,qAdrszMessRight
ldr x4,qAdrsZoneConv1
ldr x5,qAdrszCarriageReturn
bl displayStrings
add x11,x11,#1
cmp x11,#NBPOINTS / 2
blt 1b
100:
ldp x12,x13,[sp],16 // restaur registers
ldp x10,x11,[sp],16 // restaur registers
ldp x8,x9,[sp],16 // restaur registers
ldp x6,x7,[sp],16 // restaur registers
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
qAdrszMessAddressTree: .quad szMessAddressTree
qAdrszMessTreeValue: .quad szMessTreeValue
qAdrsBufferConv16: .quad sBufferConv16
qAdrszMessLeft: .quad szMessLeft
qAdrszMessRight: .quad szMessRight
qAdrsZoneConv1: .quad sZoneConv1
/***************************************************/
/* display node coordonnées */
/***************************************************/
/* x0 node address */
/* x1 dimension */
displayCoord:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
add x4,x0,#nodeKD_val
mov x3,x1 // save dimension
mov x5,#0
1:
ldr x0,[x4,x5,lsl #3]
ldr x1,qAdrsZoneConv
bl conversion10
mov x0,#2
ldr x1,qAdrszMessTreeValue
ldr x2,qAdrsZoneConv
bl displayStrings
add x5,x5,#1
cmp x5,x3
blt 1b
ldr x0,qAdrszCarriageReturn
bl affichageMess
100:
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* display multi strings */
/* new version 24/05/2023 */
/***************************************************/
/* x0 contains number strings address */
/* x1 address string1 */
/* x2 address string2 */
/* x3 address string3 */
/* x4 address string4 */
/* x5 address string5 */
/* x6 address string5 */
/* x7 address string6 */
displayStrings: // INFO: displayStrings
stp x8,lr,[sp,-16]! // save registers
stp x2,fp,[sp,-16]! // save registers
add fp,sp,#32 // save paraméters address (4 registers saved * 8 bytes)
mov x8,x0 // save strings number
cmp x8,#0 // 0 string -> end
ble 100f
mov x0,x1 // string 1
bl affichageMess
cmp x8,#1 // number > 1
ble 100f
mov x0,x2
bl affichageMess
cmp x8,#2
ble 100f
mov x0,x3
bl affichageMess
cmp x8,#3
ble 100f
mov x0,x4
bl affichageMess
cmp x8,#4
ble 100f
mov x0,x5
bl affichageMess
cmp x8,#5
ble 100f
mov x0,x6
bl affichageMess
cmp x8,#6
ble 100f
mov x0,x7
bl affichageMess
100:
ldp x2,fp,[sp],16 // restaur registers
ldp x8,lr,[sp],16 // restaur registers
ret
/******************************************************************/
/* random number */
/******************************************************************/
/* x0 contains inferior value */
/* x1 contains maxi value */
/* x0 return random number */
extRandom:
stp x1,lr,[sp,-16]! // save registers
stp x2,x8,[sp,-16]! // save registers
stp x19,x20,[sp,-16]! // save registers
sub sp,sp,16 // reserve 16 octets on stack
mov x19,x0
add x20,x1,1
mov x0,sp // store result on stack
mov x1,8 // length 8 bytes
mov x2,0
mov x8,278 // call system Linux 64 bits Urandom
svc 0
mov x0,sp // load résult on stack
ldr x0,[x0]
sub x2,x20,x19 // calculation of the range of values
udiv x1,x0,x2 // calculation range modulo
msub x0,x1,x2,x0
add x0,x0,x19 // and add inferior value
100:
add sp,sp,16 // alignement stack
ldp x19,x20,[sp],16 // restaur 2 registers
ldp x2,x8,[sp],16 // restaur 2 registers
ldp x1,lr,[sp],16 // restaur 2 registers
ret // retour adresse lr x30
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
.include "../includeARM64.inc"- Output:
Program 64 bits start.
Node address : 0000000000420628 Value : 2 Value : 3
Left : FFFFFFFFFFFFFFFF Right : FFFFFFFFFFFFFFFF
Node address : 0000000000420650 Value : 9 Value : 6
Left : 0000000000420628 Right : 0000000000420678
Node address : 0000000000420678 Value : 5 Value : 4
Left : FFFFFFFFFFFFFFFF Right : FFFFFFFFFFFFFFFF
Node address : 00000000004206A0 Value : 7 Value : 2
Left : 0000000000420650 Right : 00000000004206F0
Node address : 00000000004206C8 Value : 8 Value : 1
Left : FFFFFFFFFFFFFFFF Right : FFFFFFFFFFFFFFFF
Node address : 00000000004206F0 Value : 4 Value : 7
Left : 00000000004206C8 Right : FFFFFFFFFFFFFFFF
Nearest point = Value : 8 Value : 1
Node visited = 3
square distance :2
Random point coordonnés = Value : 6981 Value : 1946 Value : 4127
Nearest point = Value : 6561 Value : 1995 Value : 5172
Node visited = 95
square distance :1270826
ARM Assembly
/* ARM assembly Raspberry PI */
/* program kdtree.s */
/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly
for the routine affichageMess conversion10
see at end of this program the instruction include */
/* REMARK 2 : In order not to use floating point numbers,
the coordinates of the points are integers numbers.
The displayed distance represents the square of the distance between 2 points */
/* This program draws heavily from the published C program. Thanks to its creator. */
/* for constantes see task include a file in arm assembly */
/************************************/
/* Constantes */
/************************************/
.include "../constantes.inc"
.equ MAXI, 3
.equ NBPOINTSRAND, 2000
.equ MAXCOORD, 10000
/***********************************************/
/* structures */
/**********************************************/
/* Définition node */
.struct 0
nodeKD_val: // value array
.struct nodeKD_val + (4 * MAXI)
nodeKD_left: // left pointer
.struct nodeKD_left + 4
nodeKD_right: // right pointer
.struct nodeKD_right + 4
nodeKD_end:
/*********************************/
/* Initialized data */
/*********************************/
.data
szMessAddressTree: .asciz "Node address : "
szMessTreeValue: .asciz " Value : "
szMessLeft: .asciz " Left : "
szMessRight: .asciz "Right : "
szMessResult: .asciz "Nearest point = "
szMessRandCoor: .asciz "\n\nRandom point coordonnés = "
szMessVisited: .asciz "Node visited = "
szMessDistance: .asciz "square distance :"
szMessStart: .asciz "Program 32 bits start.\n"
szCarriageReturn: .asciz "\n"
.align 4
tabPoint1: .int 2,3, 5,4, 9,6, 4,7, 8,1, 7,2
//tabPoint1: .int 1,7, 3,4, 4,6, 6,2, 8,12, 10,9, 12,3, 14,1
//tabPoint1: .int 3,5, 1,3, 2,8, 4,6, 5,4
.equ NBPOINTS, (. - tabPoint1) / 4
tabPoint2: .int 9,2
//tabPoint2: .int 3,7
.equ NBPOINTS2, (. - tabPoint2) / 4
iGraine: .int 123456 // random init
/*********************************/
/* UnInitialized data */
/*********************************/
.bss
stNode1: .skip nodeKD_end
KDtree1: .skip nodeKD_end * NBPOINTS
KDtree2: .skip nodeKD_end * NBPOINTS2
KdtreeRand: .skip nodeKD_end * NBPOINTSRAND
tabPointsRand: .skip nodeKD_end
sZoneConv: .skip 24 // conversion buffer
sZoneConv1: .skip 24 // conversion buffer
sBufferConv16: .skip 16
iDistance: .skip 4 // best distance
stNearest: .skip nodeKD_end // best node
nbNodeVi: .skip 4 // visited node
/*********************************/
/* code section */
/*********************************/
.text
.global main
main: @ entry of program
ldr r0,iAdrszMessStart
bl affichageMess
ldr r0,iAdrtabPoint1 @ points array address
ldr r1,iAdrKDtree1 @ KD tree address
mov r2,#NBPOINTS @ array points size
mov r3,#2 @ domension 2
bl initKDTree @ init tree
ldr r0,iAdrKDtree1 @ KD tree address
mov r1,#0 @ start index
mov r2,#NBPOINTS / 2 @ end = array points size
mov r3,#0 @ level
mov r4,#2 @ dimension 2
bl createTree
mov r8,r0 @ save median
cmp r0,#-1
beq 100f
ldr r0,iAdrKDtree1 @ KD tree address
mov r1,#2 @ dimension 2
bl displayTree
ldr r0,iAdrtabPoint2 @ points array address
ldr r1,iAdrKDtree2 @ search KDtree address
mov r2,#NBPOINTS2 @ search array points size
mov r3,#2 @ dimension 2
bl initKDTree @ init search tree
ldr r0,iAdrKDtree1 @ KDtree address
mov r1,#nodeKD_end
mla r0,r8,r1,r0 @ compute median address
ldr r1,iAdrKDtree2 @ search KDtree address
mov r2,#0 @ first index
mov r3,#2 @ dimension 2
bl searchNearest @ search nearest point
ldr r0,iAdrszMessResult @ display résult
bl affichageMess
ldr r0,iAdrstNearest @ nearest point address
ldr r0,[r0]
mov r1,#2
bl displayCoord @ coordonnés display
ldr r0,iAdrnbNodeVi @ visited nodes
ldr r0,[r0]
ldr r1,iAdrsZoneConv
bl conversion10
mov r0,#3
ldr r1,iAdrszMessVisited
ldr r2,iAdrsZoneConv
ldr r3,iAdrszCarriageReturn
bl displayStrings
ldr r0,iAdriDistance @ best distance address
ldr r0,[r0]
ldr r1,iAdrsZoneConv
bl conversion10
mov r0,#3
ldr r1,iAdrszMessDistance
ldr r2,iAdrsZoneConv
ldr r3,iAdrszCarriageReturn
bl displayStrings
bl traitRandom
100: @ standard end of the program
mov r0, #0 @ return code
mov r7, #EXIT @ request to exit program
svc #0 @ perform the system call
iAdrszCarriageReturn: .int szCarriageReturn
iAdrsZoneConv: .int sZoneConv
iAdrszMessDistance: .int szMessDistance
iAdrszMessResult: .int szMessResult
iAdrszMessVisited: .int szMessVisited
iAdrszMessStart: .int szMessStart
iAdrtabPoint1: .int tabPoint1
iAdrtabPoint2: .int tabPoint2
iAdrKDtree1: .int KDtree1
iAdrKDtree2: .int KDtree2
/***************************************************/
/* traitement random points */
/***************************************************/
traitRandom:
push {r1-r10,lr} @ save des registres
ldr r8,iAdrKdtreeRand @ KD tree address
mov r7,#nodeKD_end
ldr r10,iNbPointsRand
mov r3,#0
1: @ loop generate random coordonnes
mov r0,#MAXCOORD
bl genereraleas @ X
mov r4,r0
mov r0,#MAXCOORD
bl genereraleas @ Y
mov r5,r0
mov r0,#MAXCOORD
bl genereraleas @ Z
mov r6,r0
mla r0,r3,r7,r8
add r1,r0,#nodeKD_val @ node address
str r4,[r1] @ store X,Y,Z
str r5,[r1,#4]
str r6,[r1,#8]
mov r4,#-1 @ init pointer left and right
str r4,[r0,#nodeKD_left]
str r4,[r0,#nodeKD_right]
add r3,r3,#1
cmp r3,r10
blt 1b
mov r0,r8 @ KD tree address
mov r1,#0 @ start indice
mov r2,r10 @ array points size
mov r3,#0 @ level
mov r4,#3 @ dimension 2
bl createTree
mov r9,r0 @ save median
@ create random search point
mov r0,#MAXCOORD
bl genereraleas
mov r4,r0
mov r0,#MAXCOORD
bl genereraleas
mov r5,r0
mov r0,#MAXCOORD
bl genereraleas
mov r6,r0
ldr r3,iAdrtabPointsRand
add r0,r3,#nodeKD_val
str r4,[r0]
str r5,[r0,#4]
str r6,[r0,#8]
ldr r0,iAdrszMessRandCoor
bl affichageMess
mov r0,r3
mov r1,#3
bl displayCoord @ search
ldr r0,iAdriDistance @ init best distance address
mov r1,#0
str r1,[r0]
mla r0,r9,r7,r8 @ median KD tree address
mov r1,r3 @ search point address
mov r2,#0 @ index
mov r3,#3 @ dimension 2
bl searchNearest
ldr r0,iAdrszMessResult
bl affichageMess
ldr r0,iAdrstNearest
ldr r0,[r0]
mov r1,#3
bl displayCoord
ldr r0,iAdrnbNodeVi
ldr r0,[r0]
ldr r1,iAdrsZoneConv
bl conversion10
mov r0,#3
ldr r1,iAdrszMessVisited
ldr r2,iAdrsZoneConv
ldr r3,iAdrszCarriageReturn
bl displayStrings
ldr r0,iAdriDistance @ best distance address
ldr r0,[r0]
ldr r1,iAdrsZoneConv
bl conversion10
mov r0,#3
ldr r1,iAdrszMessDistance
ldr r2,iAdrsZoneConv
ldr r3,iAdrszCarriageReturn
bl displayStrings
100:
pop {r1-r10,pc}
iAdrKdtreeRand: .int KdtreeRand
iAdrtabPointsRand: .int tabPointsRand
iAdrszMessRandCoor: .int szMessRandCoor
iNbPointsRand: .int NBPOINTSRAND
/***************************************************/
/* init KN tree */
/***************************************************/
/* r0 array points */
/* r1 array tree address */
/* r2 points number */
/* r3 dimension */
initKDTree:
push {r1-r9,lr} @ save des registres
mov r8,#0 @ points indice
mov r4,#0 @ node tree indice
mov r6,#nodeKD_end @ structure size
1:
mla r5,r4,r6,r1 @ compute node address
mov r9,#0 @ index value
2:
ldr r7,[r0,r8,lsl #2] @ load one point coord
str r7,[r5,r9,lsl #2] @ store in node value
add r8,r8,#1 @ next point
add r9,r9,#1 @ next node value
cmp r9,r3 @ = dimension ?
blt 2b @ no loop
mov r7,#-1 @ init pointer left and right
str r7,[r5,#nodeKD_left]
str r7,[r5,#nodeKD_right]
add r4,r4,#1 @ increment node tree indice
cmp r8,r2 @ end points ?
blt 1b
100:
pop {r1-r9,pc}
/***************************************************/
/* create KD tree */
/***************************************************/
/* r0 array tree address */
/* r1 start index */
/* r2 end index
/* r3 level */
/* r4 dimention */
createTree:
push {r1-r12,lr} @ save des registres
cmp r1,r2 @ if start = end -> return -1
movge r0,#-1
bge 100f
add r5,r1,#1 @ if start + 1 = end -> return start
cmp r5,r2
moveq r0,r1
beq 100f
mov r8,r0 @ save address
mov r7,r1 @ save start index
mov r12,r2 @ save end index
mov r5,r3 @ save level
mov r10,r4 @ save dimension
mov r9,#nodeKD_end @ node structure size
mov r1,r7 @ start
mov r2,r12 @ end
bl findMedian
cmp r0,#0 @
movlt r0,#-1
blt 100f
mov r6,r0 @ save indice median
add r5,r5,#1 @ compute new value index
cmp r5,r10 @ => dimension ?
movge r5,#0
mov r0,r8 @ tree address
mov r1,r7 @ start
mov r2,r6 @ end = median
mov r3,r5 @ index
mov r4,r10 @ dimension
bl createTree
mla r1,r6,r9,r8 @ compute median address
cmp r0,#-1 @ left address is ok ?
mlane r0,r9,r0,r8 @ yes compute address
str r0,[r1,#nodeKD_left] @ and store
mov r0,r8 @ KDtree address
add r1,r6,#1 @ start = indice median + 1
mov r2,r12 @ end
mov r3,r5 @ index
mov r4,r10 @ dimension
bl createTree
mla r1,r6,r9,r8 @ compute median address
cmp r0,#-1 @ indice right ok ?
mlane r0,r9,r0,r8 @ yes compute address
str r0,[r1,#nodeKD_right] @ and store in pointer
mov r0,r6 @ return indice median node
100:
pop {r1-r12,pc}
/***************************************************/
/* find median and sort points */
/***************************************************/
/* r0 tree address */
/* r1 start tree indice
/* r2 end tree indice */
/* r3 indice */
/* r0 return median point index */
findMedian:
push {r1-r12,lr} @ save des registres
cmp r2,r1
movle r0,#-1
ble 100f
mov r7,#nodeKD_end @ node size
add r5,r1,#1 @ next node address
cmp r2,r5 @ equal to end ?
moveq r0,r1
beq 100f @ yes return
mov r8,r1 @ save start
mov r12,r0 @ save tree address
add r4,r2,r1 @ end + start
lsr r9,r4,#1 @ divide by 2 = median index
mla r10,r7,r9,r0 @ mediam md address
lsl r5,r3,#2 @ index * 4
1:
cmp r2,r8 @ end <= start
movle r0,r2 @ stop ?
ble 100f
add r6,r10,#nodeKD_val
add r11,r6,r5 @ add shift index
ldr r6,[r11] @ load pivot value
mov r0,r10 @ median address
sub r1,r2,#1 @ compute address end - 1
mul r1,r7,r1
add r1,r1,r12
bl inversion @ inversion median and end - 1
mov r11,r8 @ store=indice start
mov r4,r8 @ p =indice start
2:
cmp r4,r2 @ compare p and end
bge 5f
mla r3,r4,r7,r12 @ compute p address
add r3,r3,r5 @ add shift index
ldr r0,[r3] @ load value
cmp r0,r6 @ compare to pivot
bge 4f
cmp r4,r11 @ compare p et store
beq 3f
mla r0,r4,r7,r12 @ compute p address
mla r1,r11,r7,r12 @ compute store address
bl inversion @ inversion p and store
3:
add r11,r11,#1 @ increment store
4:
add r4,r4,#1 @ increment p
b 2b
5:
moveq r0,r9 @ equal return median index
beq 100f
mla r0,r11,r7,r12 @ store address
sub r1,r2,#1 @ end - 1
mla r1,r7,r1,r12 @ compute address
bl inversion @ inversion store and end - 1
ldr r0,[r0,#nodeKD_val] @ load store value
add r0,r0,r5 @ add shift index
ldr r1,[r10,#nodeKD_val] @ load median value
add r1,r1,r5 @ add shift index
cmp r0,r1 @ compare values
moveq r0,r9 @ equal return median index
beq 100f
cmp r11,r9 @ compare index store and median
movgt r2,r11 @ new end
movle r8,r11 @ new start
b 1b @ and loop
100:
pop {r1-r12,pc}
/***************************************************/
/* compute distance into 2 points */
/***************************************************/
/* r0 tree node address */
/* r1 search points address */
/* r2 dimension */
distance:
push {r3-r7,lr} @ save des registres
add r0,#nodeKD_val @ root node values array address
add r1,#nodeKD_val @ search node values array addresse
mov r3,#0 @ first value index
mov r7,#0 @ init distance
1:
ldr r4,[r0,r3,lsl #2] @ load value
ldr r5,[r1,r3,lsl #2] @ load value
sub r6,r5,r4 @ différence
add r3,r3,#1
mla r7,r6,r6,r7 @ compute square TODO: overflow
cmp r3,r2 @ end ?
blt 1b @ no -> loop
mov r0,r7 @ return distance
100:
pop {r3-r7,pc}
/***************************************************/
/* search nearest point */
/***************************************************/
/* r0 tree address */
/* r1 search points address */
/* r2 index */
/* r3 dimension */
searchNearest:
push {r1-r12,lr} @ save des registres
cmp r0,#-1
beq 100f
mov r7,r0 @ start with median
mov r8,r1 @ save serach point address
lsl r9,r2,#2 @ shift index
mov r10,r3 @ save dimension
mov r2,r3 @ dimension
bl distance @ compute distance
mov r11,r0
ldr r1,[r7,r9]
ldr r12,[r8,r9]
sub r12,r1
mul r6,r12,r12 @ distance axis
ldr r4,iAdriDistance
ldr r5,[r4]
cmp r5,#0 @ first distance ?
moveq r5,r11 @ new best distance
moveq r3,r7 @ new best node
beq 1f
cmp r11,r5 @ compare new distance and best distance
bgt 2f
mov r5,r11 @ new best distance
mov r3,r7 @ new best node
1:
str r5,[r4] @ store new best distance
ldr r4,iAdrstNearest @ and store best node address
str r3,[r4]
2:
ldr r1,iAdrnbNodeVi @ increment visited node
ldr r3,[r1]
add r3,r3,#1
str r3,[r1]
cmp r5,#0 @ distance = 0 -> end @
beq 100f
cmp r12,#0 @ else search in childen tree
ldrlt r0,[r7,#nodeKD_left]
ldrge r0,[r7,#nodeKD_right]
mov r1,r8
lsr r2,r9,#2
add r2,r2,#1
cmp r2,r10
movge r2,#0
mov r3,r10
bl searchNearest
ldr r4,iAdriDistance @ best distance
ldr r5,[r4]
cmp r6,r5 @ compare distance
bge 100f
cmp r12,#0 @ else search in childen tree
ldrge r0,[r7,#nodeKD_left]
ldrlt r0,[r7,#nodeKD_right]
bl searchNearest
100:
pop {r1-r12,pc}
iAdrstNearest: .int stNearest
iAdriDistance: .int iDistance
iAdrnbNodeVi: .int nbNodeVi
/***************************************************/
/* inversion */
/***************************************************/
/* r0 tree address */
/* r1 tree */
inversion:
push {r0-r4,lr} @ save des registres
add r0,#nodeKD_val
add r1,#nodeKD_val
mov r2,#0
1:
ldr r4,[r0,r2,lsl #2]
ldr r3,[r1,r2,lsl #2]
str r3,[r0,r2,lsl #2]
str r4,[r1,r2,lsl #2]
add r2,r2,#1
cmp r2,#MAXI
blt 1b
100:
pop {r0-r4,pc}
/***************************************************/
/* display tree */
/***************************************************/
/* r0 tree address */
/* r1 dimension */
displayTree:
push {r2-r12,lr} @ save des registres
mov r10,r0
mov r7,r1
mov r11,#0
mov r12,#nodeKD_end
1:
mla r9,r12,r11,r10
mov r0,r9
ldr r1,iAdrsBufferConv16
bl conversion16
mov r0,#2
ldr r1,iAdrszMessAddressTree
ldr r2,iAdrsBufferConv16
bl displayStrings
mov r8,#0
2:
add r4,r9,#nodeKD_val
ldr r0,[r4,r8,lsl #2]
ldr r1,iAdrsZoneConv
bl conversion10
mov r0,#2
ldr r1,iAdrszMessTreeValue
ldr r2,iAdrsZoneConv
bl displayStrings
add r8,r8,#1
cmp r8,r7
blt 2b
ldr r0,iAdrszCarriageReturn
bl affichageMess
add r0,r9,#nodeKD_left
ldr r0,[r0]
ldr r1,iAdrsZoneConv
bl conversion16
add r0,r9,#nodeKD_right
ldr r0,[r0]
ldr r1,iAdrsZoneConv1
bl conversion16
mov r0,#5
ldr r1,iAdrszMessLeft
ldr r2,iAdrsZoneConv
ldr r3,iAdrszMessRight
ldr r4,iAdrsZoneConv1
push {r4}
ldr r4,iAdrszCarriageReturn
push {r4}
bl displayStrings
add sp,sp,#8
add r11,r11,#1
cmp r11,#NBPOINTS / 2
blt 1b
100:
pop {r2-r12,pc}
iAdrszMessAddressTree: .int szMessAddressTree
iAdrszMessTreeValue: .int szMessTreeValue
iAdrsBufferConv16: .int sBufferConv16
iAdrszMessLeft: .int szMessLeft
iAdrszMessRight: .int szMessRight
iAdrsZoneConv1: .int sZoneConv1
/***************************************************/
/* display node coordonnées */
/***************************************************/
/* r0 node address */
/* r1 dimension */
displayCoord:
push {r1-r5,lr} @ save des registres
add r4,r0,#nodeKD_val
mov r3,r1 @ save dimension
mov r5,#0
1:
ldr r0,[r4,r5,lsl #2]
ldr r1,iAdrsZoneConv
bl conversion10
mov r0,#2
ldr r1,iAdrszMessTreeValue
ldr r2,iAdrsZoneConv
bl displayStrings
add r5,r5,#1
cmp r5,r3
blt 1b
ldr r0,iAdrszCarriageReturn
bl affichageMess
100:
pop {r1-r5,pc}
/***************************************************/
/* display multi strings */
/***************************************************/
/* r0 contains number strings address */
/* r1 address string1 */
/* r2 address string2 */
/* r3 address string3 */
/* other address on the stack */
/* thinck to add number other address * 4 to add to the stack */
displayStrings: @ INFO: displayStrings
push {r1-r4,fp,lr} @ save des registres
add fp,sp,#24 @ save paraméters address (6 registers saved * 4 bytes)
mov r4,r0 @ save strings number
cmp r4,#0 @ 0 string -> end
ble 100f
mov r0,r1 @ string 1
bl affichageMess
cmp r4,#1 @ number > 1
ble 100f
mov r0,r2
bl affichageMess
cmp r4,#2
ble 100f
mov r0,r3
bl affichageMess
cmp r4,#3
ble 100f
mov r3,#3
sub r2,r4,#4
1: @ loop extract address string on stack
ldr r0,[fp,r2,lsl #2]
bl affichageMess
subs r2,#1
bge 1b
100:
pop {r1-r4,fp,pc}
/***************************************************/
/* Generation random number */
/***************************************************/
/* r0 contains limit */
genereraleas:
push {r1-r4,lr} @ save registers
ldr r4,iAdriGraine
ldr r2,[r4]
ldr r3,iNbDep1
mul r2,r3,r2
ldr r3,iNbDep1
add r2,r2,r3
str r2,[r4] @ maj de la graine pour l appel suivant
cmp r0,#0
beq 100f
mov r1,r0 @ divisor
mov r0,r2 @ dividende
bl division
mov r0,r3 @ résult = remainder
100: @ end function
pop {r1-r4,lr} @ restaur registers
bx lr @ return
iAdriGraine: .int iGraine
iNbDep1: .int 0x343FD
iNbDep2: .int 0x269EC3
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
.include "../affichage.inc"- Output:
Program 32 bits start.
Node address : 00012494 Value : 2 Value : 3
Left : FFFFFFFF Right : FFFFFFFF
Node address : 000124A8 Value : 9 Value : 6
Left : 00012494 Right : 000124BC
Node address : 000124BC Value : 5 Value : 4
Left : FFFFFFFF Right : FFFFFFFF
Node address : 000124D0 Value : 7 Value : 2
Left : 000124A8 Right : 000124F8
Node address : 000124E4 Value : 8 Value : 1
Left : FFFFFFFF Right : FFFFFFFF
Node address : 000124F8 Value : 4 Value : 7
Left : 000124E4 Right : FFFFFFFF
Nearest point = Value : 8 Value : 1
Node visited = 3
square distance :2
Random point coordonnés = Value : 4877 Value : 3830 Value : 3035
Nearest point = Value : 4947 Value : 4836 Value : 2401
Node visited = 122
square distance :1418892
C
Using a Quickselectesque median algorithm. Compared to unbalanced trees (random insertion), it takes slightly longer (maybe half a second or so) to construct a million-node tree, though average look up visits about 1/3 fewer nodes.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <time.h>
#define MAX_DIM 3
struct kd_node_t{
double x[MAX_DIM];
struct kd_node_t *left, *right;
};
inline double
dist(struct kd_node_t *a, struct kd_node_t *b, int dim)
{
double t, d = 0;
while (dim--) {
t = a->x[dim] - b->x[dim];
d += t * t;
}
return d;
}
inline void swap(struct kd_node_t *x, struct kd_node_t *y) {
double tmp[MAX_DIM];
memcpy(tmp, x->x, sizeof(tmp));
memcpy(x->x, y->x, sizeof(tmp));
memcpy(y->x, tmp, sizeof(tmp));
}
/* see quickselect method */
struct kd_node_t*
find_median(struct kd_node_t *start, struct kd_node_t *end, int idx)
{
if (end <= start) return NULL;
if (end == start + 1)
return start;
struct kd_node_t *p, *store, *md = start + (end - start) / 2;
double pivot;
while (1) {
pivot = md->x[idx];
swap(md, end - 1);
for (store = p = start; p < end; p++) {
if (p->x[idx] < pivot) {
if (p != store)
swap(p, store);
store++;
}
}
swap(store, end - 1);
/* median has duplicate values */
if (store->x[idx] == md->x[idx])
return md;
if (store > md) end = store;
else start = store;
}
}
struct kd_node_t*
make_tree(struct kd_node_t *t, int len, int i, int dim)
{
struct kd_node_t *n;
if (!len) return 0;
if ((n = find_median(t, t + len, i))) {
i = (i + 1) % dim;
n->left = make_tree(t, n - t, i, dim);
n->right = make_tree(n + 1, t + len - (n + 1), i, dim);
}
return n;
}
/* global variable, so sue me */
int visited;
void nearest(struct kd_node_t *root, struct kd_node_t *nd, int i, int dim,
struct kd_node_t **best, double *best_dist)
{
double d, dx, dx2;
if (!root) return;
d = dist(root, nd, dim);
dx = root->x[i] - nd->x[i];
dx2 = dx * dx;
visited ++;
if (!*best || d < *best_dist) {
*best_dist = d;
*best = root;
}
/* if chance of exact match is high */
if (!*best_dist) return;
if (++i >= dim) i = 0;
nearest(dx > 0 ? root->left : root->right, nd, i, dim, best, best_dist);
if (dx2 >= *best_dist) return;
nearest(dx > 0 ? root->right : root->left, nd, i, dim, best, best_dist);
}
#define N 1000000
#define rand1() (rand() / (double)RAND_MAX)
#define rand_pt(v) { v.x[0] = rand1(); v.x[1] = rand1(); v.x[2] = rand1(); }
int main(void)
{
int i;
struct kd_node_t wp[] = {
{{2, 3}}, {{5, 4}}, {{9, 6}}, {{4, 7}}, {{8, 1}}, {{7, 2}}
};
struct kd_node_t testNode = {{9, 2}};
struct kd_node_t *root, *found, *million;
double best_dist;
root = make_tree(wp, sizeof(wp) / sizeof(wp[1]), 0, 2);
visited = 0;
found = 0;
nearest(root, &testNode, 0, 2, &found, &best_dist);
printf(">> WP tree\nsearching for (%g, %g)\n"
"found (%g, %g) dist %g\nseen %d nodes\n\n",
testNode.x[0], testNode.x[1],
found->x[0], found->x[1], sqrt(best_dist), visited);
million =(struct kd_node_t*) calloc(N, sizeof(struct kd_node_t));
srand(time(0));
for (i = 0; i < N; i++) rand_pt(million[i]);
root = make_tree(million, N, 0, 3);
rand_pt(testNode);
visited = 0;
found = 0;
nearest(root, &testNode, 0, 3, &found, &best_dist);
printf(">> Million tree\nsearching for (%g, %g, %g)\n"
"found (%g, %g, %g) dist %g\nseen %d nodes\n",
testNode.x[0], testNode.x[1], testNode.x[2],
found->x[0], found->x[1], found->x[2],
sqrt(best_dist), visited);
/* search many random points in million tree to see average behavior.
tree size vs avg nodes visited:
10 ~ 7
100 ~ 16.5
1000 ~ 25.5
10000 ~ 32.8
100000 ~ 38.3
1000000 ~ 42.6
10000000 ~ 46.7 */
int sum = 0, test_runs = 100000;
for (i = 0; i < test_runs; i++) {
found = 0;
visited = 0;
rand_pt(testNode);
nearest(root, &testNode, 0, 3, &found, &best_dist);
sum += visited;
}
printf("\n>> Million tree\n"
"visited %d nodes for %d random findings (%f per lookup)\n",
sum, test_runs, sum/(double)test_runs);
// free(million);
return 0;
}
- Output:
>> WP tree searching for (9, 2) found (8, 1) dist 1.41421 seen 3 nodes >> Million tree searching for (0.29514, 0.897237, 0.941998) found (0.296093, 0.896173, 0.948082) dist 0.00624896 seen 44 nodes >> Million tree visited 4271442 nodes for 100000 random findings (42.714420 per lookup)
C++
This code is based on the C version. A significant difference is the use of the standard library function nth_element, which replaces the find_median function in the C version.
#include <algorithm>
#include <array>
#include <cmath>
#include <iostream>
#include <random>
#include <vector>
/**
* Class for representing a point. coordinate_type must be a numeric type.
*/
template<typename coordinate_type, size_t dimensions>
class point {
public:
point(std::array<coordinate_type, dimensions> c) : coords_(c) {}
point(std::initializer_list<coordinate_type> list) {
size_t n = std::min(dimensions, list.size());
std::copy_n(list.begin(), n, coords_.begin());
}
/**
* Returns the coordinate in the given dimension.
*
* @param index dimension index (zero based)
* @return coordinate in the given dimension
*/
coordinate_type get(size_t index) const {
return coords_[index];
}
/**
* Returns the distance squared from this point to another
* point.
*
* @param pt another point
* @return distance squared from this point to the other point
*/
double distance(const point& pt) const {
double dist = 0;
for (size_t i = 0; i < dimensions; ++i) {
double d = get(i) - pt.get(i);
dist += d * d;
}
return dist;
}
private:
std::array<coordinate_type, dimensions> coords_;
};
template<typename coordinate_type, size_t dimensions>
std::ostream& operator<<(std::ostream& out, const point<coordinate_type, dimensions>& pt) {
out << '(';
for (size_t i = 0; i < dimensions; ++i) {
if (i > 0)
out << ", ";
out << pt.get(i);
}
out << ')';
return out;
}
/**
* C++ k-d tree implementation, based on the C version at rosettacode.org.
*/
template<typename coordinate_type, size_t dimensions>
class kdtree {
public:
typedef point<coordinate_type, dimensions> point_type;
private:
struct node {
node(const point_type& pt) : point_(pt), left_(nullptr), right_(nullptr) {}
coordinate_type get(size_t index) const {
return point_.get(index);
}
double distance(const point_type& pt) const {
return point_.distance(pt);
}
point_type point_;
node* left_;
node* right_;
};
node* root_ = nullptr;
node* best_ = nullptr;
double best_dist_ = 0;
size_t visited_ = 0;
std::vector<node> nodes_;
struct node_cmp {
node_cmp(size_t index) : index_(index) {}
bool operator()(const node& n1, const node& n2) const {
return n1.point_.get(index_) < n2.point_.get(index_);
}
size_t index_;
};
node* make_tree(size_t begin, size_t end, size_t index) {
if (end <= begin)
return nullptr;
size_t n = begin + (end - begin)/2;
auto i = nodes_.begin();
std::nth_element(i + begin, i + n, i + end, node_cmp(index));
index = (index + 1) % dimensions;
nodes_[n].left_ = make_tree(begin, n, index);
nodes_[n].right_ = make_tree(n + 1, end, index);
return &nodes_[n];
}
void nearest(node* root, const point_type& point, size_t index) {
if (root == nullptr)
return;
++visited_;
double d = root->distance(point);
if (best_ == nullptr || d < best_dist_) {
best_dist_ = d;
best_ = root;
}
if (best_dist_ == 0)
return;
double dx = root->get(index) - point.get(index);
index = (index + 1) % dimensions;
nearest(dx > 0 ? root->left_ : root->right_, point, index);
if (dx * dx >= best_dist_)
return;
nearest(dx > 0 ? root->right_ : root->left_, point, index);
}
public:
kdtree(const kdtree&) = delete;
kdtree& operator=(const kdtree&) = delete;
/**
* Constructor taking a pair of iterators. Adds each
* point in the range [begin, end) to the tree.
*
* @param begin start of range
* @param end end of range
*/
template<typename iterator>
kdtree(iterator begin, iterator end) : nodes_(begin, end) {
root_ = make_tree(0, nodes_.size(), 0);
}
/**
* Constructor taking a function object that generates
* points. The function object will be called n times
* to populate the tree.
*
* @param f function that returns a point
* @param n number of points to add
*/
template<typename func>
kdtree(func&& f, size_t n) {
nodes_.reserve(n);
for (size_t i = 0; i < n; ++i)
nodes_.push_back(f());
root_ = make_tree(0, nodes_.size(), 0);
}
/**
* Returns true if the tree is empty, false otherwise.
*/
bool empty() const { return nodes_.empty(); }
/**
* Returns the number of nodes visited by the last call
* to nearest().
*/
size_t visited() const { return visited_; }
/**
* Returns the distance between the input point and return value
* from the last call to nearest().
*/
double distance() const { return std::sqrt(best_dist_); }
/**
* Finds the nearest point in the tree to the given point.
* It is not valid to call this function if the tree is empty.
*
* @param pt a point
* @return the nearest point in the tree to the given point
*/
const point_type& nearest(const point_type& pt) {
if (root_ == nullptr)
throw std::logic_error("tree is empty");
best_ = nullptr;
visited_ = 0;
best_dist_ = 0;
nearest(root_, pt, 0);
return best_->point_;
}
};
void test_wikipedia() {
typedef point<int, 2> point2d;
typedef kdtree<int, 2> tree2d;
point2d points[] = { { 2, 3 }, { 5, 4 }, { 9, 6 }, { 4, 7 }, { 8, 1 }, { 7, 2 } };
tree2d tree(std::begin(points), std::end(points));
point2d n = tree.nearest({ 9, 2 });
std::cout << "Wikipedia example data:\n";
std::cout << "nearest point: " << n << '\n';
std::cout << "distance: " << tree.distance() << '\n';
std::cout << "nodes visited: " << tree.visited() << '\n';
}
typedef point<double, 3> point3d;
typedef kdtree<double, 3> tree3d;
struct random_point_generator {
random_point_generator(double min, double max)
: engine_(std::random_device()()), distribution_(min, max) {}
point3d operator()() {
double x = distribution_(engine_);
double y = distribution_(engine_);
double z = distribution_(engine_);
return point3d({x, y, z});
}
std::mt19937 engine_;
std::uniform_real_distribution<double> distribution_;
};
void test_random(size_t count) {
random_point_generator rpg(0, 1);
tree3d tree(rpg, count);
point3d pt(rpg());
point3d n = tree.nearest(pt);
std::cout << "Random data (" << count << " points):\n";
std::cout << "point: " << pt << '\n';
std::cout << "nearest point: " << n << '\n';
std::cout << "distance: " << tree.distance() << '\n';
std::cout << "nodes visited: " << tree.visited() << '\n';
}
int main() {
try {
test_wikipedia();
std::cout << '\n';
test_random(1000);
std::cout << '\n';
test_random(1000000);
} catch (const std::exception& e) {
std::cerr << e.what() << '\n';
}
return 0;
}
- Output:
Wikipedia example data: nearest point: (8, 1) distance: 1.41421 nodes visited: 3 Random data (1000 points): point: (0.740311, 0.290258, 0.832057) nearest point: (0.761247, 0.294663, 0.83404) distance: 0.0214867 nodes visited: 15 Random data (1000000 points): point: (0.646712, 0.555327, 0.596551) nearest point: (0.642795, 0.552513, 0.599618) distance: 0.00571496 nodes visited: 46
Common Lisp
The 3D data set is the set of coordinates from (0,0,0) to (9,9,9) (ie. uniformly distributed), while the ordinates of the random target are positive real < 10.
(in-package cl-user)
(defvar *random-target*
(list (float (/ (random 1000) 100))
(float (/ (random 1000) 100))
(float (/ (random 1000) 100))))
(defun distance (n target &key (semi nil))
"distance node to target:
returns squared euclidean distance, or squared semi distance if option set"
(if semi
(expt (- (nth (first n) (second n))
(nth (first n) target))
2)
(reduce #'+
(mapcar (lambda (x y) (* (- x y) (- x y)))
(second n)
target))))
(defun target< (n target)
"returns true if target is to its left in axis dim"
(< (nth (first n) target)
(nth (first n) (second n))))
(defun next-node (n target &key (opposite nil))
"return the next child when nn searching, return opposing child if option
oppose set"
(if (or (and (target< n target) (not opposite))
(and (not (target< n target)) opposite))
(third n)
(fourth n)))
(defun make-kdtree (axis data dims)
"a kdtree is a binary tree where nodes are:
terminal: (axis data-point), or
branch: (axis split-point (left-kdtree) (right-kdtree))"
(cond ((null data) nil)
((eql (length data) 1) ; singleton?
(list axis (first data))) ; terminal node
;; else branch node:
;; #pts=odd splits list into 2 even parts with sp in middle
;; #pts=even splits list into 2 uneven parts with shorter length first (but never nil)
(t
(let ((sd (sort (copy-list data)
#'<
:key (lambda (x) (nth axis x)))) ; sort the axis ordinates
(sp (truncate (length data) 2)) ; get mid pt
(nxta (mod (1+ axis) dims)))
(list axis
(nth sp sd)
(make-kdtree nxta
(subseq sd 0 sp)
dims)
(make-kdtree nxta
(subseq sd (1+ sp))
dims))))))
(defun visit-kdtree (kdt &key (node-function nil))
"depth first visit all nodes in kdtree and optionally apply a function to
each node visited"
(when kdt
(when node-function (funcall node-function kdt))
(visit-kdtree (third kdt) :node-function node-function)
(visit-kdtree (fourth kdt) :node-function node-function)))
(defun count-nodes (kdt)
"count of the terminal nodes"
(cond ((null kdt) 0)
((= (length kdt) 2) 1)
(t (+ 1
(count-nodes (third kdt))
(count-nodes (fourth kdt))))))
(defvar *hits*)
(defun nn-kdtree (kdt node-stack target)
"nearest neighbour search"
(when kdt
;; stage 1 - find the 'closest' terminal node using insertion logic
(let ((best (do ((node kdt (next-node node target)))
((not (next-node node target))
(incf *hits*)
node) ; return first best est.
(push node node-stack)
(incf *hits*))))
;; stage 2 - unwind the path, at each node if node is closer then make it best
(do ((node (pop node-stack) (pop node-stack)))
((null node) best) ; return nearest pt
;; iteration: update best if node is closer
(when (< (distance node target) (distance best target))
(setf best node))
;; venture down opposing side if split point is inside HS
(let ((opposing-best
(if (< (distance node target :semi t) (distance best target))
(nn-kdtree (next-node node target :opposite t)
(list)
target)
nil))) ; otherwise ignore this subtree
(when (and opposing-best
(< (distance opposing-best target) (distance best target)))
(setf best opposing-best)))))))
(defun process (data target
&key (render nil)
&aux (dims (length target)))
"process one set of data & optionally display tree"
(let* ((*hits* 0)
(kdt (make-kdtree 0 data dims))
(nn (nn-kdtree kdt (list) target)))
(when render
(visit-kdtree kdt
:node-function
(lambda (n)
(format t
"~A node: axis:~A point: ~A target:~A semi-distance-sqd:~A euclidean-distance-sqd:~A~%"
(if (not (next-node n target))
"TERMINAL"
"BRANCH")
(first n)
(second n)
target
(distance n target :semi t)
(distance n target)))))
(format t "~%NN to ~A is ~A, distance ~A [tree has ~A nodes, ~A were visited.]~%"
target
(second nn)
(sqrt (distance nn target))
(count-nodes kdt)
*hits*)))
(defun main ()
;; TASK 1 - nn search small set of 2D points
(process '((2 3) (5 4) (9 6) (4 7) (8 1) (7 2))
'(9 2)
:render t)
;; TASK 2 - nn search 1000 coordinate points in 3D space
(process (let ((ll (list)))
(dotimes (i 10)
(dotimes (j 10)
(dotimes (k 10)
(push (list i j k) ll))))
ll)
*random-target*))
- Output:
CL-USER(21): (main) BRANCH node: axis:0 point: (7 2) target:(9 2) semi-distance-sqd:4 euclidean-distance-sqd:4 BRANCH node: axis:1 point: (5 4) target:(9 2) semi-distance-sqd:4 euclidean-distance-sqd:20 TERMINAL node: axis:0 point: (2 3) target:(9 2) semi-distance-sqd:49 euclidean-distance-sqd:50 TERMINAL node: axis:0 point: (4 7) target:(9 2) semi-distance-sqd:25 euclidean-distance-sqd:50 BRANCH node: axis:1 point: (9 6) target:(9 2) semi-distance-sqd:16 euclidean-distance-sqd:16 TERMINAL node: axis:0 point: (8 1) target:(9 2) semi-distance-sqd:1 euclidean-distance-sqd:2 NN to (9 2) is (8 1), distance 1.4142135 [tree has 6 nodes, 3 were visited.] NN to (7.52 8.82 2.55) is (8 9 3), distance 0.68212914 [tree has 1000 nodes, 51 were visited.] NIL CL-USER(22):
D
Points are values, the code is templated on the the dimensionality of the points and the floating point type of the coordinate. Instead of sorting it uses the faster topN, that partitions the points array in two halves around their median.
// Implmentation following pseudocode from
// "An introductory tutorial on kd-trees" by Andrew W. Moore,
// Carnegie Mellon University, PDF accessed from:
// http://www.autonlab.org/autonweb/14665
import std.typecons, std.math, std.algorithm, std.random, std.range,
std.traits, core.memory;
/// k-dimensional point.
struct Point(size_t k, F) if (isFloatingPoint!F) {
F[k] data;
alias data this; // Kills DMD std.algorithm.swap inlining.
// Define opIndexAssign and opIndex for dmd.
enum size_t length = k;
/// Square of the euclidean distance.
double sqd(in ref Point!(k, F) q) const pure nothrow @nogc {
double sum = 0;
foreach (immutable dim, immutable pCoord; data)
sum += (pCoord - q[dim]) ^^ 2;
return sum;
}
}
// Following field names in the paper.
// rangeElt would be whatever data is associated with the Point.
// We don't bother with it for this example.
struct KdNode(size_t k, F) {
Point!(k, F) domElt;
immutable int split;
typeof(this)* left, right;
}
struct Orthotope(size_t k, F) { /// k-dimensional rectangle.
Point!(k, F) min, max;
}
struct KdTree(size_t k, F) {
KdNode!(k, F)* n;
Orthotope!(k, F) bounds;
// Constructs a KdTree from a list of points, also associating the
// bounds of the tree. The bounds could be computed of course, but
// in this example we know them already. The algorithm is table
// 6.3 in the paper.
this(Point!(k, F)[] pts, in Orthotope!(k, F) bounds_) pure {
static KdNode!(k, F)* nk2(size_t split)(Point!(k, F)[] exset)
pure {
if (exset.empty)
return null;
if (exset.length == 1)
return new KdNode!(k, F)(exset[0], split, null, null);
// Pivot choosing procedure. We find median, then find
// largest index of points with median value. This
// satisfies the inequalities of steps 6 and 7 in the
// algorithm.
auto m = exset.length / 2;
topN!((p, q) => p[split] < q[split])(exset, m);
immutable d = exset[m];
while (m+1 < exset.length && exset[m+1][split] == d[split])
m++;
enum nextSplit = (split + 1) % d.length;//cycle coordinates
return new KdNode!(k, F)(d, split,
nk2!nextSplit(exset[0 .. m]),
nk2!nextSplit(exset[m + 1 .. $]));
}
this.n = nk2!0(pts);
this.bounds = bounds_;
}
}
/**
Find nearest neighbor. Return values are:
nearest neighbor--the ooint within the tree that is nearest p.
square of the distance to that point.
a count of the nodes visited in the search.
*/
auto findNearest(size_t k, F)(KdTree!(k, F) t, in Point!(k, F) p)
pure nothrow @nogc {
// Algorithm is table 6.4 from the paper, with the addition of
// counting the number nodes visited.
static Tuple!(Point!(k, F), "nearest",
F, "distSqd",
int, "nodesVisited")
nn(KdNode!(k, F)* kd, in Point!(k, F) target,
Orthotope!(k, F) hr, F maxDistSqd) pure nothrow @nogc {
if (kd == null)
return typeof(return)(Point!(k, F)(), F.infinity, 0);
int nodesVisited = 1;
immutable s = kd.split;
auto pivot = kd.domElt;
auto leftHr = hr;
auto rightHr = hr;
leftHr.max[s] = pivot[s];
rightHr.min[s] = pivot[s];
KdNode!(k, F)* nearerKd, furtherKd;
Orthotope!(k, F) nearerHr, furtherHr;
if (target[s] <= pivot[s]) {
//nearerKd, nearerHr = kd.left, leftHr;
//furtherKd, furtherHr = kd.right, rightHr;
nearerKd = kd.left;
nearerHr = leftHr;
furtherKd = kd.right;
furtherHr = rightHr;
} else {
//nearerKd, nearerHr = kd.right, rightHr;
//furtherKd, furtherHr = kd.left, leftHr;
nearerKd = kd.right;
nearerHr = rightHr;
furtherKd = kd.left;
furtherHr = leftHr;
}
auto n1 = nn(nearerKd, target, nearerHr, maxDistSqd);
auto nearest = n1.nearest;
auto distSqd = n1.distSqd;
nodesVisited += n1.nodesVisited;
if (distSqd < maxDistSqd)
maxDistSqd = distSqd;
auto d = (pivot[s] - target[s]) ^^ 2;
if (d > maxDistSqd)
return typeof(return)(nearest, distSqd, nodesVisited);
d = pivot.sqd(target);
if (d < distSqd) {
nearest = pivot;
distSqd = d;
maxDistSqd = distSqd;
}
immutable n2 = nn(furtherKd, target, furtherHr, maxDistSqd);
nodesVisited += n2.nodesVisited;
if (n2.distSqd < distSqd) {
nearest = n2.nearest;
distSqd = n2.distSqd;
}
return typeof(return)(nearest, distSqd, nodesVisited);
}
return nn(t.n, p, t.bounds, F.infinity);
}
void showNearest(size_t k, F)(in string heading, KdTree!(k, F) kd,
in Point!(k, F) p) {
import std.stdio: writeln;
writeln(heading, ":");
writeln("Point: ", p);
immutable n = kd.findNearest(p);
writeln("Nearest neighbor: ", n.nearest);
writeln("Distance: ", sqrt(n.distSqd));
writeln("Nodes visited: ", n.nodesVisited, "\n");
}
void main() {
static Point!(k, F) randomPoint(size_t k, F)() {
typeof(return) result;
foreach (immutable i; 0 .. k)
result[i] = uniform(F(0), F(1));
return result;
}
static Point!(k, F)[] randomPoints(size_t k, F)(in size_t n) {
return n.iota.map!(_ => randomPoint!(k, F)).array;
}
import std.stdio, std.conv, std.datetime, std.typetuple;
rndGen.seed(1); // For repeatable outputs.
alias D2 = TypeTuple!(2, double);
alias P = Point!D2;
auto kd1 = KdTree!D2([P([2, 3]), P([5, 4]), P([9, 6]),
P([4, 7]), P([8, 1]), P([7, 2])],
Orthotope!D2(P([0, 0]), P([10, 10])));
showNearest("Wikipedia example data", kd1, P([9, 2]));
enum int N = 400_000;
alias F3 = TypeTuple!(3, float);
alias Q = Point!F3;
StopWatch sw;
GC.disable;
sw.start;
auto kd2 = KdTree!F3(randomPoints!F3(N),
Orthotope!F3(Q([0, 0, 0]), Q([1, 1, 1])));
sw.stop;
GC.enable;
showNearest(text("k-d tree with ", N,
" random 3D ", F3[1].stringof,
" points (construction time: ",
sw.peek.msecs, " ms)"), kd2, randomPoint!F3);
sw.reset;
sw.start;
enum int M = 10_000;
size_t visited = 0;
foreach (immutable _; 0 .. M) {
immutable n = kd2.findNearest(randomPoint!F3);
visited += n.nodesVisited;
}
sw.stop;
writefln("Visited an average of %0.2f nodes on %d searches " ~
"in %d ms.", visited / double(M), M, sw.peek.msecs);
}
- Output:
using the ldc2 compiler
Wikipedia example data: Point: [9, 2] Nearest neighbor: [8, 1] Distance: 1.41421 Nodes visited: 3 k-d tree with 400000 random 3D float points (construction time: 250 ms): Point: [0.22012, 0.984514, 0.698782] Nearest neighbor: [0.225766, 0.978981, 0.69885] Distance: 0.00790531 Nodes visited: 54 Visited an average of 43.10 nodes on 10000 searches in 33 ms.
Faster Alternative Version
This version performs less lookups. Compiled with DMD this version is two times slower than the C version. Compiled with ldc2 it's a little faster than the C version compiled with gcc.
import std.stdio, std.algorithm, std.math, std.random;
struct KdNode(size_t dim) {
double[dim] x;
KdNode* left, right;
}
// See QuickSelect method.
KdNode!dim* findMedian(size_t idx, size_t dim)(KdNode!dim[] nodes) pure nothrow @nogc {
auto start = nodes.ptr;
auto end = &nodes[$ - 1] + 1;
if (end <= start)
return null;
if (end == start + 1)
return start;
auto md = start + (end - start) / 2;
while (true) {
immutable double pivot = md.x[idx];
swap(md.x, (end - 1).x); // Swaps the whole arrays x.
auto store = start;
foreach (p; start .. end) {
if (p.x[idx] < pivot) {
if (p != store)
swap(p.x, store.x);
store++;
}
}
swap(store.x, (end - 1).x);
// Median has duplicate values.
if (store.x[idx] == md.x[idx])
return md;
if (store > md)
end = store;
else
start = store;
}
}
KdNode!dim* makeTree(size_t dim, size_t i = 0)(KdNode!dim[] nodes)
pure nothrow @nogc {
if (!nodes.length)
return null;
auto n = nodes.findMedian!i;
if (n != null) {
enum i2 = (i + 1) % dim;
immutable size_t nPos = n - nodes.ptr;
n.left = makeTree!(dim, i2)(nodes[0 .. nPos]);
n.right = makeTree!(dim, i2)(nodes[nPos + 1 .. $]);
}
return n;
}
void nearest(size_t dim)(in KdNode!dim* root,
in ref KdNode!dim nd,
in size_t i,
ref const(KdNode!dim)* best,
ref double bestDist,
ref size_t nVisited) pure nothrow @safe @nogc {
static double dist(in ref KdNode!dim a, in ref KdNode!dim b)
pure nothrow @nogc {
double result = 0;
static foreach (i; 0 .. dim)
result += (a.x[i] - b.x[i]) ^^ 2;
return result;
}
if (root == null)
return;
immutable double d = dist(*root, nd);
immutable double dx = root.x[i] - nd.x[i];
immutable double dx2 = dx ^^ 2;
nVisited++;
if (!best || d < bestDist) {
bestDist = d;
best = root;
}
// If chance of exact match is high.
if (!bestDist)
return;
immutable i2 = (i + 1 >= dim) ? 0 : i + 1;
nearest!dim(dx > 0 ? root.left : root.right,
nd, i2, best, bestDist, nVisited);
if (dx2 >= bestDist)
return;
nearest!dim(dx > 0 ? root.right : root.left,
nd, i2, best, bestDist, nVisited);
}
void randPt(size_t dim)(ref KdNode!dim v, ref Xorshift rng)
pure nothrow @safe @nogc {
static foreach (i; 0 .. dim)
v.x[i] = rng.uniform01;
}
/// smallTest
unittest {
KdNode!2[] wp = [{[2, 3]}, {[5, 4]}, {[9, 6]},
{[4, 7]}, {[8, 1]}, {[7, 2]}];
KdNode!2 thisPt = {[9, 2]};
auto root = makeTree(wp);
const(KdNode!2)* found = null;
double bestDist = 0;
size_t nVisited = 0;
root.nearest(thisPt, 0, found, bestDist, nVisited);
writefln("WP tree:\n Searching for %s\n" ~
" Found %s, dist = %g\n Seen %d nodes.\n",
thisPt.x, found.x, sqrt(bestDist), nVisited);
}
/// bigTest
unittest {
enum N = 1_000_000;
enum testRuns = 100_000;
auto bigTree = new KdNode!3[N];
auto rng = 1.Xorshift;
foreach (ref node; bigTree)
randPt(node, rng);
auto root = makeTree(bigTree);
KdNode!3 thisPt;
randPt(thisPt, rng);
const(KdNode!3)* found = null;
double bestDist = 0;
size_t nVisited = 0;
root.nearest(thisPt, 0, found, bestDist, nVisited);
writefln("Big tree (%d nodes):\n Searching for %s\n" ~ " Found %s, dist = %g\n Seen %d nodes.", N, thisPt.x, found.x, sqrt(bestDist), nVisited);
size_t sum = 0;
foreach (immutable _; 0 .. testRuns) {
found = null;
nVisited = 0;
randPt(thisPt, rng);
nearest!3(root, thisPt, 0, found, bestDist, nVisited);
sum += nVisited;
}
writefln("\nBig tree:\n Visited %d nodes for %d random " ~
"searches (%.2f per lookup).",
sum, testRuns, sum / double(testRuns));
}
- Output:
WP tree: Searching for [9, 2] Found [8, 1], dist = 1.41421 Seen 3 nodes. Big tree (1000000 nodes): Searching for [0.225893, 0.725471, 0.486279] Found [0.220761, 0.729613, 0.489134], dist = 0.00718703 Seen 35 nodes. Big tree: Visited 4267592 nodes for 100000 random searches (42.68 per lookup).
Go
// Implmentation following pseudocode from "An intoductory tutorial on kd-trees"
// by Andrew W. Moore, Carnegie Mellon University, PDF accessed from
// http://www.autonlab.org/autonweb/14665
package main
import (
"fmt"
"math"
"math/rand"
"sort"
"time"
)
// point is a k-dimensional point.
type point []float64
// sqd returns the square of the euclidean distance.
func (p point) sqd(q point) float64 {
var sum float64
for dim, pCoord := range p {
d := pCoord - q[dim]
sum += d * d
}
return sum
}
// kdNode following field names in the paper.
// rangeElt would be whatever data is associated with the point. we don't
// bother with it for this example.
type kdNode struct {
domElt point
split int
left, right *kdNode
}
type kdTree struct {
n *kdNode
bounds hyperRect
}
type hyperRect struct {
min, max point
}
// Go slices are reference objects. The data must be copied if you want
// to modify one without modifying the original.
func (hr hyperRect) copy() hyperRect {
return hyperRect{append(point{}, hr.min...), append(point{}, hr.max...)}
}
// newKd constructs a kdTree from a list of points, also associating the
// bounds of the tree. The bounds could be computed of course, but in this
// example we know them already. The algorithm is table 6.3 in the paper.
func newKd(pts []point, bounds hyperRect) kdTree {
var nk2 func([]point, int) *kdNode
nk2 = func(exset []point, split int) *kdNode {
if len(exset) == 0 {
return nil
}
// pivot choosing procedure. we find median, then find largest
// index of points with median value. this satisfies the
// inequalities of steps 6 and 7 in the algorithm.
sort.Sort(part{exset, split})
m := len(exset) / 2
d := exset[m]
for m+1 < len(exset) && exset[m+1][split] == d[split] {
m++
}
// next split
s2 := split + 1
if s2 == len(d) {
s2 = 0
}
return &kdNode{d, split, nk2(exset[:m], s2), nk2(exset[m+1:], s2)}
}
return kdTree{nk2(pts, 0), bounds}
}
// a container type used for sorting. it holds the points to sort and
// the dimension to use for the sort key.
type part struct {
pts []point
dPart int
}
// satisfy sort.Interface
func (p part) Len() int { return len(p.pts) }
func (p part) Less(i, j int) bool {
return p.pts[i][p.dPart] < p.pts[j][p.dPart]
}
func (p part) Swap(i, j int) { p.pts[i], p.pts[j] = p.pts[j], p.pts[i] }
// nearest. find nearest neighbor. return values are:
// nearest neighbor--the point within the tree that is nearest p.
// square of the distance to that point.
// a count of the nodes visited in the search.
func (t kdTree) nearest(p point) (best point, bestSqd float64, nv int) {
return nn(t.n, p, t.bounds, math.Inf(1))
}
// algorithm is table 6.4 from the paper, with the addition of counting
// the number nodes visited.
func nn(kd *kdNode, target point, hr hyperRect,
maxDistSqd float64) (nearest point, distSqd float64, nodesVisited int) {
if kd == nil {
return nil, math.Inf(1), 0
}
nodesVisited++
s := kd.split
pivot := kd.domElt
leftHr := hr.copy()
rightHr := hr.copy()
leftHr.max[s] = pivot[s]
rightHr.min[s] = pivot[s]
targetInLeft := target[s] <= pivot[s]
var nearerKd, furtherKd *kdNode
var nearerHr, furtherHr hyperRect
if targetInLeft {
nearerKd, nearerHr = kd.left, leftHr
furtherKd, furtherHr = kd.right, rightHr
} else {
nearerKd, nearerHr = kd.right, rightHr
furtherKd, furtherHr = kd.left, leftHr
}
var nv int
nearest, distSqd, nv = nn(nearerKd, target, nearerHr, maxDistSqd)
nodesVisited += nv
if distSqd < maxDistSqd {
maxDistSqd = distSqd
}
d := pivot[s] - target[s]
d *= d
if d > maxDistSqd {
return
}
if d = pivot.sqd(target); d < distSqd {
nearest = pivot
distSqd = d
maxDistSqd = distSqd
}
tempNearest, tempSqd, nv := nn(furtherKd, target, furtherHr, maxDistSqd)
nodesVisited += nv
if tempSqd < distSqd {
nearest = tempNearest
distSqd = tempSqd
}
return
}
func main() {
rand.Seed(time.Now().Unix())
kd := newKd([]point{{2, 3}, {5, 4}, {9, 6}, {4, 7}, {8, 1}, {7, 2}},
hyperRect{point{0, 0}, point{10, 10}})
showNearest("WP example data", kd, point{9, 2})
kd = newKd(randomPts(3, 1000), hyperRect{point{0, 0, 0}, point{1, 1, 1}})
showNearest("1000 random 3d points", kd, randomPt(3))
}
func randomPt(dim int) point {
p := make(point, dim)
for d := range p {
p[d] = rand.Float64()
}
return p
}
func randomPts(dim, n int) []point {
p := make([]point, n)
for i := range p {
p[i] = randomPt(dim)
}
return p
}
func showNearest(heading string, kd kdTree, p point) {
fmt.Println()
fmt.Println(heading)
fmt.Println("point: ", p)
nn, ssq, nv := kd.nearest(p)
fmt.Println("nearest neighbor:", nn)
fmt.Println("distance: ", math.Sqrt(ssq))
fmt.Println("nodes visited: ", nv)
}
- Output:
WP example data point: [9 2] nearest neighbor: [8 1] distance: 1.4142135623730951 nodes visited: 3 1000 random 3d points point: [0.314731890562714 0.5908890147906868 0.2657722255021785] nearest neighbor: [0.2541611609533609 0.5781168738628141 0.27829000365095274] distance: 0.06315564613771865 nodes visited: 25
Haskell
There is a space leak when creating the trees which will lead to terrible performance on massive trees. This can probably be quelled with strictness annotations.
import System.Random
import Data.List (sortBy, genericLength, minimumBy)
import Data.Ord (comparing)
-- A finite list of dimensional accessors tell a KDTree how to get a
-- Euclidean dimensional value 'b' out of an arbitrary datum 'a'.
type DimensionalAccessors a b = [a -> b]
-- A binary tree structure of 'a'.
data Tree a = Node a (Tree a) (Tree a)
| Empty
instance Show a => Show (Tree a) where
show Empty = "Empty"
show (Node value left right) =
"(" ++ show value ++ " " ++ show left ++ " " ++ show right ++ ")"
-- A k-d tree structure of 'a' with Euclidean dimensions of 'b'.
data KDTree a b = KDTree (DimensionalAccessors a b) (Tree a)
instance Show a => Show (KDTree a b) where
show (KDTree _ tree) = "KDTree " ++ show tree
-- The squared Euclidean distance formula.
sqrDist :: Num b => DimensionalAccessors a b -> a -> a -> b
sqrDist dims a b = sum $ map square $ zipWith (-) a' b'
where
a' = map ($ a) dims
b' = map ($ b) dims
square :: Num a => a -> a
square = (^ 2)
-- Insert a value into a k-d tree.
insert :: Ord b => KDTree a b -> a -> KDTree a b
insert (KDTree dims tree) value = KDTree dims $ ins (cycle dims) tree
where
ins _ Empty = Node value Empty Empty
ins (d:ds) (Node split left right) =
if d value < d split
then Node split (ins ds left) right
else Node split left (ins ds right)
-- Produce an empty k-d tree.
empty :: DimensionalAccessors a b -> KDTree a b
empty dims = KDTree dims Empty
-- Produce a k-d tree with one value.
singleton :: Ord b => DimensionalAccessors a b -> a -> KDTree a b
singleton dims value = insert (empty dims) value
-- Create a k-d tree from a list of values using the median-finding algorithm.
fromList :: Ord b => DimensionalAccessors a b -> [a] -> KDTree a b
fromList dims values = KDTree dims $ fList (cycle dims) values
where
fList _ [] = Empty
fList (d:ds) values =
let sorted = sortBy (comparing d) values
(lower, higher) = splitAt (genericLength sorted `div` 2) sorted
in case higher of
[] -> Empty
median:rest -> Node median (fList ds lower) (fList ds rest)
-- Create a k-d tree from a list of values by repeatedly inserting the values
-- into a tree. Faster than median-finding, but can create unbalanced trees.
fromListLinear :: Ord b => DimensionalAccessors a b -> [a] -> KDTree a b
fromListLinear dims values = foldl insert (empty dims) values
-- Given a k-d tree, find the nearest value to a given value.
-- Also report how many nodes were visited.
nearest :: (Ord b, Num b, Integral c) => KDTree a b -> a -> (Maybe a, c)
nearest (KDTree dims tree) value = near (cycle dims) tree
where
dist = sqrDist dims
-- If we have an empty tree, then return nothing.
near _ Empty = (Nothing, 1)
-- We hit a leaf node, so it is the current best.
near _ (Node split Empty Empty) = (Just split, 1)
near (d:ds) (Node split left right) =
-- Move down the tree in the fashion of insertion.
let dimdist x y = square (d x - d y)
splitDist = dist value split
hyperPlaneDist = dimdist value split
bestLeft = near ds left
bestRight = near ds right
-- maybeThisBest is the node of the side of the split where the value
-- resides, and maybeOtherBest is the node on the other side of the split.
((maybeThisBest, thisCount), (maybeOtherBest, otherCount)) =
if d value < d split
then (bestLeft, bestRight)
else (bestRight, bestLeft)
in case maybeThisBest of
Nothing ->
-- From the search point (in this case), the hypersphere radius to
-- the split node is always >= the hyperplane distance, so we will
-- always check the other side.
let count = 1 + thisCount + otherCount
in case maybeOtherBest of
-- We are currently at a leaf node, so this is the only choice.
-- It is not strictly necessary to take care of this case
-- because of the above pattern matching in near.
Nothing -> (Just split, count)
-- We have a node on the other side, so compare
-- it to the split point to see which is closer.
Just otherBest ->
if dist value otherBest < splitDist
then (maybeOtherBest, count)
else (Just split, count)
Just thisBest ->
let thisBestDist = dist value thisBest
best =
-- Determine which is the closer node of this side.
if splitDist < thisBestDist
then split
else thisBest
bestDist = dist value best
in
if bestDist < hyperPlaneDist
-- If the distance to the best node is less than the distance
-- to the splitting hyperplane, then the current best node is the
-- only choice.
then (Just best, 1 + thisCount)
-- There is a chance that a node on the other side is closer
-- than the current best.
else
let count = 1 + thisCount + otherCount
in case maybeOtherBest of
Nothing -> (Just best, count)
Just otherBest ->
if bestDist < dist value otherBest
then (Just best, count)
else (maybeOtherBest, count)
-- Dimensional accessors for a 2-tuple
tuple2D :: [(a, a) -> a]
tuple2D = [fst, snd]
-- Dimensional accessors for a 3-tuple
tuple3D :: [(a, a, a) -> a]
tuple3D = [d1, d2, d3]
where
d1 (a, _, _) = a
d2 (_, b, _) = b
d3 (_, _, c) = c
-- Random 3-tuple generation
instance (Random a, Random b, Random c) => Random (a, b, c) where
random gen =
let (vA, genA) = random gen
(vB, genB) = random genA
(vC, genC) = random genB
in ((vA, vB, vC), genC)
randomR ((lA, lB, lC), (hA, hB, hC)) gen =
let (vA, genA) = randomR (lA, hA) gen
(vB, genB) = randomR (lB, hB) genA
(vC, genC) = randomR (lC, hC) genB
in ((vA, vB, vC), genC)
printResults :: (Show a, Show b, Show c, Floating c) =>
a -> (Maybe a, b) -> DimensionalAccessors a c -> IO ()
printResults point result dims = do
let (nearest, visited) = result
case nearest of
Nothing -> putStrLn "Could not find nearest."
Just value -> do
let dist = sqrt $ sqrDist dims point value
putStrLn $ "Point: " ++ show point
putStrLn $ "Nearest: " ++ show value
putStrLn $ "Distance: " ++ show dist
putStrLn $ "Visited: " ++ show visited
putStrLn ""
-- Naive nearest search used to confirm results.
linearNearest :: (Ord b, Num b) => DimensionalAccessors a b -> a -> [a] -> Maybe a
linearNearest _ _ [] = Nothing
linearNearest dims value xs = Just $ minimumBy (comparing $ sqrDist dims value) xs
main :: IO ()
main = do
let wikiValues :: [(Double, Double)]
wikiValues = [(2, 3), (5, 4), (9, 6), (4, 7), (8, 1), (7, 2)]
wikiTree = fromList tuple2D wikiValues
wikiSearch = (9, 2)
wikiNearest = nearest wikiTree wikiSearch
putStrLn "Wikipedia example:"
printResults wikiSearch wikiNearest tuple2D
let stdGen = mkStdGen 0
randRange :: ((Double, Double, Double), (Double, Double, Double))
randRange = ((0, 0, 0), (1000, 1000, 1000))
(randSearch, stdGenB) = randomR randRange stdGen
randValues = take 1000 $ randomRs randRange stdGenB
randTree = fromList tuple3D randValues
randNearest = nearest randTree randSearch
randNearestLinear = linearNearest tuple3D randSearch randValues
putStrLn "1000 random 3D points on the range of [0, 1000):"
printResults randSearch randNearest tuple3D
putStrLn "Confirm naive nearest:"
print randNearestLinear
- Output:
Wikipedia example: Point: (9.0,2.0) Nearest: (8.0,1.0) Distance: 1.4142135623730951 Visited: 3 1000 random 3D points on the range of [0, 1000): Point: (992.9251340102518,993.3624439225405,464.8305261946105) Nearest: (939.1965739740829,980.2876583283734,452.4829965078272) Distance: 56.658359235505955 Visited: 23 Confirm naive nearest: Just (939.1965739740829,980.2876583283734,452.4829965078272)
J
As a general rule, tree algorithms are a bad idea in J. That said, here's an implementation:
coclass 'kdnode'
create=:3 :0
Axis=: ({:$y)|<.2^.#y
Mask=: Axis~:i.{:$y
if. 3>#y do.
Leaf=:1
Points=: y
else.
Leaf=:0
data=. y /: Axis|."1 y
n=. <.-:#data
Points=: ,:n{data
Left=: conew&'kdnode' n{.data
Right=: conew&'kdnode' (1+n)}.data
end.
)
distance=: +/&.:*:@:-"1
nearest=:3 :0
_ 0 nearest y
:
n=.' ',~":N_base_=:N_base_+1
dists=. Points distance y
ndx=. (i. <./) dists
nearest=. ndx { Points
range=. ndx { dists
if. Leaf do.
range;nearest return.
else.
d0=. x <. range
p0=. nearest
if. d0=0 do. 0;y return. end.
if. 0={./:Axis|."1 y,Points do.
'dist pnt'=.d0 nearest__Left y
if. dist > d0 do.
d0;p0 return.
end.
if. dist < d0 do.
if. dist > (Mask#pnt) distance Mask#,Points do.
'dist2 pnt2'=. d0 nearest__Right y
if. dist2 < dist do. dist2;pnt2 return. end.
end.
end.
else.
'dist pnt'=. d0 nearest__Right y
if. dist > d0 do.
d0;p0 return.
end.
if. dist < d0 do.
if. dist > (Mask#pnt) distance Mask#,Points do.
'dist2 pnt2'=. d0 nearest__Left y
if. dist2 < dist do. dist2;pnt2 return. end.
end.
end.
end.
end.
dist;pnt return.
)
coclass 'kdtree'
create=:3 :0
root=: conew&'kdnode' y
)
nearest=:3 :0
N_base_=:0
'dist point'=. nearest__root y
dist;N_base_;point
)
And here's example use:
tree=:conew&'kdtree' (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
nearest__tree 9 2
┌───────┬─┬───┐
│1.41421│4│8 1│
└───────┴─┴───┘
The first box is distance from argument point to selected point. The second box is the number of nodes visited. The third box is the selected point.
Here's the bigger problem:
tree=:conew&'kdtree' dataset=:?1000 3$0
nearest__tree pnt
┌─────────┬──┬──────────────────────────┐
│0.0387914│12│0.978082 0.767632 0.392523│
└─────────┴──┴──────────────────────────┘
So, why are trees "generally a bad idea in J"?
First off, that's a lot of code, it took time to write. Let's assume that that time was free. Let's also assume that the time taken to build the tree structure was free. We're going to use this tree billions of times. Now what?
Well, let's compare the above implementation to a brute force implementation for time. Here's a "visit all nodes" implementation. It should give us the same kinds of results but we will claim that each candidate point is a node so we'll be visiting a lot more "nodes":
build0=:3 :0
data=: y
)
distance=: +/&.:*:@:-"1
nearest0=:3 :0
nearest=. data {~ (i. <./) |data distance y
(nearest distance y);(#data);nearest
)
Here's the numbers we get:
build0 (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
nearest0 9 2
┌───────┬─┬───┐
│1.41421│6│8 1│
└───────┴─┴───┘
build0 dataset
nearest0 pnt
┌─────────┬────┬──────────────────────────┐
│0.0387914│1000│0.978082 0.767632 0.392523│
└─────────┴────┴──────────────────────────┘
But what about timing?
tree=:conew&'kdtree' (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
timespacex 'nearest__tree 9 2'
0.000487181 19328
build0 (2,3), (5,4), (9,6), (4,7), (8,1),: (7,2)
timespacex 'nearest0 9 2'
3.62419e_5 6016
The kdtree implementation is over ten times slower than the brute force implementation for this small dataset. How about the bigger dataset?
tree=:conew&'kdtree' dataset
timespacex 'nearest__tree pnt'
0.00141408 45312
build0 dataset
timespacex 'nearest0 pnt'
0.00140702 22144
On the bigger dataset, the kdtree implementation is about the same speed as the brute force implementation.
For a more practical approach to this kind of problem, see https://github.com/locklin/j-nearest-neighbor (that is: link to a high performance implementation).
See also: wp:KISS_principle
Java
Based on the C++ solution. File KdTree.java:
import java.util.*;
public class KdTree {
private int dimensions_;
private Node root_ = null;
private Node best_ = null;
private double bestDistance_ = 0;
private int visited_ = 0;
public KdTree(int dimensions, List<Node> nodes) {
dimensions_ = dimensions;
root_ = makeTree(nodes, 0, nodes.size(), 0);
}
public Node findNearest(Node target) {
if (root_ == null)
throw new IllegalStateException("Tree is empty!");
best_ = null;
visited_ = 0;
bestDistance_ = 0;
nearest(root_, target, 0);
return best_;
}
public int visited() {
return visited_;
}
public double distance() {
return Math.sqrt(bestDistance_);
}
private void nearest(Node root, Node target, int index) {
if (root == null)
return;
++visited_;
double d = root.distance(target);
if (best_ == null || d < bestDistance_) {
bestDistance_ = d;
best_ = root;
}
if (bestDistance_ == 0)
return;
double dx = root.get(index) - target.get(index);
index = (index + 1) % dimensions_;
nearest(dx > 0 ? root.left_ : root.right_, target, index);
if (dx * dx >= bestDistance_)
return;
nearest(dx > 0 ? root.right_ : root.left_, target, index);
}
private Node makeTree(List<Node> nodes, int begin, int end, int index) {
if (end <= begin)
return null;
int n = begin + (end - begin)/2;
Node node = QuickSelect.select(nodes, begin, end - 1, n, new NodeComparator(index));
index = (index + 1) % dimensions_;
node.left_ = makeTree(nodes, begin, n, index);
node.right_ = makeTree(nodes, n + 1, end, index);
return node;
}
private static class NodeComparator implements Comparator<Node> {
private int index_;
private NodeComparator(int index) {
index_ = index;
}
public int compare(Node n1, Node n2) {
return Double.compare(n1.get(index_), n2.get(index_));
}
}
public static class Node {
private double[] coords_;
private Node left_ = null;
private Node right_ = null;
public Node(double[] coords) {
coords_ = coords;
}
public Node(double x, double y) {
this(new double[]{x, y});
}
public Node(double x, double y, double z) {
this(new double[]{x, y, z});
}
double get(int index) {
return coords_[index];
}
double distance(Node node) {
double dist = 0;
for (int i = 0; i < coords_.length; ++i) {
double d = coords_[i] - node.coords_[i];
dist += d * d;
}
return dist;
}
public String toString() {
StringBuilder s = new StringBuilder("(");
for (int i = 0; i < coords_.length; ++i) {
if (i > 0)
s.append(", ");
s.append(coords_[i]);
}
s.append(')');
return s.toString();
}
}
}
File QuickSelect.java:
import java.util.*;
//
// Java implementation of quickselect algorithm.
// See https://en.wikipedia.org/wiki/Quickselect
//
public class QuickSelect {
private static final Random random = new Random();
public static <T> T select(List<T> list, int n, Comparator<? super T> cmp) {
return select(list, 0, list.size() - 1, n, cmp);
}
public static <T> T select(List<T> list, int left, int right, int n, Comparator<? super T> cmp) {
for (;;) {
if (left == right)
return list.get(left);
int pivot = pivotIndex(left, right);
pivot = partition(list, left, right, pivot, cmp);
if (n == pivot)
return list.get(n);
else if (n < pivot)
right = pivot - 1;
else
left = pivot + 1;
}
}
private static <T> int partition(List<T> list, int left, int right, int pivot, Comparator<? super T> cmp) {
T pivotValue = list.get(pivot);
swap(list, pivot, right);
int store = left;
for (int i = left; i < right; ++i) {
if (cmp.compare(list.get(i), pivotValue) < 0) {
swap(list, store, i);
++store;
}
}
swap(list, right, store);
return store;
}
private static <T> void swap(List<T> list, int i, int j) {
T value = list.get(i);
list.set(i, list.get(j));
list.set(j, value);
}
private static int pivotIndex(int left, int right) {
return left + random.nextInt(right - left + 1);
}
}
File KdTreeTest.java:
import java.util.*;
public class KdTreeTest {
public static void main(String[] args) {
testWikipedia();
System.out.println();
testRandom(1000);
System.out.println();
testRandom(1000000);
}
private static void testWikipedia() {
double[][] coords = {
{ 2, 3 }, { 5, 4 }, { 9, 6 }, { 4, 7 }, { 8, 1 }, { 7, 2 }
};
List<KdTree.Node> nodes = new ArrayList<>();
for (int i = 0; i < coords.length; ++i)
nodes.add(new KdTree.Node(coords[i]));
KdTree tree = new KdTree(2, nodes);
KdTree.Node nearest = tree.findNearest(new KdTree.Node(9, 2));
System.out.println("Wikipedia example data:");
System.out.println("nearest point: " + nearest);
System.out.println("distance: " + tree.distance());
System.out.println("nodes visited: " + tree.visited());
}
private static KdTree.Node randomPoint(Random random) {
double x = random.nextDouble();
double y = random.nextDouble();
double z = random.nextDouble();
return new KdTree.Node(x, y, z);
}
private static void testRandom(int points) {
Random random = new Random();
List<KdTree.Node> nodes = new ArrayList<>();
for (int i = 0; i < points; ++i)
nodes.add(randomPoint(random));
KdTree tree = new KdTree(3, nodes);
KdTree.Node target = randomPoint(random);
KdTree.Node nearest = tree.findNearest(target);
System.out.println("Random data (" + points + " points):");
System.out.println("target: " + target);
System.out.println("nearest point: " + nearest);
System.out.println("distance: " + tree.distance());
System.out.println("nodes visited: " + tree.visited());
}
}
- Output:
Wikipedia example data: nearest point: (8.0, 1.0) distance: 1.4142135623730951 nodes visited: 3 Random data (1000 points): target: (0.3616716857847091, 0.9408264735245401, 0.7566330710858733) nearest point: (0.39544672834842864, 0.9130081104715686, 0.7408845015796762) distance: 0.04650411019060053 nodes visited: 13 Random data (1000000 points): target: (0.5383987287951542, 0.43077475564298007, 0.05452587970614786) nearest point: (0.5415653686393387, 0.42874081149927457, 0.056730119253074474) distance: 0.004361560347252592 nodes visited: 34
jq
Works with gojq, the Go implementation of jq
Since jq does not include a PRNG, the following assumes that an external source of entropy such as /dev/urandom is available. See the "Invocation" section below for details.
In the following, a Point is represented by a numeric JSON array.
### Generic utilities
# Output: a PRN in range(0; .)
def prn:
if . == 1 then 0
else . as $n
| (($n-1)|tostring|length) as $w
| [limit($w; inputs)] | join("") | tonumber
| if . < $n then . else ($n | prn) end
end;
def rand: 1000 | prn / 1000;
def randomPt($dim):
[ range(0; $dim) | rand ] ;
def randomPts($dim; $n):
[ range(0;$n) | randomPt($dim) ];
def sq: .*.;
# $p and $q should be numeric arrays of the same length
def PtSqd($p; $q):
[$p, $q] | transpose | map((.[0] - .[1]) | sq) | add;
### K-d trees
def HyperRect($min; $max):
{ $min, $max };
def NearestNeighbor($nearest; $distSqd; $nodesVisited):
{ $nearest, $distSqd, $nodesVisited };
def KdNode($domElt; $split; $left; $right):
{$domElt, $split, $left, $right};
def KdTree($pts; $bounds):
def nk2($exset; $split):
if ($exset|length == 0) then null
else { $exset }
| .exset |= sort_by(.[$split])
| .m = ((.exset|length/2)|floor)
| .exset[.m] as $d
| until (.m + 1 >= (.exset|length) or .exset[.m + 1][$split] != $d[$split];
.m += 1)
| .s2 = $split + 1
| if .s2 == ($d|length) then .s2 = 0 else . end
| KdNode($d;
$split;
nk2(.exset[: .m]; .s2);
nk2(.exset[.m + 1 :]; .s2)
)
end ;
{ n: nk2($pts; 0),
$bounds } ;
# Input: a KdTree
def nearest($p):
# Input: a KdTree
def nn_($target; $hr; $maxDistSqd):
if not then NearestNeighbor(null; infinite; 0)
else .split as $s
| .domElt as $pivot
| .left as $left
| .right as $right
| ($target[$s] <= $pivot[$s]) as $targetInLeft
| { nodesVisited: 1,
leftHr: $hr,
rightHr: $hr }
| .leftHr.max[$s] = $pivot[$s]
| .rightHr.min[$s] = $pivot[$s]
| (if $targetInLeft then $left else $right end) as $nearerKd
| (if $targetInLeft then .leftHr else .rightHr end) as $nearerHr
| (if $targetInLeft then $right else $left end) as $furtherKd
| (if $targetInLeft then .rightHr else .leftHr end) as $furtherHr
| ($nearerKd | nn_($target; $nearerHr; $maxDistSqd)) as $res
| .nearest = $res.nearest
| .distSqd = $res.distSqd
| .nodesVisited += $res.nodesVisited
| .maxDistSqd2 = ([.distSqd, $maxDistSqd] | min)
| .d = (($pivot[$s] - $target[$s]) | sq)
| if .d > .maxDistSqd2 then NearestNeighbor(.nearest; .distSqd; .nodesVisited)
else .d = PtSqd($pivot; $target)
| if .d < .distSqd
then .nearest = $pivot
| .distSqd = .d
| .maxDistSqd2 = .distSqd
else .
end
| ($furtherKd | nn_($target; $furtherHr; .maxDistSqd2)) as $temp
| .nodesVisited += $temp.nodesVisited
| if $temp.distSqd < .distSqd
then .nearest = $temp.nearest
| .distSqd = $temp.distSqd
else .
end
| NearestNeighbor(.nearest; .distSqd; .nodesVisited)
end
end ;
.n | nn_($p; .bounds; infinite);
def example($heading; $hr; $points; $p):
KdTree($points; .hr)
| nearest($p)
| "\(heading):",
"Point : \($p)",
"Nearest neighbor : \(.nearest)",
"Distance : \(.distSqd | sqrt)",
"Nodes visited : \(.nodesVisited)",
"";
### Examples
def points: [[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]];
example("WP example data";
HyperRect([0, 0]; [10, 10]); points; [9,2] ),
example("1,000 random 3D points";
HyperRect([0, 0, 0]; [1, 1, 1]); randomPts(3; 1000); randomPt(3)),
example("400,000 random 3D points";
HyperRect([0, 0, 0]; [1, 1, 1]); randomPts(3; 400,000); randomPt(3))- Output:
WP example data: Point : [9,2] Nearest neighbor : [8,1] Distance : 1.4142135623730951 Nodes visited : 3 1,000 random 3D points: Point : [0.625,0.521,0.996] Nearest neighbor : [0.597,0.492,0.997] Distance : 0.040323690307311935 Nodes visited : 28 400,000 random 3D points: Point : [0.847,0.237,0.702] Nearest neighbor : [0.848,0.237,0.706] Distance : 0.004123105625617665 Nodes visited : 29
Julia
ENV["NN_DEBUG"] = true # or change DEBUG = true in NearestNeighbors.jl file
using NearestNeighbors
const data = [[2, 3] [5, 4] [9, 6] [4, 7] [8, 1] [7, 2]]
NearestNeighbors.reset_stats()
const kdtree = KDTree(Float64.(data))
const indexpoint = [9,2]
idx, dist = knn(kdtree, indexpoint, 1)
println("Wikipedia example: The nearest neighbor to $indexpoint is ",
"$(data[1:2, idx[1]]) at distance $(dist).")
NearestNeighbors.print_stats()
NearestNeighbors.reset_stats()
const cubedis = rand(3, 1000)
const kdcubetree = KDTree(cubedis)
const rand3point = rand(3, 1)
idx, dist = knn(kdcubetree, rand3point, 1)
println("\n\n1000 cube points: The point $rand3point is closest to the point $(cubedis[1:3, idx[1]])",
" at distance $(dist[1]).")
NearestNeighbors.print_stats()
NearestNeighbors.reset_stats()
const cubedis2 = rand(3, 500000)
const kdcubetree2 = KDTree(cubedis2)
const rand3point2 = rand(3, 1)
idx, dist = knn(kdcubetree2, rand3point2, 1)
println("\n\nExtra: The point $rand3point2 is closest to the point $(cubedis2[1:3, idx[1]]) out of $(size(cubedis2)[2]) points,",
" at distance $(dist[1]).")
NearestNeighbors.print_stats()
- Output:
Wikipedia example: The nearest neighbor to [9, 2] is [8, 1] at distance [1.41421]. Nodes visited: 1 Points visited: 6, out of these: 0 unchecked. 1000 cube points: The point [0.386894; 0.465532; 0.710547] is closest to the point [0.389152; 0.459995; 0.639123] at distance [0.0716743]. Nodes visited: 13 Points visited: 30, out of these: 0 unchecked. Extra: The point [0.373496; 0.740711; 0.417431] is closest to the point [0.377841; 0.744084; 0.420237] out of 500000 points, at distance [0.00617466]. Nodes visited: 28 Points visited: 40, out of these: 0 unchecked.
Kotlin
// version 1.1.51
import java.util.Random
typealias Point = DoubleArray
fun Point.sqd(p: Point) = this.zip(p) { a, b -> (a - b) * (a - b) }.sum()
class HyperRect (val min: Point, val max: Point) {
fun copy() = HyperRect(min.copyOf(), max.copyOf())
}
data class NearestNeighbor(val nearest: Point?, val distSqd: Double, val nodesVisited: Int)
class KdNode(
val domElt: Point,
val split: Int,
var left: KdNode?,
var right: KdNode?
)
class KdTree {
val n: KdNode?
val bounds: HyperRect
constructor(pts: MutableList<Point>, bounds: HyperRect) {
fun nk2(exset: MutableList<Point>, split: Int): KdNode? {
if (exset.size == 0) return null
val exset2 = exset.sortedBy { it[split] }
for (i in 0 until exset.size) exset[i] = exset2[i]
var m = exset.size / 2
val d = exset[m]
while (m + 1 < exset.size && exset[m + 1][split] == d[split]) m++
var s2 = split + 1
if (s2 == d.size) s2 = 0
return KdNode(
d,
split,
nk2(exset.subList(0, m), s2),
nk2(exset.subList(m + 1, exset.size), s2)
)
}
this.n = nk2(pts, 0)
this.bounds = bounds
}
fun nearest(p: Point) = nn(n, p, bounds, Double.POSITIVE_INFINITY)
private fun nn(
kd: KdNode?,
target: Point,
hr: HyperRect,
maxDistSqd: Double
): NearestNeighbor {
if (kd == null) return NearestNeighbor(null, Double.POSITIVE_INFINITY, 0)
var nodesVisited = 1
val s = kd.split
val pivot = kd.domElt
val leftHr = hr.copy()
val rightHr = hr.copy()
leftHr.max[s] = pivot[s]
rightHr.min[s] = pivot[s]
val targetInLeft = target[s] <= pivot[s]
val nearerKd = if (targetInLeft) kd.left else kd.right
val nearerHr = if (targetInLeft) leftHr else rightHr
val furtherKd = if (targetInLeft) kd.right else kd.left
val furtherHr = if (targetInLeft) rightHr else leftHr
var (nearest, distSqd, nv) = nn(nearerKd, target, nearerHr, maxDistSqd)
nodesVisited += nv
var maxDistSqd2 = if (distSqd < maxDistSqd) distSqd else maxDistSqd
var d = pivot[s] - target[s]
d *= d
if (d > maxDistSqd2) return NearestNeighbor(nearest, distSqd, nodesVisited)
d = pivot.sqd(target)
if (d < distSqd) {
nearest = pivot
distSqd = d
maxDistSqd2 = distSqd
}
val temp = nn(furtherKd, target, furtherHr, maxDistSqd2)
nodesVisited += temp.nodesVisited
if (temp.distSqd < distSqd) {
nearest = temp.nearest
distSqd = temp.distSqd
}
return NearestNeighbor(nearest, distSqd, nodesVisited)
}
}
val rand = Random()
fun randomPt(dim: Int) = Point(dim) { rand.nextDouble() }
fun randomPts(dim: Int, n: Int) = MutableList<Point>(n) { randomPt(dim) }
fun showNearest(heading: String, kd: KdTree, p: Point) {
println("$heading:")
println("Point : ${p.asList()}")
val (nn, ssq, nv) = kd.nearest(p)
println("Nearest neighbor : ${nn?.asList()}")
println("Distance : ${Math.sqrt(ssq)}")
println("Nodes visited : $nv")
println()
}
fun main(args: Array<String>) {
val points = mutableListOf(
doubleArrayOf(2.0, 3.0),
doubleArrayOf(5.0, 4.0),
doubleArrayOf(9.0, 6.0),
doubleArrayOf(4.0, 7.0),
doubleArrayOf(8.0, 1.0),
doubleArrayOf(7.0, 2.0)
)
var hr = HyperRect(doubleArrayOf(0.0, 0.0), doubleArrayOf(10.0, 10.0))
var kd = KdTree(points, hr)
showNearest("WP example data", kd, doubleArrayOf(9.0, 2.0))
hr = HyperRect(doubleArrayOf(0.0, 0.0, 0.0), doubleArrayOf(1.0, 1.0, 1.0))
kd = KdTree(randomPts(3, 1000), hr)
showNearest("1000 random 3D points", kd, randomPt(3))
hr = hr.copy()
kd = KdTree(randomPts(3, 400_000), hr)
showNearest("400,000 random 3D points", kd, randomPt(3))
}
Sample output:
WP example data: Point : [9.0, 2.0] Nearest neighbor : [8.0, 1.0] Distance : 1.4142135623730951 Nodes visited : 3 1000 random 3D points: Point : [0.5682408811963585, 0.6635465099329054, 0.9023807963146424] Nearest neighbor : [0.5552585116046708, 0.6604379445683164, 0.8609079250093159] Distance : 0.04356838478930882 Nodes visited : 17 400,000 random 3D points: Point : [0.72025478016928, 0.4029017493035246, 0.9535502838653142] Nearest neighbor : [0.7200735279619622, 0.3974244838953903, 0.9528380574924709] Distance : 0.005526350976908346 Nodes visited : 43
Nim
This Nim version is an adaptation of the D fast version which is a translation of the C version. And on some points our version is closer to the C++ version, also an adaptation of the C version.
We have tried to be as generic as possible, accepting nodes of any numerical type (and tested for integers and floats).
type
Point[Dim: static Natural; T: SomeNumber] = array[Dim, T]
KdNode[Dim: static Natural; T: SomeNumber] = ref object
x: Point[Dim, T]
left, right: KdNode[Dim, T]
func toKdNodes[N, Dim: static Natural; T](a: array[N, array[Dim, T]]): array[N, KdNode[Dim, T]] =
## Create an array of KdNodes from an array of list of numerical values.
for i in 0..<N:
result[i] = KdNode[Dim, T](x: a[i])
func dist(a, b: Point): Point.T =
## Return the squared distance between two points.
for i in 0..<Point.Dim:
let t = a[i] - b[i]
result += t * t
func findMedian(nodes: openArray[KdNode]; slice: Slice; idx: Natural): int =
## Return the index of the median node in a list of nodes.
## The list is defined by the full list of nodes and an slice.
var first = slice.a
var last = slice.b
if last < first: return -1
if last == first: return first
let md = first + (last - first + 1) div 2
while true:
let pivot = nodes[md].x[idx]
swap nodes[md].x, nodes[last].x
var store = first
for i in first..last:
if nodes[i].x[idx] < pivot:
if i != store:
swap nodes[i].x, nodes[store].x
inc store
swap nodes[store].x, nodes[last].x
if nodes[store].x[idx] == nodes[md].x[idx]:
return md
if store > md:
last = store
else:
first = store
func makeTree(nodes: openArray[KdNode]; slice: Slice; idx: Natural = 0): KdNode =
## Build a tree from a list of nodes. Return the root of the tree.
if slice.b < slice.a: return nil
let n = nodes.findMedian(slice, idx)
if n < 0: return nil
let idx = (idx + 1) mod result.Dim
nodes[n].left = nodes.makeTree(slice.a..<n, idx)
nodes[n].right = nodes.makeTree((n + 1)..slice.b, idx)
result = nodes[n]
func nearest(root,: KdNode; point: Point; idx: Natural;
best: var KdNode; bestDist: var root.T; nVisited: var int) =
## Return the node of a tree which is the nearest to a given point.
if root.isNil: return
let d = dist(root.x, point)
let dx = root.x[idx] - point[idx]
inc nVisited
if best.isNil or d < bestDist:
bestDist = d
best = root
if bestDist == 0: return
let idx = (idx + 1) mod root.Dim
nearest(if dx > 0: root.left else: root.right, point, idx, best, bestDist, nVisited)
if dx * dx >= bestDist: return
nearest(if dx > 0: root.right else: root.left, point, idx, best, bestDist, nVisited)
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
import math, random, strformat
proc displayResult(title: string; thisPt: Point;
found: KdNode; bestDist: thisPt.T; nVisited: int) =
echo title & ':'
echo &" Searching for {thisPt}"
echo &" Found {found.x}, dist = {sqrt(float(bestDist)):.5f}"
echo &" Seen {nVisited} nodes."
echo ""
proc initRandom(point: var Point) =
for item in point.mitems:
item = rand(1.0)
proc runSmallTest() =
let
wp = [[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]].toKdNodes()
#thisPt = newPoint([9, 2])
thisPt = Point([9, 2])
root = wp.makeTree(0..wp.high)
var
found: KdNode[root.Dim, root.T]
bestDist = root.T(0)
nVisited = 0
root.nearest(thisPt, 0, found, bestDist, nVisited)
displayResult("WP tree", thisPt, found, bestDist, nVisited)
proc runBigTest() =
const N = 1_000_000
const TestRuns = 100_000
randomize()
var bigTree: array[N, KdNode[3, float]]
for node in bigTree.mitems:
new(node)
node.x.initRandom()
let root = bigTree.makeTree(0..bigTree.high)
var thisPt: Point[3, float]
thisPt.initRandom()
var
found: KdNode[3, float]
bestDist = 0.0
nVisited = 0
root.nearest(thisPt, 0, found, bestDist, nVisited)
displayResult("Big Tree", thisPt, found, bestDist, nVisited)
var sum = 0
for _ in 0..<TestRuns:
found = nil
nVisited = 0
thisPt.initRandom()
root.nearest(thisPt, 0, found, bestDist, nVisited)
sum += nVisited
echo "Big tree:"
echo &" Visited {sum} nodes for {TestRuns} random searches ({sum / TestRuns:.2f} per lookup)."
runSmallTest()
runBigTest()
- Output:
WP tree: Searching for [9, 2] Found [8, 1], dist = 1.41421 Seen 3 nodes. Big Tree: Searching for [0.4851429304889259, 0.3418258981276956, 0.8257271960979238] Found [0.4805897150826164, 0.3388812898131908, 0.8295219237899825], dist = 0.00662 Seen 47 nodes. Big tree: Visited 4271480 nodes for 100000 random searches (42.71 per lookup).
Phix
Added the 3D cube of 1000 points (0,0,0 to 9,9,9), which was not present in the C code, then inspired by the single "naughty global" of C (visited) I decided to outdo that as well, and have four of them ;-). [With enough motivation, many functions would probably accept more parameters and return multiple values instead.]
-- demo\rosetta\kd_tree.exw with javascript_semantics enum X,LEFT,RIGHT -- (keeping the IDX on each node too would not be a bad idea..) -- (the code below deduces it from the (unbalanced) tree depth) sequence kd_nodes -- nb probably not best coding style function sqdist(sequence p,q) return sum(sq_power(sq_sub(p,q),2)) end function procedure swap(integer x,y) {kd_nodes[x],kd_nodes[y]} = {kd_nodes[y],kd_nodes[x]} end procedure function find_median(integer first, last, idx) if last<=first then return NULL end if if last==first+1 then return first end if integer p, stor, md = first+floor((last-first)/2) while true do atom pivot = kd_nodes[md][X][idx] swap(md, last-1) stor = first for p=first to last-1 do if kd_nodes[p][X][idx]<pivot then if p!=stor then swap(p, stor) end if stor += 1 end if end for swap(stor, last-1) /* median has duplicate values */ atom sx = kd_nodes[stor][X][idx], mx = kd_nodes[md][X][idx] if sx == mx then exit end if if stor>md then last = stor else first = stor end if end while return md end function function make_tree(object t, len, i, dim) if sequence(t) then kd_nodes = deep_copy(t) t = 1 end if integer n = iff(len=0?0:find_median(t, t+len, i)) if n then i = mod(i,dim)+1 if length(kd_nodes[n])!=1 then ?9/0 end if -- (once) kd_nodes[n] = {kd_nodes[n][X],0,0} -- (add l/r slots) kd_nodes[n][LEFT] = make_tree(t, n-t, i, dim) kd_nodes[n][RIGHT] = make_tree(n+1, t+len-(n+1), i, dim) end if return n end function integer visited, best atom best_dist procedure nearest(integer root, sequence nd, integer i, dim) if root=0 then return end if atom d = sqdist(kd_nodes[root][X],nd[X]), dx = kd_nodes[root][X][i] - nd[X][i], dx2 = dx * dx; visited += 1 if best=0 or d<best_dist then best_dist = d; best = root; end if /* if chance of exact match is high */ if best_dist=0 then return end if i = mod(i,dim)+1 integer {l,r} = kd_nodes[root][LEFT..RIGHT] if dx<=0 then {l,r} = {r,l} end if nearest(l, nd, i, dim) if (dx2 >= best_dist) then return end if nearest(r, nd, i, dim) end procedure function rand_pt() return sq_rnd({{0,0,0}}) end function function make_3D() sequence s = {} for i=0 to 9 do for j=0 to 9 do for k=0 to 9 do s = append(s,{{i,j,k}}) end for end for end for return s end function // aside: shave a little off under pwa/p2js, but 1e6 build still takes ~20s constant test_runs = iff(platform()=JS?10_000:100_000) procedure test(string id, sequence nodes, test_node, bool show_average_behaviour=false) integer dim = length(nodes[1][X]), root = make_tree(nodes, length(nodes), 1, dim) best = 0 visited = 0 nearest(root, test_node, 1, dim) printf(1,"\n>> %s tree\nsearching for %v\nfound %v dist %g\nseen %d nodes\n", {id, test_node[X], kd_nodes[best][X], sqrt(best_dist), visited}) if show_average_behaviour then -- -- search many random points to see average behavior. -- -- tree size vs avg nodes visited: -- 10 ~ 7 -- 100 ~ 16.5 -- 1000 ~ 25.5 -- 10000 ~ 32.8 -- 100000 ~ 38.3 -- 1000000 ~ 42.6 -- 10000000 ~ 46.7 -- integer total = 0 for i=1 to test_runs do best = 0 visited = 0 test_node = rand_pt() nearest(root, test_node, 1, 3) total += visited end for printf(1,"average behaviour: "& "visited %d nodes for %,d random findings (%f per lookup)\n", {total, test_runs, total/test_runs}); end if end procedure atom t0 = time() sequence wp = {{{2, 3}}, {{5, 4}}, {{9, 6}}, {{4, 7}}, {{8, 1}}, {{7, 2}}}, test_node = {{9, 2}} test("WP",wp,test_node) sequence cube = make_3D() test_node = sq_mul(rand_pt(),10) test("3D",cube,test_node,true) constant N = 1_000_000 sequence million = repeat(0,N) for i=1 to N do million[i] = rand_pt() end for test_node = rand_pt() test("Million",million,test_node,true) ?elapsed(time()-t0) if platform()!=JS then ?"done" {} = wait_key() end if
- Output:
>> WP tree
searching for {9,2}
found {8,1} dist 1.41421
seen 3 nodes
>> 3D tree
searching for {8.641182192,5.703920147,9.893391353}
found {9,6,9} dist 1.00725
seen 27 nodes
average behaviour: visited 2171052 nodes for 100,000 random findings (21.710520 per lookup)
>> Million tree
searching for {0.4157659043,0.6029827275,0.8408004809}
found {0.4204315508,0.6014092206,0.8489383544} dist 0.00951153
seen 43 nodes
average behaviour: visited 4267089 nodes for 100,000 random findings (42.670890 per lookup)
"35.8s"
(Most of the time is taken up building the million tree.)
Python
from random import seed, random
from time import time
from operator import itemgetter
from collections import namedtuple
from math import sqrt
from copy import deepcopy
def sqd(p1, p2):
return sum((c1 - c2) ** 2 for c1, c2 in zip(p1, p2))
class KdNode(object):
__slots__ = ("dom_elt", "split", "left", "right")
def __init__(self, dom_elt, split, left, right):
self.dom_elt = dom_elt
self.split = split
self.left = left
self.right = right
class Orthotope(object):
__slots__ = ("min", "max")
def __init__(self, mi, ma):
self.min, self.max = mi, ma
class KdTree(object):
__slots__ = ("n", "bounds")
def __init__(self, pts, bounds):
def nk2(split, exset):
if not exset:
return None
exset.sort(key=itemgetter(split))
m = len(exset) // 2
d = exset[m]
while m + 1 < len(exset) and exset[m + 1][split] == d[split]:
m += 1
d = exset[m]
s2 = (split + 1) % len(d) # cycle coordinates
return KdNode(d, split, nk2(s2, exset[:m]),
nk2(s2, exset[m + 1:]))
self.n = nk2(0, pts)
self.bounds = bounds
T3 = namedtuple("T3", "nearest dist_sqd nodes_visited")
def find_nearest(k, t, p):
def nn(kd, target, hr, max_dist_sqd):
if kd is None:
return T3([0.0] * k, float("inf"), 0)
nodes_visited = 1
s = kd.split
pivot = kd.dom_elt
left_hr = deepcopy(hr)
right_hr = deepcopy(hr)
left_hr.max[s] = pivot[s]
right_hr.min[s] = pivot[s]
if target[s] <= pivot[s]:
nearer_kd, nearer_hr = kd.left, left_hr
further_kd, further_hr = kd.right, right_hr
else:
nearer_kd, nearer_hr = kd.right, right_hr
further_kd, further_hr = kd.left, left_hr
n1 = nn(nearer_kd, target, nearer_hr, max_dist_sqd)
nearest = n1.nearest
dist_sqd = n1.dist_sqd
nodes_visited += n1.nodes_visited
if dist_sqd < max_dist_sqd:
max_dist_sqd = dist_sqd
d = (pivot[s] - target[s]) ** 2
if d > max_dist_sqd:
return T3(nearest, dist_sqd, nodes_visited)
d = sqd(pivot, target)
if d < dist_sqd:
nearest = pivot
dist_sqd = d
max_dist_sqd = dist_sqd
n2 = nn(further_kd, target, further_hr, max_dist_sqd)
nodes_visited += n2.nodes_visited
if n2.dist_sqd < dist_sqd:
nearest = n2.nearest
dist_sqd = n2.dist_sqd
return T3(nearest, dist_sqd, nodes_visited)
return nn(t.n, p, t.bounds, float("inf"))
def show_nearest(k, heading, kd, p):
print(heading + ":")
print("Point: ", p)
n = find_nearest(k, kd, p)
print("Nearest neighbor:", n.nearest)
print("Distance: ", sqrt(n.dist_sqd))
print("Nodes visited: ", n.nodes_visited, "\n")
def random_point(k):
return [random() for _ in range(k)]
def random_points(k, n):
return [random_point(k) for _ in range(n)]
if __name__ == "__main__":
seed(1)
P = lambda *coords: list(coords)
kd1 = KdTree([P(2, 3), P(5, 4), P(9, 6), P(4, 7), P(8, 1), P(7, 2)],
Orthotope(P(0, 0), P(10, 10)))
show_nearest(2, "Wikipedia example data", kd1, P(9, 2))
N = 400000
t0 = time()
kd2 = KdTree(random_points(3, N), Orthotope(P(0, 0, 0), P(1, 1, 1)))
t1 = time()
text = lambda *parts: "".join(map(str, parts))
show_nearest(2, text("k-d tree with ", N,
" random 3D points (generation time: ",
t1-t0, "s)"),
kd2, random_point(3))
- Output:
Wikipedia example data: Point: [9, 2] Nearest neighbor: [8, 1] Distance: 1.41421356237 Nodes visited: 3 k-d tree with 400000 random 3D points (generation time: 14.8755565302s): Point: [0.066694022911324868, 0.13692213852082813, 0.94939167224227283] Nearest neighbor: [0.067027753280507252, 0.14407354836507069, 0.94543775920177597] Distance: 0.00817847583914 Nodes visited: 33
Racket
The following code is optimized for readability.
#lang racket
; A tree consists of a point, a left and a right subtree.
(struct tree (p l r) #:transparent)
; If the node is in depth d, then the points in l has
; the (d mod k)'th coordinate less than the same coordinate in p.
(define (kdtree d k ps)
(cond [(empty? ps) #f] ; #f represents an empty subtree
[else (define-values (p l r) (split-points ps (modulo d k)))
(tree p (kdtree (+ d 1) k l) (kdtree (+ d 1) k r))]))
(define (split-points ps d)
(define (ref p) (vector-ref p d))
(define sorted-ps (sort ps < #:key ref))
(define mid (quotient (+ (length ps)) 2))
(define median (ref (list-ref sorted-ps mid)))
(define-values (l r) (partition(λ(x)(< (ref x) median))sorted-ps))
(values (first r) l (rest r)))
; The bounding box of a subtree:
(struct bb (mins maxs) #:transparent)
(define (infinite-bb k)
(bb (make-vector k -inf.0) (make-vector k +inf.0)))
(define/match (copy-bb h)
[((bb mins maxs))
(bb (vector-copy mins) (vector-copy maxs))])
(define (dist v w) (for/sum ([x v] [y w]) (sqr (- x y))))
(define (intersects? g r hr) (<= (dist (closest-in-hr g hr) g) r))
(define (closest-in-hr g hr)
(for/vector ([gi g] [mini (bb-mins hr)] [maxi (bb-maxs hr)])
(cond [(<= gi mini) mini]
[(< mini gi maxi) gi]
[else maxi])))
(define (split-bb hr d x)
(define left (copy-bb hr))
(define right (copy-bb hr))
(vector-set! (bb-maxs left) d x)
(vector-set! (bb-mins right) d x)
(values left right))
(define visits 0) ; for statistics only
(define (visit) (set! visits (+ visits 1)))
(define (reset-visits) (set! visits 0))
(define (regret-visit) (set! visits (- visits 1)))
(define (nearest-neighbor g t k)
(define (nearer? p q) (< (dist p g) (dist q g)))
(define (nearest p q) (if (nearer? p q) p q))
(define (nn d t bb) (visit)
(define (ref p) (vector-ref p (modulo d k)))
(match t
[#f (regret-visit) #(+inf.0 +inf.0 +inf.0)]
[(tree p l r)
(define-values (lbb rbb) (split-bb bb (modulo d k) (ref p)))
(define-values (near near-bb far far-bb)
(if (< (ref g) (ref p))
(values l lbb r rbb)
(values r rbb l lbb)))
(define n (nearest p (nn (+ d 1) near near-bb)))
(if (intersects? g (dist n g) far-bb)
(nearest n (nn (+ d 1) far far-bb))
n)]))
(nn 0 t (infinite-bb k)))
Tests:
(define (wikipedia-test)
(define t (kdtree 0 2 '(#(2 3) #(5 4) #(9 6) #(4 7) #(8 1) #(7 2))))
(reset-visits)
(define n (nearest-neighbor #(9 2) t 2))
(displayln "Wikipedia Test")
(displayln (~a "Nearest neighbour to (9,2) is: " n))
(displayln (~a "Distance: " (dist n #(9 2))))
(displayln (~a "Visits: " visits "\n")))
(define (test k n)
(define (random!) (for/vector ([_ k]) (random)))
(define points (for/list ([_ n]) (random!)))
(define t (kdtree 0 k points))
(reset-visits)
(define target (for/vector ([_ k]) 0.75))
(define nb (nearest-neighbor target t k))
(define nb-control (argmin (λ (p) (dist p target)) points))
(displayln (~a n " points in R^3 test"))
(displayln (~a "Nearest neighbour to " target " is: \n\t\t" nb))
(displayln (~a "Control: \t" nb-control))
(displayln (~a "Distance: \t" (dist nb target)))
(displayln (~a "Control: \t" (dist nb-control target)))
(displayln (~a "Visits: \t" visits)))
(wikipedia-test)
(test 3 1000)
(test 3 1000)
- Output:
Wikipedia Test
Nearest neighbour to (9,2) is: #(8 1)
Distance: 2
Visits: 3
1000 points in R^3 test
Nearest neighbour to #(0.75 0.75 0.75) is:
#(0.8092534479975508 0.7507095851813429 0.7706494651024903)
Control: #(0.8092534479975508 0.7507095851813429 0.7706494651024903)
Distance: 0.003937875019747008
Control: 0.003937875019747008
Visits: 83
1000 points in R^3 test
Nearest neighbour to #(0.75 0.75 0.75) is:
#(0.7775581478448806 0.7806612633582072 0.7396664367640902)
Control: #(0.7775581478448806 0.7806612633582072 0.7396664367640902)
Distance: 0.0018063471125121851
Control: 0.0018063471125121851
Visits: 39
Raku
(formerly Perl 6)
class Kd_node {
has $.d;
has $.split;
has $.left;
has $.right;
}
class Orthotope {
has $.min;
has $.max;
}
class Kd_tree {
has $.n;
has $.bounds;
method new($pts, $bounds) { self.bless(n => nk2(0,$pts), bounds => $bounds) }
sub nk2($split, @e) {
return () unless @e;
my @exset = @e.sort(*.[$split]);
my $m = +@exset div 2;
my @d = @exset[$m][];
while $m+1 < @exset and @exset[$m+1][$split] eqv @d[$split] {
++$m;
}
my $s2 = ($split + 1) % @d; # cycle coordinates
Kd_node.new: :@d, :$split,
left => nk2($s2, @exset[0 ..^ $m]),
right => nk2($s2, @exset[$m ^.. *]);
}
}
class T3 {
has $.nearest;
has $.dist_sqd = Inf;
has $.nodes_visited = 0;
}
sub find_nearest($k, $t, @p) {
return nn($t.n, @p, $t.bounds, Inf);
sub nn($kd, @target, $hr, $max_dist_sqd is copy) {
return T3.new(:nearest([0.0 xx $k])) unless $kd;
my $nodes_visited = 1;
my $s = $kd.split;
my $pivot = $kd.d;
my $left_hr = $hr.clone;
my $right_hr = $hr.clone;
$left_hr.max[$s] = $pivot[$s];
$right_hr.min[$s] = $pivot[$s];
my $nearer_kd;
my $further_kd;
my $nearer_hr;
my $further_hr;
if @target[$s] <= $pivot[$s] {
($nearer_kd, $nearer_hr) = $kd.left, $left_hr;
($further_kd, $further_hr) = $kd.right, $right_hr;
}
else {
($nearer_kd, $nearer_hr) = $kd.right, $right_hr;
($further_kd, $further_hr) = $kd.left, $left_hr;
}
my $n1 = nn($nearer_kd, @target, $nearer_hr, $max_dist_sqd);
my $nearest = $n1.nearest;
my $dist_sqd = $n1.dist_sqd;
$nodes_visited += $n1.nodes_visited;
if $dist_sqd < $max_dist_sqd {
$max_dist_sqd = $dist_sqd;
}
my $d = ($pivot[$s] - @target[$s]) ** 2;
if $d > $max_dist_sqd {
return T3.new(:$nearest, :$dist_sqd, :$nodes_visited);
}
$d = [+] (@$pivot Z- @target) X** 2;
if $d < $dist_sqd {
$nearest = $pivot;
$dist_sqd = $d;
$max_dist_sqd = $dist_sqd;
}
my $n2 = nn($further_kd, @target, $further_hr, $max_dist_sqd);
$nodes_visited += $n2.nodes_visited;
if $n2.dist_sqd < $dist_sqd {
$nearest = $n2.nearest;
$dist_sqd = $n2.dist_sqd;
}
T3.new(:$nearest, :$dist_sqd, :$nodes_visited);
}
}
sub show_nearest($k, $heading, $kd, @p) {
print qq:to/END/;
$heading:
Point: [@p.join(',')]
END
my $n = find_nearest($k, $kd, @p);
print qq:to/END/;
Nearest neighbor: [$n.nearest.join(',')]
Distance: &sqrt($n.dist_sqd)
Nodes visited: $n.nodes_visited()
END
}
sub random_point($k) { [rand xx $k] }
sub random_points($k, $n) { [random_point($k) xx $n] }
sub MAIN {
my $kd1 = Kd_tree.new([[2, 3],[5, 4],[9, 6],[4, 7],[8, 1],[7, 2]],
Orthotope.new(:min([0, 0]), :max([10, 10])));
show_nearest(2, "Wikipedia example data", $kd1, [9, 2]);
my $N = 1000;
my $t0 = now;
my $kd2 = Kd_tree.new(random_points(3, $N), Orthotope.new(:min([0,0,0]), :max([1,1,1])));
my $t1 = now;
show_nearest(2,
"k-d tree with $N random 3D points (generation time: {$t1 - $t0}s)",
$kd2, random_point(3));
}
- Output:
Wikipedia example data: Point: [9,2] Nearest neighbor: [8,1] Distance: 1.4142135623731 Nodes visited: 3 k-d tree with 1000 random 3D points (generation time: 67.0934954s): Point: [0.765565651400664,0.223251226280109,0.00536717765240979] Nearest neighbor: [0.758919336088656,0.228895111242011,0.0383284709862686] Distance: 0.0340950700678338 Nodes visited: 23
Rust
use std::cmp::Ordering;
use std::cmp::Ordering::Less;
use std::ops::Sub;
use std::time::Instant;
use rand::prelude::*;
#[derive(Clone, PartialEq, Debug)]
struct Point {
pub coords: Vec<f32>,
}
impl<'a, 'b> Sub<&'b Point> for &'a Point {
type Output = Point;
fn sub(self, rhs: &Point) -> Point {
assert_eq!(self.coords.len(), rhs.coords.len());
Point {
coords: self
.coords
.iter()
.zip(rhs.coords.iter())
.map(|(&x, &y)| x - y)
.collect(),
}
}
}
impl Point {
fn norm_sq(&self) -> f32 {
self.coords.iter().map(|n| n * n).sum()
}
}
struct KDTreeNode {
point: Point,
dim: usize,
// Construction could become faster if we use an arena allocator,
// but this is easier to use.
left: Option<Box<KDTreeNode>>,
right: Option<Box<KDTreeNode>>,
}
impl KDTreeNode {
/// Create a new KDTreeNode around the `dim`th dimension.
/// Alternatively, we could dynamically determine the dimension to
/// split on by using the longest dimension.
pub fn new(points: &mut [Point], dim: usize) -> KDTreeNode {
let points_len = points.len();
if points_len == 1 {
return KDTreeNode {
point: points[0].clone(),
dim,
left: None,
right: None,
};
}
// Split around the median
let pivot = quickselect_by(points, points_len / 2, &|a, b| {
a.coords[dim].partial_cmp(&b.coords[dim]).unwrap()
});
let left = Some(Box::new(KDTreeNode::new(
&mut points[0..points_len / 2],
(dim + 1) % pivot.coords.len(),
)));
let right = if points.len() >= 3 {
Some(Box::new(KDTreeNode::new(
&mut points[points_len / 2 + 1..points_len],
(dim + 1) % pivot.coords.len(),
)))
} else {
None
};
KDTreeNode {
point: pivot,
dim,
left,
right,
}
}
pub fn find_nearest_neighbor<'a>(&'a self, point: &Point) -> (&'a Point, usize) {
self.find_nearest_neighbor_helper(point, &self.point, (point - &self.point).norm_sq(), 1)
}
fn find_nearest_neighbor_helper<'a>(
&'a self,
point: &Point,
best: &'a Point,
best_dist_sq: f32,
n_visited: usize,
) -> (&'a Point, usize) {
let mut my_best = best;
let mut my_best_dist_sq = best_dist_sq;
let mut my_n_visited = n_visited;
// We should always examine the near side
if self.point.coords[self.dim] < point.coords[self.dim] && self.right.is_some() {
let (a, b) = self.right.as_ref().unwrap().find_nearest_neighbor_helper(
point,
my_best,
my_best_dist_sq,
my_n_visited,
);
my_best = a;
my_n_visited = b;
} else if self.left.is_some() {
let (a, b) = self.left.as_ref().unwrap().find_nearest_neighbor_helper(
point,
my_best,
my_best_dist_sq,
my_n_visited,
);
my_best = a;
my_n_visited = b;
}
// distance along this node's axis
let axis_dist_sq = (self.point.coords[self.dim] - point.coords[self.dim]).powi(2);
if axis_dist_sq <= my_best_dist_sq {
// self can only be nearer than best if axis_dist_sq is less than
// best_dist_sq because axis_dist_sq is a lower bound for
// self_dist_sq
let self_dist_sq = (point - &self.point).norm_sq();
if self_dist_sq < my_best_dist_sq {
my_best = &self.point;
my_best_dist_sq = self_dist_sq;
}
// bookkeeping
my_n_visited += 1;
// same reasoning applies for the far side of the split
if self.point.coords[self.dim] < point.coords[self.dim] && self.left.is_some() {
let (a, b) = self.left.as_ref().unwrap().find_nearest_neighbor_helper(
point,
my_best,
my_best_dist_sq,
my_n_visited,
);
my_best = a;
my_n_visited = b;
} else if self.right.is_some() {
let (a, b) = self.right.as_ref().unwrap().find_nearest_neighbor_helper(
point,
my_best,
my_best_dist_sq,
my_n_visited,
);
my_best = a;
my_n_visited = b;
}
}
(my_best, my_n_visited)
}
}
pub fn main() {
let mut rng = thread_rng();
// wordpress
let mut wp_points: Vec<Point> = [
[2.0, 3.0],
[5.0, 4.0],
[9.0, 6.0],
[4.0, 7.0],
[8.0, 1.0],
[7.0, 2.0],
]
.iter()
.map(|x| Point { coords: x.to_vec() })
.collect();
let wp_tree = KDTreeNode::new(&mut wp_points, 0);
let wp_target = Point {
coords: vec![9.0, 2.0],
};
let (point, n_visited) = wp_tree.find_nearest_neighbor(&wp_target);
println!("Wikipedia example data:");
println!("Point: [9, 2]");
println!("Nearest neighbor: {:?}", point);
println!("Distance: {}", (point - &wp_target).norm_sq().sqrt());
println!("Nodes visited: {}", n_visited);
// randomly generated 3D
let n_random = 1000;
let mut make_random_point = || Point {
coords: (0..3).map(|_| (rng.gen::<f32>() - 0.5) * 1000.0).collect(),
};
let mut random_points: Vec<Point> = (0..n_random).map(|_| make_random_point()).collect();
let start_cons_time = Instant::now();
let random_tree = KDTreeNode::new(&mut random_points, 0);
let cons_time = start_cons_time.elapsed();
println!(
"1,000 3d points (Construction time: {}ms)",
cons_time.as_millis(),
);
let random_target = make_random_point();
let (point, n_visited) = random_tree.find_nearest_neighbor(&random_target);
println!("Point: {:?}", random_target);
println!("Nearest neighbor: {:?}", point);
println!("Distance: {}", (point - &random_target).norm_sq().sqrt());
println!("Nodes visited: {}", n_visited);
// benchmark search time
let n_searches = 1000;
let random_targets: Vec<Point> = (0..n_searches).map(|_| make_random_point()).collect();
let start_search_time = Instant::now();
let mut total_n_visited = 0;
for target in &random_targets {
let (_, n_visited) = random_tree.find_nearest_neighbor(target);
total_n_visited += n_visited;
}
let search_time = start_search_time.elapsed();
println!(
"Visited an average of {} nodes on {} searches in {} ms",
total_n_visited as f32 / n_searches as f32,
n_searches,
search_time.as_millis(),
);
}
fn quickselect_by<T>(arr: &mut [T], position: usize, cmp: &dyn Fn(&T, &T) -> Ordering) -> T
where
T: Clone,
{
// We use `thread_rng` here because it was already initialized in `main`.
let mut pivot_index = thread_rng().gen_range(0, arr.len());
// Need to wrap in another closure or we get ownership complaints.
// Tried using an unboxed closure to get around this but couldn't get it to work.
pivot_index = partition_by(arr, pivot_index, &|a: &T, b: &T| cmp(a, b));
let array_len = arr.len();
match position.cmp(&pivot_index) {
Ordering::Equal => arr[position].clone(),
Ordering::Less => quickselect_by(&mut arr[0..pivot_index], position, cmp),
Ordering::Greater => quickselect_by(
&mut arr[pivot_index + 1..array_len],
position - pivot_index - 1,
cmp,
),
}
}
fn partition_by<T>(arr: &mut [T], pivot_index: usize, cmp: &dyn Fn(&T, &T) -> Ordering) -> usize {
let array_len = arr.len();
arr.swap(pivot_index, array_len - 1);
let mut store_index = 0;
for i in 0..array_len - 1 {
if cmp(&arr[i], &arr[array_len - 1]) == Less {
arr.swap(i, store_index);
store_index += 1;
}
}
arr.swap(array_len - 1, store_index);
store_index
}
- Output:
Wikipedia example data:
Point: [9, 2]
Nearest neighbor: Point { coords: [8.0, 1.0] }
Distance: 1.4142135
Nodes visited: 4
1,000 3d points (Construction time: 6ms)
Point: Point { coords: [-353.30945, 277.02594, -260.73093] }
Nearest neighbor: Point { coords: [-345.98798, -24.195671, -350.5432] }
Distance: 314.41107
Nodes visited: 183
Visited an average of 351.573 nodes on 1000 searches in 415 ms
Scala
This example works for sequences of Int, Double, etc, so it is non-minimal due to its type-safe Numeric parameterisation.
object KDTree {
import Numeric._
// Task 1A. Build tree of KDNodes. Translated from Wikipedia.
def apply[T](points: Seq[Seq[T]], depth: Int = 0)(implicit num: Numeric[T]): Option[KDNode[T]] = {
val dim = points.headOption.map(_.size) getOrElse 0
if (points.isEmpty || dim < 1) None
else {
val axis = depth % dim
val sorted = points.sortBy(_(axis))
val median = sorted(sorted.size / 2)(axis)
val (left, right) = sorted.partition(v => num.lt(v(axis), median))
Some(KDNode(right.head, apply(left, depth + 1), apply(right.tail, depth + 1), axis))
}
}
// Task 1B. Find the nearest node in this subtree. Translated from Wikipedia.
case class KDNode[T](value: Seq[T], left: Option[KDNode[T]], right: Option[KDNode[T]], axis: Int)(implicit num: Numeric[T]) {
def nearest(to: Seq[T]): Nearest[T] = {
val default = Nearest(value, to, Set(this))
compare(to, value) match {
case 0 => default // exact match
case t =>
lazy val bestL = left.map(_ nearest to).getOrElse(default)
lazy val bestR = right.map(_ nearest to).getOrElse(default)
val branch1 = if (t < 0) bestL else bestR
val best = if (num.lt(branch1.distsq, default.distsq)) branch1 else default
val splitDist = num.minus(to(axis), value(axis))
if (num.lt(num.times(splitDist, splitDist), best.distsq)) {
val branch2 = if (t < 0) bestR else bestL
val visited = branch2.visited ++ best.visited + this
if (num.lt(branch2.distsq, best.distsq))
branch2.copy(visited = visited)
else best.copy(visited = visited)
} else best.copy(visited = best.visited + this)
}
}
}
// Keep track of nodes visited, as per task. Pretty-printable.
case class Nearest[T](value: Seq[T], to: Seq[T], visited: Set[KDNode[T]] = Set[KDNode[T]]())(implicit num: Numeric[T]) {
lazy val distsq = KDTree.distsq(value, to)
override def toString = f"Searched for=${to} found=${value} distance=${math.sqrt(num.toDouble(distsq))}%.4f visited=${visited.size}"
}
// Numeric utilities
def distsq[T](a: Seq[T], b: Seq[T])(implicit num: Numeric[T]) =
a.zip(b).map(c => num.times(num.minus(c._1, c._2), num.minus(c._1, c._2))).sum
def compare[T](a: Seq[T], b: Seq[T])(implicit num: Numeric[T]): Int =
a.zip(b).find(c => num.compare(c._1, c._2) != 0).map(c => num.compare(c._1, c._2)).getOrElse(0)
}
Task test:
object KDTreeTest extends App {
def test[T](haystack: Seq[Seq[T]], needles: Seq[T]*)(implicit num: Numeric[T]) = {
println
val tree = KDTree(haystack)
if (haystack.size < 20) tree.foreach(println)
for (kd <- tree; needle <- needles; nearest = kd nearest needle) {
println(nearest)
// Brute force proof
val better = haystack
.map(KDTree.Nearest(_, needle))
.filter(n => num.lt(n.distsq, nearest.distsq))
.sortBy(_.distsq)
assert(better.isEmpty, s"Found ${better.size} closer than ${nearest.value} e.g. ${better.head}")
}
}
// Results 1
val wikitest = List(List(2,3), List(5,4), List(9,6), List(4,7), List(8,1), List(7,2))
test(wikitest, List(9,2))
// Results 2 (1000 points uniformly distributed in 3-d cube coordinates, sides 2 to 20)
val uniform = for(x <- 1 to 10; y <- 1 to 10; z <- 1 to 10) yield List(x*2, y*2, z*2)
assume(uniform.size == 1000)
test(uniform, List(0, 0, 0), List(2, 2, 20), List(9, 10, 11))
// Results 3 (1000 points randomly distributed in 3-d cube coordinates, sides -1.0 to 1.0)
scala.util.Random.setSeed(0)
def random(n: Int) = (1 to n).map(_ => (scala.util.Random.nextDouble - 0.5)* 2)
test((1 to 1000).map(_ => random(3)), random(3))
// Results 4 (27 points uniformly distributed in 3-d cube coordinates, sides 3...9)
val small = for(x <- 1 to 3; y <- 1 to 3; z <- 1 to 3) yield List(x*3, y*3, z*3)
assume(small.size == 27)
test(small, List(0, 0, 0), List(4, 5, 6))
}
- Output:
KDNode(List(7, 2),Some(KDNode(List(5, 4),Some(KDNode(List(2, 3),None,None,0)),Some(KDNode(List(4, 7),None,None,0)),1)),Some(KDNode(List(9, 6),Some(KDNode(List(8, 1),None,None,0)),None,1)),0) Searched for=List(9, 2) found=List(8, 1) distance=1.4142 visited=3 Searched for=List(0, 0, 0) found=List(2, 2, 2) distance=3.4641 visited=10 Searched for=List(2, 2, 20) found=List(2, 2, 20) distance=0.0000 visited=9 Searched for=List(9, 10, 11) found=List(8, 10, 12) distance=1.4142 visited=134 Searched for=Vector(0.19269603520919643, -0.25958512078298535, -0.2572864045762784) found=Vector(0.07811099409527977, -0.2477618820196814, -0.20252227622550611) distance=0.1275 visited=25 Searched for=List(0, 0, 0) found=List(3, 3, 3) distance=5.1962 visited=4 Searched for=List(4, 5, 6) found=List(3, 6, 6) distance=1.4142 visited=6
Sidef
struct Kd_node {
d,
split,
left,
right,
}
struct Orthotope {
min,
max,
}
class Kd_tree(n, bounds) {
method init {
n = self.nk2(0, n);
}
method nk2(split, e) {
return(nil) if (e.len <= 0);
var exset = e.sort_by { _[split] }
var m = (exset.len // 2);
var d = exset[m];
while ((m+1 < exset.len) && (exset[m+1][split] == d[split])) {
++m;
}
var s2 = ((split + 1) % d.len); # cycle coordinates
Kd_node(d: d, split: split,
left: self.nk2(s2, exset.first(m)),
right: self.nk2(s2, exset.last(m-1)));
}
}
struct T3 {
nearest,
dist_sqd = Inf,
nodes_visited = 0,
}
func find_nearest(k, t, p) {
func nn(kd, target, hr, max_dist_sqd) {
kd || return T3(nearest: [0]*k);
var nodes_visited = 1;
var s = kd.split;
var pivot = kd.d;
var left_hr = Orthotope(hr.min, hr.max);
var right_hr = Orthotope(hr.min, hr.max);
left_hr.max[s] = pivot[s];
right_hr.min[s] = pivot[s];
var nearer_kd;
var further_kd;
var nearer_hr;
var further_hr;
if (target[s] <= pivot[s]) {
(nearer_kd, nearer_hr) = (kd.left, left_hr);
(further_kd, further_hr) = (kd.right, right_hr);
}
else {
(nearer_kd, nearer_hr) = (kd.right, right_hr);
(further_kd, further_hr) = (kd.left, left_hr);
}
var n1 = nn(nearer_kd, target, nearer_hr, max_dist_sqd);
var nearest = n1.nearest;
var dist_sqd = n1.dist_sqd;
nodes_visited += n1.nodes_visited;
if (dist_sqd < max_dist_sqd) {
max_dist_sqd = dist_sqd;
}
var d = (pivot[s] - target[s] -> sqr);
if (d > max_dist_sqd) {
return T3(nearest: nearest, dist_sqd: dist_sqd, nodes_visited: nodes_visited);
}
d = (pivot ~Z- target »sqr»() «+»);
if (d < dist_sqd) {
nearest = pivot;
dist_sqd = d;
max_dist_sqd = dist_sqd;
}
var n2 = nn(further_kd, target, further_hr, max_dist_sqd);
nodes_visited += n2.nodes_visited;
if (n2.dist_sqd < dist_sqd) {
nearest = n2.nearest;
dist_sqd = n2.dist_sqd;
}
T3(nearest: nearest, dist_sqd: dist_sqd, nodes_visited: nodes_visited);
}
return nn(t.n, p, t.bounds, Inf);
}
func show_nearest(k, heading, kd, p) {
print <<-"END"
#{heading}:
Point: [#{p.join(',')}]
END
var n = find_nearest(k, kd, p);
print <<-"END"
Nearest neighbor: [#{n.nearest.join(',')}]
Distance: #{sqrt(n.dist_sqd)}
Nodes visited: #{n.nodes_visited()}
END
}
func random_point(k) { k.of { 1.rand } }
func random_points(k, n) { n.of { random_point(k) } }
var kd1 = Kd_tree([[2, 3],[5, 4],[9, 6],[4, 7],[8, 1],[7, 2]],
Orthotope(min: [0, 0], max: [10, 10]));
show_nearest(2, "Wikipedia example data", kd1, [9, 2]);
var N = 1000
var t0 = Time.micro
var kd2 = Kd_tree(random_points(3, N), Orthotope(min: [0,0,0], max: [1,1,1]))
var t1 = Time.micro
show_nearest(2,
"k-d tree with #{N} random 3D points (generation time: #{t1 - t0}s)",
kd2, random_point(3))
- Output:
Wikipedia example data: Point: [9,2] Nearest neighbor: [8,1] Distance: 1.41421356237309504880168872420969807856967187537695 Nodes visited: 3 k-d tree with 1000 random 3D points (generation time: 0.25858s): Point: [0.28961099389483532140941908632585463245703951185091,0.53383735570157521548169561846919925542828568155241,0.06543742264584889854604603500785549847809104064543] Nearest neighbor: [0.28344130897803248636407083474733485442276225490765,0.54224255944924304382130637584184680424540291089424,0.12494797195998536910186918458377312162395340543897] Distance: 0.06041703353924870133411659885818573345011076349573 Nodes visited: 30
Tcl
package require TclOO
oo::class create KDTree {
variable t dim
constructor {points} {
set t [my Build 0 $points 0 end]
set dim [llength [lindex $points 0]]
}
method Build {split exset from to} {
set exset [lsort -index $split -real [lrange $exset $from $to]]
if {![llength $exset]} {return 0}
set m [expr {[llength $exset] / 2}]
set d [lindex $exset $m]
while {[set mm $m;incr mm] < [llength $exset] && \
[lindex $exset $mm $split] == [lindex $d $split]} {
set m $mm
}
set s [expr {($split + 1) % [llength $d]}]
list 1 $d $split [my Build $s $exset 0 [expr {$m-1}]] \
[my Build $s $exset [expr {$m+1}] end]
}
method findNearest {p} {
lassign [my FN $t $p inf] p d2 count
return [list $p [expr {sqrt($d2)}] $count]
}
method FN {kd target maxDist2} {
if {[lindex $kd 0] == 0} {
return [list [lrepeat $dim 0.0] inf 0]
}
set nodesVisited 1
lassign $kd -> pivot s
if {[lindex $target $s] <= [lindex $pivot $s]} {
set nearerKD [lindex $kd 3]
set furtherKD [lindex $kd 4]
} else {
set nearerKD [lindex $kd 4]
set furtherKD [lindex $kd 3]
}
lassign [my FN $nearerKD $target $maxDist2] nearest dist2 count
incr nodesVisited $count
if {$dist2 < $maxDist2} {
set maxDist2 $dist2
}
set d2 [expr {([lindex $pivot $s]-[lindex $target $s])**2}]
if {$d2 > $maxDist2} {
return [list $nearest $dist2 $nodesVisited]
}
set d2 0.0
foreach pp $pivot tp $target {set d2 [expr {$d2+($pp-$tp)**2}]}
if {$d2 < $dist2} {
set nearest $pivot
set maxDist2 [set dist2 $d2]
}
lassign [my FN $furtherKD $target $maxDist2] fNearest fDist2 count
incr nodesVisited $count
if {$fDist2 < $dist2} {
set nearest $fNearest
set dist2 $fDist2
}
return [list $nearest $dist2 $nodesVisited]
}
}
Demonstration code:
proc showNearest {heading tree point} {
puts ${heading}:
puts "Point: \[[join $point ,]\]"
lassign [$tree findNearest $point] nearest distance count
puts "Nearest neighbor: \[[join $nearest ,]\]"
puts "Distance: $distance"
puts "Nodes visited: $count"
}
proc randomPoint k {
for {set j 0} {$j < $k} {incr j} {lappend p [::tcl::mathfunc::rand]}
return $p
}
proc randomPoints {k n} {
for {set i 0} {$i < $n} {incr i} {
set p {}
for {set j 0} {$j < $k} {incr j} {
lappend p [::tcl::mathfunc::rand]
}
lappend ps $p
}
return $ps
}
KDTree create kd1 {{2 3} {5 4} {9 6} {4 7} {8 1} {7 2}}
showNearest "Wikipedia example data" kd1 {9 2}
puts ""
set N 1000
set t [time {KDTree create kd2 [randomPoints 3 $N]}]
showNearest "k-d tree with $N random 3D points (generation time: [lindex $t 0] us)" kd2 [randomPoint 3]
kd2 destroy
puts ""
set N 1000000
set t [time {KDTree create kd2 [randomPoints 3 $N]}]
showNearest "k-d tree with $N random 3D points (generation time: [lindex $t 0] us)" kd2 [randomPoint 3]
puts "Search time: [time {kd2 findNearest [randomPoint 3]} 10000]"
- Output:
Wikipedia example data: Point: [9,2] Nearest neighbor: [8,1] Distance: 1.4142135623730951 Nodes visited: 3 k-d tree with 1000 random 3D points (generation time: 11908 us): Point: [0.8480196329057308,0.6659702466176685,0.961934903153188] Nearest neighbor: [0.8774737389187672,0.7011300077201472,0.8920397525150514] Distance: 0.0836007490668497 Nodes visited: 29 k-d tree with 1000000 random 3D points (generation time: 19643366 us): Point: [0.10923849936073576,0.9714587558859301,0.30731017482807405] Nearest neighbor: [0.10596616664247875,0.9733627601402638,0.3079096774141815] Distance: 0.0038331184393709545 Nodes visited: 22 Search time: 289.894755 microseconds per iteration
Wren
import "./dynamic" for Tuple
import "./math" for Nums
import "./seq" for Lst
import "./sort" for Sort
import "random" for Random
// A Point is represented by a 2 element List of Nums
var PtSqd = Fn.new { |p, q| Nums.sum(Lst.zip(p, q).map { |r| (r[0] - r[1]) * (r[0] - r[1]) }) }
var HyperRect = Tuple.create("HyperRect", ["min", "max"])
var HrCopy = Fn.new { |hr| HyperRect.new(hr.min.toList, hr.max.toList) }
var NearestNeighbor = Tuple.create("NearestNeighbor", ["nearest", "distSqd", "nodesVisited"])
var KdNode = Tuple.create("KdNode", ["domElt", "split", "left", "right"])
class KdTree {
construct new(pts, bounds) {
var nk2 // recursive closure
nk2 = Fn.new { |exset, split|
if (exset.count == 0) return null
var cmp = Fn.new { |p, q| (p[split] - q[split]).sign }
Sort.quick(exset, 0, exset.count - 1, cmp)
var m = (exset.count/2).floor
var d = exset[m]
while (m + 1 < exset.count && exset[m + 1][split] == d[split]) m = m + 1
var s2 = split + 1
if (s2 == d.count) s2 = 0
return KdNode.new(
d,
split,
nk2.call(exset[0...m], s2),
nk2.call(exset[m + 1..-1], s2)
)
}
_n = nk2.call(pts, 0)
_bounds = bounds
}
nearest(p) { nn_(_n, p, _bounds, 1/0) }
nn_(kd, target, hr, maxDistSqd) {
if (!kd) return NearestNeighbor.new(null, 1/0, 0)
var nodesVisited = 1
var s = kd.split
var pivot = kd.domElt
var leftHr = HrCopy.call(hr)
var rightHr = HrCopy.call(hr)
leftHr.max[s] = pivot[s]
rightHr.min[s] = pivot[s]
var targetInLeft = target[s] <= pivot[s]
var nearerKd = (targetInLeft) ? kd.left : kd.right
var nearerHr = (targetInLeft) ? leftHr : rightHr
var furtherKd = (targetInLeft) ? kd.right : kd.left
var furtherHr = (targetInLeft) ? rightHr : leftHr
var res = nn_(nearerKd, target, nearerHr, maxDistSqd)
var nearest = res.nearest
var distSqd = res.distSqd
var nv = res.nodesVisited
nodesVisited = nodesVisited + nv
var maxDistSqd2 = (distSqd < maxDistSqd) ? distSqd : maxDistSqd
var d = pivot[s] - target[s]
d = d * d
if (d > maxDistSqd2) return NearestNeighbor.new(nearest, distSqd, nodesVisited)
d = PtSqd.call(pivot, target)
if (d < distSqd) {
nearest = pivot
distSqd = d
maxDistSqd2 = distSqd
}
var temp = nn_(furtherKd, target, furtherHr, maxDistSqd2)
nodesVisited = nodesVisited + temp.nodesVisited
if (temp.distSqd < distSqd) {
nearest = temp.nearest
distSqd = temp.distSqd
}
return NearestNeighbor.new(nearest, distSqd, nodesVisited)
}
}
var rand = Random.new()
var randomPt = Fn.new { |dim|
var pt = List.filled(dim, 0)
for (i in 0...dim) pt[i] = rand.float()
return pt
}
var randomPts = Fn.new { |dim, n|
var pts = List.filled(n, null)
for (i in 0...n) pts[i] = randomPt.call(dim)
return pts
}
var showNearest = Fn.new { |heading, kd, p|
System.print("%(heading):")
System.print("Point : %(p)")
var res = kd.nearest(p)
System.print("Nearest neighbor : %(res.nearest)")
System.print("Distance : %(res.distSqd.sqrt)")
System.print("Nodes visited : %(res.nodesVisited)")
System.print()
}
var points = [[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]]
var hr = HyperRect.new([0, 0], [10, 10])
var kd = KdTree.new(points, hr)
showNearest.call("WP example data", kd, [9, 2])
hr = HyperRect.new([0, 0, 0], [1, 1, 1])
kd = KdTree.new(randomPts.call(3, 1000), hr)
showNearest.call("1,000 random 3D points", kd, randomPt.call(3))
hr = HrCopy.call(hr)
kd = KdTree.new(randomPts.call(3, 400000), hr)
showNearest.call("400,000 random 3D points", kd, randomPt.call(3))
- Output:
Sample output:
WP example data: Point : [9, 2] Nearest neighbor : [8, 1] Distance : 1.4142135623731 Nodes visited : 3 1000 random 3D points: Point : [0.12939208666935, 0.59363898087134, 0.055267216633677] Nearest neighbor : [0.11864823194614, 0.56792856962105, 0.064465652254435] Distance : 0.029343941092531 Nodes visited : 17 400,000 random 3D points: Point : [0.90399249951976, 0.76753089149634, 0.81889978227179] Nearest neighbor : [0.90977751130258, 0.77238453898072, 0.82103471724632] Distance : 0.007847432865301 Nodes visited : 30
zkl
class KdTree{
const NEAREST=0, DIST_SQD=1, NODES_VISITED=2;
class KdNode{
fcn init(_dom_elt,_split,_left,_right){
var dom_elt=_dom_elt.apply("toFloat"),
split=_split, left=_left, right=_right;
}
}
fcn init(pts,_bounds){ // pts is points is ( (x,y,..),..)
var n=fcn(split, exset){
if(not exset) return(Void);
exset=exset.copy() // mutable lists sort much quicker than read only
.sort('wrap(abc,xyz){ abc[split]<xyz[split] });
m,d:=exset.len()/2, exset[m];
while(m+1<exset.len() and exset[m + 1][split]==d[split]){ m+=1 }
s2:=(split+1)%d.len(); # cycle coordinates
KdNode(d,split,self.fcn(s2,exset[0,m]),
self.fcn(s2,exset[m+1,*]))
}(0,pts);
var bounds=_bounds;
}
fcn findNearest(k,p){
fcn(node,target,hr,max_dist_sqd,k){
if(not node) return(k.pump(List,0.0), (0.0).MAX, 0); // fake node far away
nodes_visited,s,pivot:=1, node.split, node.dom_elt;
left_hr,right_hr:=hr.copy(True), hr.copy(True); // deep-ish copy
left_hr.max[s]=right_hr.min[s]=pivot[s];
reg nearer_node, nearer_hr, further_node, further_hr;
if(target[s]<=pivot[s]){
nearer_node, nearer_hr = node.left, left_hr;
further_node, further_hr = node.right, right_hr;
}else{
nearer_node, nearer_hr = node.right, right_hr;
further_node, further_hr = node.left, left_hr;
}
n1:=self.fcn(nearer_node, target, nearer_hr, max_dist_sqd, k);
nearest,dist_sqd:=n1; // n1 is ( (a,b),distance,#visited )
nodes_visited+=n1[NODES_VISITED];
if(dist_sqd<max_dist_sqd) max_dist_sqd=dist_sqd;
d:=(pivot[s] - target[s]).pow(2);
if(d>max_dist_sqd) return(nearest, dist_sqd, nodes_visited);
d:=sqd(pivot,target);
if(d<dist_sqd) nearest,dist_sqd,max_dist_sqd=pivot,d,dist_sqd;
n2:=self.fcn(further_node, target, further_hr, max_dist_sqd,k);
nodes_visited+=n2[NODES_VISITED];
if(n2[DIST_SQD]<dist_sqd) nearest,dist_sqd=n2;
return(nearest, dist_sqd, nodes_visited)
}(n, p, bounds, (0.0).MAX,k)
}
fcn [private] sqd(p1,p2){ // point deltas squared and summed
p1.zipWith(fcn(a,b){ (a-b).pow(2) }, p2).sum(0.0)
}
fcn show_nearest(k, heading, p){
println(heading,":");
println("Point: ",p);
n:=findNearest(k,p);
println("Nearest neighbor:", n[NEAREST]);
println("Distance: ", n[DIST_SQD].sqrt());
println("Nodes visited: ", n[NODES_VISITED], "\n");
}
}class Orthotope{ fcn init(mi,ma){ var min=mi,max=ma; } }
kd1:=KdTree(T(T(2,3), T(5,4), T(9,6), T(4,7), T(8,1), T(7,2)),
Orthotope(List(0,0), List(10,10)));
kd1.show_nearest(2, "Wikipedia example data", T(9,2));- Output:
Wikipedia example data: Point: L(9,2) Nearest neighbor:L(8,1) Distance: 1.41421 Nodes visited: 3
fcn randomPoint(k){ k.pump(List,(0.0).random(1)) } // ( (p1,p2,,pk), ... n of them), p in [0..1]
fcn randomPoints(k,n){ // ( (p1,p2,,pk), ... n of them), p in [0..1]
n.pump(List,randomPoint.fp(k))
}
N:=40000;
kd2:=KdTree(randomPoints(3,N), Orthotope(L(0,0,0), L(1,1,1)));
kd2.show_nearest(2, String("k-d tree with ", N," random 3D points"),
randomPoint(3));- Output:
k-d tree with 40000 random 3D points: Point: L(0.186009,0.186009,0.186009) Nearest neighbor:L(0.186012,0.186012,0.186012) Distance: 3.91942e-06 Nodes visited: 16