Matrix chain multiplication
You are encouraged to solve this task according to the task description, using any language you may know.
- Problem
Using the most straightfoward algorithm (which we assume here), computing the product of two matrices of dimensions (n1,n2) and (n2,n3) requires n1*n2*n3 FMA operations. The number of operations required to compute the product of matrices A1, A2... An depends on the order of matrix multiplications, hence on where parens are put. Remember that the matrix product is associative, but not commutative, hence only the parens can be moved.
For instance, with four matrices, one can compute A(B(CD)), A((BC)D), (AB)(CD), (A(BC))D, (AB)C)D. The number of different ways to put the parens is a Catalan number, and grows exponentially with the number of factors.
Here is an example of computation of the total cost, for matrices A(5,6), B(6,3), C(3,1):
- AB costs 5*6*3=90 and produces a matrix of dimensions (5,3), then (AB)C costs 5*3*1=15. The total cost is 105.
- BC costs 6*3*1=18 and produces a matrix of dimensions (6,1), then A(BC) costs 5*6*1=30. The total cost is 48.
In this case, computing (AB)C requires more than twice as many operations as A(BC). The difference can be much more dramatic in real cases.
- Task
Write a function which, given a list of the successive dimensions of matrices A1, A2... An, of arbitrary length, returns the optimal way to compute the matrix product, and the total cost. Any sensible way to describe the optimal solution is accepted. The input list does not duplicate shared dimensions: for the previous example of matrices A,B,C, one will only pass the list [5,6,3,1] (and not [5,6,6,3,3,1]) to mean the matrix dimensions are respectively (5,6), (6,3) and (3,1). Hence, a product of n matrices is represented by a list of n+1 dimensions.
Try this function on the following two lists:
- [1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2]
- [1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10]
To solve the task, it's possible, but not required, to write a function that enumerates all possible ways to parenthesize the product. This is not optimal because of the many duplicated computations, and this task is a classic application of dynamic programming.
See also Matrix chain multiplication on Wikipedia.
Contents
C[edit]
#include <stdio.h>
#include <limits.h>
#include <stdlib.h>
int **m;
int **s;
void optimal_matrix_chain_order(int *dims, int n) {
int len, i, j, k, temp, cost;
n--;
m = (int **)malloc(n * sizeof(int *));
for (i = 0; i < n; ++i) {
m[i] = (int *)calloc(n, sizeof(int));
}
s = (int **)malloc(n * sizeof(int *));
for (i = 0; i < n; ++i) {
s[i] = (int *)calloc(n, sizeof(int));
}
for (len = 1; len < n; ++len) {
for (i = 0; i < n - len; ++i) {
j = i + len;
m[i][j] = INT_MAX;
for (k = i; k < j; ++k) {
temp = dims[i] * dims[k + 1] * dims[j + 1];
cost = m[i][k] + m[k + 1][j] + temp;
if (cost < m[i][j]) {
m[i][j] = cost;
s[i][j] = k;
}
}
}
}
}
void print_optimal_chain_order(int i, int j) {
if (i == j)
printf("%c", i + 65);
else {
printf("(");
print_optimal_chain_order(i, s[i][j]);
print_optimal_chain_order(s[i][j] + 1, j);
printf(")");
}
}
int main() {
int i, j, n;
int a1[4] = {5, 6, 3, 1};
int a2[13] = {1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2};
int a3[12] = {1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10};
int *dims_list[3] = {a1, a2, a3};
int sizes[3] = {4, 13, 12};
for (i = 0; i < 3; ++i) {
printf("Dims : [");
n = sizes[i];
for (j = 0; j < n; ++j) {
printf("%d", dims_list[i][j]);
if (j < n - 1) printf(", "); else printf("]\n");
}
optimal_matrix_chain_order(dims_list[i], n);
printf("Order : ");
print_optimal_chain_order(0, n - 2);
printf("\nCost : %d\n\n", m[0][n - 2]);
for (j = 0; j <= n - 2; ++j) free(m[j]);
free(m);
for (j = 0; j <= n - 2; ++j) free(s[j]);
free(s);
}
return 0;
}
- Output:
Dims : [5, 6, 3, 1] Order : (A(BC)) Cost : 48 Dims : [1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2] Order : ((((((((AB)C)D)E)F)G)(H(IJ)))(KL)) Cost : 38120 Dims : [1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10] Order : (A((((((BC)D)(((EF)G)H))I)J)K)) Cost : 1773740
C#[edit]
using System;
class MatrixChainOrderOptimizer {
private int[,] m;
private int[,] s;
void OptimalMatrixChainOrder(int[] dims) {
int n = dims.Length - 1;
m = new int[n, n];
s = new int[n, n];
for (int len = 1; len < n; ++len) {
for (int i = 0; i < n - len; ++i) {
int j = i + len;
m[i, j] = Int32.MaxValue;
for (int k = i; k < j; ++k) {
int temp = dims[i] * dims[k + 1] * dims[j + 1];
int cost = m[i, k] + m[k + 1, j] + temp;
if (cost < m[i, j]) {
m[i, j] = cost;
s[i, j] = k;
}
}
}
}
}
void PrintOptimalChainOrder(int i, int j) {
if (i == j)
Console.Write((char)(i + 65));
else {
Console.Write("(");
PrintOptimalChainOrder(i, s[i, j]);
PrintOptimalChainOrder(s[i, j] + 1, j);
Console.Write(")");
}
}
static void Main() {
var mcoo = new MatrixChainOrderOptimizer();
var dimsList = new int[3][];
dimsList[0] = new int[4] {5, 6, 3, 1};
dimsList[1] = new int[13] {1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2};
dimsList[2] = new int[12] {1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10};
for (int i = 0; i < dimsList.Length; ++i) {
Console.Write("Dims : [");
int n = dimsList[i].Length;
for (int j = 0; j < n; ++j) {
Console.Write(dimsList[i][j]);
if (j < n - 1)
Console.Write(", ");
else
Console.WriteLine("]");
}
mcoo.OptimalMatrixChainOrder(dimsList[i]);
Console.Write("Order : ");
mcoo.PrintOptimalChainOrder(0, n - 2);
Console.WriteLine("\nCost : {0}\n", mcoo.m[0, n - 2]);
}
}
}
- Output:
Dims : [5, 6, 3, 1] Order : (A(BC)) Cost : 48 Dims : [1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2] Order : ((((((((AB)C)D)E)F)G)(H(IJ)))(KL)) Cost : 38120 Dims : [1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10] Order : (A((((((BC)D)(((EF)G)H))I)J)K)) Cost : 1773740
Fortran[edit]
This is a translation of the Python iterative solution.
module optim_m
implicit none
contains
subroutine optim(a)
implicit none
integer :: a(:), n, i, j, k
integer(8) :: c
integer, allocatable :: u(:, :)
integer(8), allocatable :: v(:, :)
n = ubound(a, 1) - 1
allocate (u(n, n), v(n, n))
v = -1
u(:, 1) = -1
v(:, 1) = 0
do j = 2, n
do i = 1, n - j + 1
do k = 1, j - 1
c = v(i, k) + v(i + k, j - k) + int(a(i), 8) * int(a(i + k), 8) * int(a(i + j), 8)
if (v(i, j) < 0 .or. c < v(i, j)) then
u(i, j) = k
v(i, j) = c
end if
end do
end do
end do
write (*, "(I0,' ')", advance="no") v(1, n)
call aux(1, n)
print *
deallocate (u, v)
contains
recursive subroutine aux(i, j)
integer :: i, j, k
k = u(i, j)
if (k < 0) then
write (*, "(I0)", advance="no") i
else
write (*, "('(')", advance="no")
call aux(i, k)
write (*, "('*')", advance="no")
call aux(i + k, j - k)
write (*, "(')')", advance="no")
end if
end subroutine
end subroutine
end module
program optim_p
use optim_m
implicit none
call optim([1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2])
call optim([1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10])
end program
Output
38120 ((((((((1*2)*3)*4)*5)*6)*7)*(8*(9*10)))*(11*12)) 1773740 (1*((((((2*3)*4)*(((5*6)*7)*8))*9)*10)*11))
Go[edit]
The first for
loop is based on the pseudo and Java code from the
Wikipedia article.
package main
import "fmt"
// PrintMatrixChainOrder prints the optimal order for chain
// multiplying matrices.
// Matrix A[i] has dimensions dims[i-1]×dims[i].
func PrintMatrixChainOrder(dims []int) {
n := len(dims) - 1
m, s := newSquareMatrices(n)
// m[i,j] will be minimum number of scalar multiplactions
// needed to compute the matrix A[i]A[i+1]…A[j] = A[i…j].
// Note, m[i,i] = zero (no cost).
// s[i,j] will be the index of the subsequence split that
// achieved minimal cost.
for lenMinusOne := 1; lenMinusOne < n; lenMinusOne++ {
for i := 0; i < n-lenMinusOne; i++ {
j := i + lenMinusOne
m[i][j] = -1
for k := i; k < j; k++ {
cost := m[i][k] + m[k+1][j] + dims[i]*dims[k+1]*dims[j+1]
if m[i][j] < 0 || cost < m[i][j] {
m[i][j] = cost
s[i][j] = k
}
}
}
}
// Format and print result.
const MatrixNames = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
var subprint func(int, int)
subprint = func(i, j int) {
if i == j {
return
}
k := s[i][j]
subprint(i, k)
subprint(k+1, j)
fmt.Printf("%*s -> %s × %s%*scost=%d\n",
n, MatrixNames[i:j+1],
MatrixNames[i:k+1],
MatrixNames[k+1:j+1],
n+i-j, "", m[i][j],
)
}
subprint(0, n-1)
}
func newSquareMatrices(n int) (m, s [][]int) {
// Allocates two n×n matrices as slices of slices but
// using only one [2n][]int and one [2n²]int backing array.
m = make([][]int, 2*n)
m, s = m[:n:n], m[n:]
tmp := make([]int, 2*n*n)
for i := range m {
m[i], tmp = tmp[:n:n], tmp[n:]
}
for i := range s {
s[i], tmp = tmp[:n:n], tmp[n:]
}
return m, s
}
func main() {
cases := [...][]int{
{1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2},
{1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10},
}
for _, tc := range cases {
fmt.Println("Dimensions:", tc)
PrintMatrixChainOrder(tc)
fmt.Println()
}
}
- Output:
Dimensions: [1 5 25 30 100 70 2 1 100 250 1 1000 2] AB -> A × B cost=125 ABC -> AB × C cost=875 ABCD -> ABC × D cost=3875 ABCDE -> ABCD × E cost=10875 ABCDEF -> ABCDE × F cost=11015 ABCDEFG -> ABCDEF × G cost=11017 IJ -> I × J cost=25000 HIJ -> H × IJ cost=25100 ABCDEFGHIJ -> ABCDEFG × HIJ cost=36118 KL -> K × L cost=2000 ABCDEFGHIJKL -> ABCDEFGHIJ × KL cost=38120 Dimensions: [1000 1 500 12 1 700 2500 3 2 5 14 10] BC -> B × C cost=6000 BCD -> BC × D cost=6012 EF -> E × F cost=1750000 EFG -> EF × G cost=1757500 EFGH -> EFG × H cost=1757506 BCDEFGH -> BCD × EFGH cost=1763520 BCDEFGHI -> BCDEFGH × I cost=1763530 BCDEFGHIJ -> BCDEFGHI × J cost=1763600 BCDEFGHIJK -> BCDEFGHIJ × K cost=1763740 ABCDEFGHIJK -> A × BCDEFGHIJK cost=1773740
Julia[edit]
Module:
module MatrixChainMultiplications
using OffsetArrays
function optim(a)
n = length(a) - 1
u = fill!(OffsetArray{Int}(0:n, 0:n), 0)
v = fill!(OffsetArray{Int}(0:n, 0:n), typemax(Int))
u[:, 1] .= -1
v[:, 1] .= 0
for j in 2:n, i in 1:n-j+1, k in 1:j-1
c = v[i, k] + v[i+k, j-k] + a[i] * a[i+k] * a[i+j]
if c < v[i, j]
u[i, j] = k
v[i, j] = c
end
end
return v[1, n], aux(u, 1, n)
end
function aux(u, i, j)
k = u[i, j]
if k < 0
return sprint(print, i)
else
return sprint(print, '(', aux(u, i, k), '×', aux(u, i + k, j - k), ")")
end
end
end # module MatrixChainMultiplications
Main:
println(MatrixChainMultiplications.optim([1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2]))
println(MatrixChainMultiplications.optim([1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10]))
- Output:
(38120, "((((((((1×2)×3)×4)×5)×6)×7)×(8×(9×10)))×(11×12))") (1773740, "(1×((((((2×3)×4)×(((5×6)×7)×8))×9)×10)×11))")
Kotlin[edit]
This is based on the pseudo-code in the Wikipedia article.
// Version 1.2.31
lateinit var m: List<IntArray>
lateinit var s: List<IntArray>
fun optimalMatrixChainOrder(dims: IntArray) {
val n = dims.size - 1
m = List(n) { IntArray(n) }
s = List(n) { IntArray(n) }
for (len in 1 until n) {
for (i in 0 until n - len) {
val j = i + len
m[i][j] = Int.MAX_VALUE
for (k in i until j) {
val temp = dims[i] * dims [k + 1] * dims[j + 1]
val cost = m[i][k] + m[k + 1][j] + temp
if (cost < m[i][j]) {
m[i][j] = cost
s[i][j] = k
}
}
}
}
}
fun printOptimalChainOrder(i: Int, j: Int) {
if (i == j)
print("${(i + 65).toChar()}")
else {
print("(")
printOptimalChainOrder(i, s[i][j])
printOptimalChainOrder(s[i][j] + 1, j)
print(")")
}
}
fun main(args: Array<String>) {
val dimsList = listOf(
intArrayOf(5, 6, 3, 1),
intArrayOf(1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2),
intArrayOf(1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10)
)
for (dims in dimsList) {
println("Dims : ${dims.asList()}")
optimalMatrixChainOrder(dims)
print("Order : ")
printOptimalChainOrder(0, s.size - 1)
println("\nCost : ${m[0][s.size - 1]}\n")
}
}
- Output:
Dims : [5, 6, 3, 1] Order : (A(BC)) Cost : 48 Dims : [1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2] Order : ((((((((AB)C)D)E)F)G)(H(IJ)))(KL)) Cost : 38120 Dims : [1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10] Order : (A((((((BC)D)(((EF)G)H))I)J)K)) Cost : 1773740
MATLAB[edit]
function [r,s] = optim(a)
n = length(a)-1;
u = zeros(n,n);
v = ones(n,n)*inf;
u(:,1) = -1;
v(:,1) = 0;
for j = 2:n
for i = 1:n-j+1
for k = 1:j-1
c = v(i,k)+v(i+k,j-k)+a(i)*a(i+k)*a(i+j);
if c<v(i,j)
u(i,j) = k;
v(i,j) = c;
end
end
end
end
r = v(1,n);
s = aux(u,1,n);
end
function s = aux(u,i,j)
k = u(i,j);
if k<0
s = sprintf("%d",i);
else
s = sprintf("(%s*%s)",aux(u,i,k),aux(u,i+k,j-k));
end
end
Output
[r,s] = optim([1,5,25,30,100,70,2,1,100,250,1,1000,2])
r =
38120
s =
"((((((((1*2)*3)*4)*5)*6)*7)*(8*(9*10)))*(11*12))"
[r,s] = optim([1000,1,500,12,1,700,2500,3,2,5,14,10])
r =
1773740
s =
"(1*((((((2*3)*4)*(((5*6)*7)*8))*9)*10)*11))"
Perl[edit]
use strict;
use feature 'say';
sub matrix_mult_chaining {
my(@dimensions) = @_;
my(@cp,@path);
# a matrix never needs to be multiplied with itself, so it has cost 0
$cp[$_][$_] = 0 for keys @dimensions;
my $n = $#dimensions;
for my $chain_length (1..$n) {
for my $start (0 .. $n - $chain_length - 1) {
my $end = $start + $chain_length;
$cp[$end][$start] = 10e10;
for my $step ($start .. $end - 1) {
my $new_cost = $cp[$step][$start]
+ $cp[$end][$step + 1]
+ $dimensions[$start] * $dimensions[$step+1] * $dimensions[$end+1];
if ($new_cost < $cp[$end][$start]) {
$cp[$end][$start] = $new_cost; # cost
$cp[$start][$end] = $step; # path
}
}
}
}
$cp[$n-1][0] . ' ' . find_path(0, $n-1, @cp);
}
sub find_path {
my($start,$end,@cp) = @_;
my $result;
if ($start == $end) {
$result .= 'A' . ($start + 1);
} else {
$result .= '(' .
find_path($start, $cp[$start][$end], @cp) .
find_path($cp[$start][$end] + 1, $end, @cp) .
')';
}
return $result;
}
say matrix_mult_chaining(<1 5 25 30 100 70 2 1 100 250 1 1000 2>);
say matrix_mult_chaining(<1000 1 500 12 1 700 2500 3 2 5 14 10>);
- Output:
38120 ((((((((A1A2)A3)A4)A5)A6)A7)(A8(A9A10)))(A11A12)) 1773740 (A1((((((A2A3)A4)(((A5A6)A7)A8))A9)A10)A11))
Perl 6[edit]
This example is based on Moritz Lenz's code, written for Carl Mäsak's Perl 6 Coding Contest, in 2010. Slightly simplified, it fulfills the Rosetta Code task as well.
sub matrix-mult-chaining(@dimensions) {
my @cp;
# @cp has a dual function:
# * the upper triangle of the diagonal matrix stores the cost (c) for
# multiplying matrices $i and $j in @cp[$j][$i], where $j > $i
# * the lower triangle stores the path (p) that was used for the lowest cost
# multiplication to get from $i to $j.
# a matrix never needs to be multiplied with itself, so it has cost 0
@cp[$_][$_] = 0 for @dimensions.keys;
my @path;
my $n = @dimensions.end;
for 1 .. $n -> $chain-length {
for 0 .. $n - $chain-length - 1 -> $start {
my $end = $start + $chain-length;
@cp[$end][$start] = Inf; # until we find a better connection
for $start .. $end - 1 -> $step {
my $new-cost = @cp[$step][$start]
+ @cp[$end][$step + 1]
+ [*] @dimensions[$start, $step+1, $end+1];
if $new-cost < @cp[$end][$start] {
@cp[$end][$start] = $new-cost; # cost
@cp[$start][$end] = $step; # path
}
}
}
}
sub find-path(Int $start, Int $end) {
if $start == $end {
take 'A' ~ ($start + 1);
} else {
take '(';
find-path($start, @cp[$start][$end]);
find-path(@cp[$start][$end] + 1, $end);
take ')';
}
}
return @cp[$n-1][0], gather { find-path(0, $n - 1) }.join;
}
say matrix-mult-chaining(<1 5 25 30 100 70 2 1 100 250 1 1000 2>);
say matrix-mult-chaining(<1000 1 500 12 1 700 2500 3 2 5 14 10>);
- Output:
(38120 ((((((((A1A2)A3)A4)A5)A6)A7)(A8(A9A10)))(A11A12))) (1773740 (A1((((((A2A3)A4)(((A5A6)A7)A8))A9)A10)A11)))
Phix[edit]
As per the wp pseudocode
function optimal_chain_order(int i, int j, sequence s)
if i==j then
return i+'A'-1
end if
return "("&optimal_chain_order(i,s[i,j],s)
&optimal_chain_order(s[i,j]+1,j,s)&")"
end function
function optimal_matrix_chain_order(sequence dims)
integer n = length(dims)-1
sequence {m,s} @= repeat(repeat(0,n),n)
for len=2 to n do
for i=1 to n-len+1 do
integer j = i+len-1
m[i][j] = -1
for k=i to j-1 do
atom cost := m[i][k] + m[k+1][j] + dims[i]*dims[k+1]*dims[j+1]
if m[i][j]<0
or cost<m[i][j] then
m[i][j] = cost;
s[i][j] = k;
end if
end for
end for
end for
return {optimal_chain_order(1,n,s),m[1,n]}
end function
constant tests = {{5, 6, 3, 1},
{1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2},
{1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10}}
for i=1 to length(tests) do
sequence ti = tests[i]
printf(1,"Dims : %s\n",{sprint(ti)})
printf(1,"Order : %s\nCost : %d\n",optimal_matrix_chain_order(ti))
end for
- Output:
Dims : {5,6,3,1} Order : (A(BC)) Cost : 48 Dims : {1,5,25,30,100,70,2,1,100,250,1,1000,2} Order : ((((((((AB)C)D)E)F)G)(H(IJ)))(KL)) Cost : 38120 Dims : {1000,1,500,12,1,700,2500,3,2,5,14,10} Order : (A((((((BC)D)(((EF)G)H))I)J)K)) Cost : 1773740
Python[edit]
We will solve the task in three steps:
1) Enumerate all ways to parenthesize (using a generator to save space), and for each one compute the cost. Then simply look up the minimal cost.
2) Merge the enumeration and the cost function in a recursive cost optimizing function. The computation is roughly the same, but it's much faster as some steps are removed.
3) The recursive solution has many duplicates computations. Memoize the previous function: this yields a dynamic programming approach.
Enumeration of parenthesizations[edit]
def parens(n):
def aux(n, k):
if n == 1:
yield k
elif n == 2:
yield [k, k + 1]
else:
a = []
for i in range(1, n):
for u in aux(i, k):
for v in aux(n - i, k + i):
yield [u, v]
yield from aux(n, 0)
Example (in the same order as in the task description)
for u in parens(4):
print(u)
[0, [1, [2, 3]]]
[0, [[1, 2], 3]]
[[0, 1], [2, 3]]
[[0, [1, 2]], 3]
[[[0, 1], 2], 3]
And here is the optimization step:
def optim1(a):
def cost(k):
if type(k) is int:
return 0, a[k], a[k + 1]
else:
s1, p1, q1 = cost(k[0])
s2, p2, q2 = cost(k[1])
assert q1 == p2
return s1 + s2 + p1 * q1 * q2, p1, q2
cmin = None
n = len(a) - 1
for u in parens(n):
c, p, q = cost(u)
if cmin is None or c < cmin:
cmin = c
umin = u
return cmin, umin
Recursive cost optimization[edit]
The previous function optim1 already used recursion, but only to compute the cost of a given parens configuration, whereas another function (a generator actually) provides these configurations. Here we will do both recursively in the same function, avoiding the computation of configurations altogether.
def optim2(a):
def aux(n, k):
if n == 1:
p, q = a[k:k + 2]
return 0, p, q, k
elif n == 2:
p, q, r = a[k:k + 3]
return p * q * r, p, r, [k, k + 1]
else:
m = None
p = a[k]
q = a[k + n]
for i in range(1, n):
s1, p1, q1, u1 = aux(i, k)
s2, p2, q2, u2 = aux(n - i, k + i)
assert q1 == p2
s = s1 + s2 + p1 * q1 * q2
if m is None or s < m:
m = s
u = [u1, u2]
return m, p, q, u
s, p, q, u = aux(len(a) - 1, 0)
return s, u
Memoized recursive call[edit]
The only difference between optim2 and optim3 is the @memoize decorator. Yet the algorithm is way faster with this. According to Wikipedia, the complexity falls from O(2^n) to O(n^3). This is confirmed by plotting log(time) vs log(n) for n up to 580 (this needs changing Python's recursion limit).
def memoize(f):
h = {}
def g(*u):
if u in h:
return h[u]
else:
r = f(*u)
h[u] = r
return r
return g
def optim3(a):
@memoize
def aux(n, k):
if n == 1:
p, q = a[k:k + 2]
return 0, p, q, k
elif n == 2:
p, q, r = a[k:k + 3]
return p * q * r, p, r, [k, k + 1]
else:
m = None
p = a[k]
q = a[k + n]
for i in range(1, n):
s1, p1, q1, u1 = aux(i, k)
s2, p2, q2, u2 = aux(n - i, k + i)
assert q1 == p2
s = s1 + s2 + p1 * q1 * q2
if m is None or s < m:
m = s
u = [u1, u2]
return m, p, q, u
s, p, q, u = aux(len(a) - 1, 0)
return s, u
Putting all together[edit]
import time
u = [[1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2],
[1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10]]
for a in u:
print(a)
print()
print("function time cost parens ")
print("-" * 90)
for f in [optim1, optim2, optim3]:
t1 = time.clock()
s, u = f(a)
t2 = time.clock()
print("%s %10.3f %10d %s" % (f.__name__, 1000 * (t2 - t1), s, u))
print()
Output (timings are in milliseconds)
[1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2] function time cost parens ------------------------------------------------------------------------------------------ optim1 838.636 38120 [[[[[[[[0, 1], 2], 3], 4], 5], 6], [7, [8, 9]]], [10, 11]] optim2 80.628 38120 [[[[[[[[0, 1], 2], 3], 4], 5], 6], [7, [8, 9]]], [10, 11]] optim3 0.373 38120 [[[[[[[[0, 1], 2], 3], 4], 5], 6], [7, [8, 9]]], [10, 11]] [1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10] function time cost parens ------------------------------------------------------------------------------------------ optim1 223.186 1773740 [0, [[[[[[1, 2], 3], [[[4, 5], 6], 7]], 8], 9], 10]] optim2 27.660 1773740 [0, [[[[[[1, 2], 3], [[[4, 5], 6], 7]], 8], 9], 10]] optim3 0.307 1773740 [0, [[[[[[1, 2], 3], [[[4, 5], 6], 7]], 8], 9], 10]]
A mean on 1000 loops to get a better precision on the optim3, yields respectively 0.365 ms and 0.287 ms.
Iterative solution[edit]
In the previous solution, memoization is done blindly with a dictionary. However, we need to compute the optimal products for all sublists. A sublist is described by its first index and length (resp. i and j+1 in the following function), hence the set of all sublists can be described by the indices of elements in a triangular array u. We first fill the "solution" (there is no product) for sublists of length 1 (u[0]), then for each successive length we optimize using what when know about smaller sublists. Instead of keeping track of the optimal solutions, the single needed one is computed in the end.
def optim4(a):
global u
n = len(a) - 1
u = [None] * n
u[0] = [[None, 0]] * n
for j in range(1, n):
v = [None] * (n - j)
for i in range(n - j):
m = None
for k in range(j):
s1, c1 = u[k][i]
s2, c2 = u[j - k - 1][i + k + 1]
c = c1 + c2 + a[i] * a[i + k + 1] * a[i + j + 1]
if m is None or c < m:
s = k
m = c
v[i] = [s, m]
u[j] = v
def aux(i, j):
s, c = u[j][i]
if s is None:
return i
else:
return [aux(i, s), aux(i + s + 1, j - s - 1)]
return u[n - 1][0][1], aux(0, n - 1)
print(optim4([1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2]))
print(optim4([1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10]))
Output
(38120, [[[[[[[[0, 1], 2], 3], 4], 5], 6], [7, [8, 9]]], [10, 11]]) (1773740, [0, [[[[[[1, 2], 3], [[[4, 5], 6], 7]], 8], 9], 10]])
Stata[edit]
Recursive solution[edit]
Here is the equivalent of optim3 in Python's solution. Memoization is done with an associative array. Multiple results are returned in a structure. The same effect as optim2 can be achieved by removing the asarray machinery.
mata
struct ans {
real scalar p,q,s
string scalar u
}
struct ans scalar function aux(n,k) {
external dim,opt
struct ans scalar r,r1,r2
real scalar s,i
if (n==1) {
r.p = dim[k]
r.q = dim[k+1]
r.s = 0
r.u = strofreal(k)
return(r)
} else if (n==2) {
r.p = dim[k]
r.q = dim[k+2]
r.s = r.p*r.q*dim[k+1]
r.u = sprintf("(%f*%f)",k,k+1)
return(r)
} else if (asarray_contains(opt,(n,k))) {
return(asarray(opt,(n,k)))
} else {
r.p = dim[k]
r.q = dim[k+n]
r.s = .
for (i=1; i<n; i++) {
r1 = aux(i,k)
r2 = aux(n-i,k+i)
s = r1.s+r2.s+r1.p*r1.q*r2.q
if (s<r.s) {
r.s = s
r.u = sprintf("(%s*%s)",r1.u,r2.u)
}
}
asarray(opt,(n,k),r)
return(r)
}
}
function optim(a) {
external dim,opt
struct ans scalar r
real scalar t
timer_clear()
dim = a
opt = asarray_create("real",2)
timer_on(1)
r = aux(length(a)-1,1)
timer_off(1)
t = timer_value(1)[1]
printf("%10.0f %10.0f %s\n",t*1000,r.s,r.u)
}
optim((1,5,25,30,100,70,2,1,100,250,1,1000,2))
optim((1000,1,500,12,1,700,2500,3,2,5,14,10))
end
Output
0 38120 ((((((((1*2)*3)*4)*5)*6)*7)*(8*(9*10)))*(11*12)) 16 1773740 (1*((((((2*3)*4)*(((5*6)*7)*8))*9)*10)*11))
The timing is in milliseconds, but the time resolution is too coarse to get a usable result. A mean on 1000 loops doing the same computation yields respectively 5.772 ms and 4.430 ms for these two cases. For comparison, the computation was made on the same machine as the Python solution.
Iterative solution[edit]
mata
function aux(u,i,j) {
k = u[i,j]
if (k<0) {
printf("%f",i)
} else {
printf("(")
aux(u,i,k)
printf("*")
aux(u,i+k,j-k)
printf(")")
}
}
function optim(a) {
n = length(a)-1
u = J(n,n,.)
v = J(n,n,.)
u[.,1] = J(n,1,-1)
v[.,1] = J(n,1,0)
for (j=2; j<=n; j++) {
for (i=1; i<=n-j+1; i++) {
for (k=1; k<j; k++) {
c = v[i,k]+v[i+k,j-k]+a[i]*a[i+k]*a[i+j]
if (c<v[i,j]) {
u[i,j] = k
v[i,j] = c
}
}
}
}
printf("%f ",v[1,n])
aux(u,1,n)
printf("\n")
}
optim((1,5,25,30,100,70,2,1,100,250,1,1000,2))
optim((1000,1,500,12,1,700,2500,3,2,5,14,10))
end
Output
38120 ((((((((1*2)*3)*4)*5)*6)*7)*(8*(9*10)))*(11*12)) 1773740 (1*((((((2*3)*4)*(((5*6)*7)*8))*9)*10)*11))
This solution is faster than the recursive one. The 1000 loops run now in 0.234 ms and 0.187 ms per loop on average.
zkl[edit]
fcn optim3(a){ // list --> (int,list)
aux:=fcn(n,k,a){ // (int,int,list) --> (int,int,int,list)
if(n==1){
p,q := a[k,2];
return(0,p,q,k);
}
if(n==2){
p,q,r := a[k,3];
return(p*q*r, p, r, T(k,k+1));
}
m,p,q,u := Void, a[k], a[k + n], Void;
foreach i in ([1..n-1]){
#if 0 // 0.70 sec for both tests
s1,p1,q1,u1 := self.fcn(i,k,a);
s2,p2,q2,u2 := self.fcn(n - i, k + i, a);
#else // 0.33 sec for both tests
s1,p1,q1,u1 := memoize(self.fcn, i,k,a);
s2,p2,q2,u2 := memoize(self.fcn, n - i, k + i, a);
#endif
_assert_(q1==p2);
s:=s1 + s2 + p1*q1*q2;
if((Void==m) or (s<m)) m,u = s,T(u1,u2);
}
return(m,p,q,u);
};
h=Dictionary(); // reset memoize
s,_,_,u := aux(a.len() - 1, 0,a);
return(s,u);
}
var h; // a Dictionary, set/reset in optim3()
fcn memoize(f,n,k,a){
key:="%d,%d".fmt(n,k); // Lists make crappy keys
if(r:=h.find(key)) return(r);
r:=f(n,k,a);
h[key]=r;
return(r);
}
fcn pp(u){ // pretty print a list of lists
var letters=["A".."Z"].pump(String);
u.pump(String,
fcn(n){ if(List.isType(n)) String("(",pp(n),")") else letters[n] })
}
fcn prnt(s,u){ "%-9,d %s\n\t-->%s\n".fmt(s,u.toString(*,*),pp(u)).println() }
s,u := optim3(T(1, 5, 25, 30, 100, 70, 2, 1, 100, 250, 1, 1000, 2));
prnt(s,u);
s,u := optim3(T(1000, 1, 500, 12, 1, 700, 2500, 3, 2, 5, 14, 10));
prnt(s,u);
optim3(T(5,6,3,1)) : prnt(_.xplode());
- Output:
38,120 L(L(L(L(L(L(L(L(0,1),2),3),4),5),6),L(7,L(8,9))),L(10,11)) -->(((((((AB)C)D)E)F)G)(H(IJ)))(KL) 1,773,740 L(0,L(L(L(L(L(L(1,2),3),L(L(L(4,5),6),7)),8),9),10)) -->A((((((BC)D)(((EF)G)H))I)J)K) 48 L(0,L(1,2)) -->A(BC)