Non-continuous subsequences: Difference between revisions
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200 => 1606938044258990275541962092341162602522202993782792835281275 |
200 => 1606938044258990275541962092341162602522202993782792835281275 |
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</pre> |
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=={{header|Kotlin}}== |
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<lang scala>// version 1.1.2 |
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fun <T> ncs(a: Array<T>) { |
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fun generate(m: Int, k: Int, c: IntArray) { |
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if (k == m) { |
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if (c[m - 1] != c[0] + m - 1) { |
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for (i in 0 until m) print("${a[c[i]]} ") |
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println() |
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} |
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} |
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else { |
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for (j in 0 until a.size) { |
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if (k == 0 || j > c[k - 1]) { |
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c[k] = j |
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generate(m, k + 1, c) |
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} |
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} |
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} |
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} |
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for (m in 2 until a.size) { |
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val c = IntArray(m) |
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generate(m, 0, c) |
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} |
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} |
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fun main(args: Array<String>) { |
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val a = arrayOf(1, 2, 3, 4) |
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ncs(a) |
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println() |
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val ca = arrayOf('a', 'b', 'c', 'd', 'e') |
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ncs(ca) |
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}</lang> |
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{{out}} |
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<pre> |
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1 3 |
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1 4 |
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2 4 |
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1 2 4 |
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1 3 4 |
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a c |
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a d |
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a e |
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b d |
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b e |
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c e |
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a b d |
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a b e |
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a c d |
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a c e |
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a d e |
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b c e |
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b d e |
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a b c e |
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a b d e |
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a c d e |
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</pre> |
</pre> |
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Revision as of 13:17, 22 May 2017
You are encouraged to solve this task according to the task description, using any language you may know.
Consider some sequence of elements. (It differs from a mere set of elements by having an ordering among members.)
A subsequence contains some subset of the elements of this sequence, in the same order.
A continuous subsequence is one in which no elements are missing between the first and last elements of the subsequence.
Note: Subsequences are defined structurally, not by their contents. So a sequence a,b,c,d will always have the same subsequences and continuous subsequences, no matter which values are substituted; it may even be the same value.
Task: Find all non-continuous subsequences for a given sequence.
Example: For the sequence 1,2,3,4, there are five non-continuous subsequences, namely:
- 1,3
- 1,4
- 2,4
- 1,3,4
- 1,2,4
Goal: There are different ways to calculate those subsequences. Demonstrate algorithm(s) that are natural for the language.
Ada
Recursive
<lang ada>with Ada.Text_IO; use Ada.Text_IO;
procedure Test_Non_Continuous is
type Sequence is array (Positive range <>) of Integer;
procedure Put_NCS
( Tail : Sequence; -- To generate subsequences of
Head : Sequence := (1..0 => 1); -- Already generated
Contiguous : Boolean := True -- It is still continuous
) is
begin
if not Contiguous and then Head'Length > 1 then
for I in Head'Range loop
Put (Integer'Image (Head (I)));
end loop;
New_Line;
end if;
if Tail'Length /= 0 then
declare
New_Head : Sequence (Head'First..Head'Last + 1);
begin
New_Head (Head'Range) := Head;
for I in Tail'Range loop
New_Head (New_Head'Last) := Tail (I);
Put_NCS
( Tail => Tail (I + 1..Tail'Last),
Head => New_Head,
Contiguous => Contiguous and then (I = Tail'First or else Head'Length = 0)
);
end loop;
end;
end if;
end Put_NCS;
begin
Put_NCS ((1,2,3)); New_Line; Put_NCS ((1,2,3,4)); New_Line; Put_NCS ((1,2,3,4,5)); New_Line;
end Test_Non_Continuous;</lang>
- Output:
1 3 1 2 4 1 3 1 3 4 1 4 2 4 1 2 3 5 1 2 4 1 2 4 5 1 2 5 1 3 1 3 4 1 3 4 5 1 3 5 1 4 1 4 5 1 5 2 3 5 2 4 2 4 5 2 5 3 5
ALGOL 68
Recursive
- note: This specimen retains the original Ada coding style.
<lang algol68>PROC test non continuous = VOID: BEGIN
MODE SEQMODE = CHAR; MODE SEQ = [1:0]SEQMODE; MODE YIELDSEQ = PROC(SEQ)VOID;
PROC gen ncs =
( SEQ tail, # To generate subsequences of #
SEQ head, # Already generated #
BOOL contiguous,# It is still continuous #
YIELDSEQ yield
) VOID:
BEGIN
IF NOT contiguous ANDTH UPB head > 1 THEN
yield (head)
FI;
IF UPB tail /= 0 THEN
[UPB head+1]SEQMODE new head;
new head [:UPB head] := head;
FOR i TO UPB tail DO
new head [UPB new head] := tail [i];
gen ncs
( tail[i + 1:UPB tail],
new head,
contiguous ANDTH (i = LWB tail OREL UPB head = 0),
yield
)
OD
FI
END # put ncs #;
# FOR SEQ seq IN # gen ncs(("a","e","i","o","u"), (), TRUE, # ) DO ( #
## (SEQ seq)VOID:
print((seq, new line))
# OD # )
END; test non continuous</lang>
- Output:
aeiu aeo aeou aeu ai aio aiou aiu ao aou au eiu eo eou eu iu
Iterative
- note: This specimen retains the original C coding style.
Note: This specimen can only handle sequences of length less than bits width of bits. <lang algol68>MODE SEQMODE = STRING; MODE SEQ = [1:0]SEQMODE; MODE YIELDSEQ = PROC(SEQ)VOID;
PROC gen ncs = (SEQ seq, YIELDSEQ yield)VOID: BEGIN
IF UPB seq - 1 > bits width THEN stop FI; [UPB seq]SEQMODE out; INT upb out;
BITS lim := 16r1 SHL UPB seq; BITS upb k := lim SHR 1; # assert(lim); #
BITS empty = 16r000000000; # const #
FOR j TO ABS lim-1 DO
INT state := 1;
BITS k1 := upb k;
WHILE k1 NE empty DO
BITS b := BIN j AND k1;
CASE state IN
# state 1 # IF b NE empty THEN state +:= 1 FI,
# state 2 # IF b EQ empty THEN state +:= 1 FI,
# state 3 #
BEGIN
IF b EQ empty THEN GO TO continue k1 FI;
upb out := 0;
BITS k2 := upb k; FOR i WHILE k2 NE empty DO
IF (BIN j AND k2) NE empty THEN out[upb out +:= 1] := seq[i] FI;
k2 := k2 SHR 1
OD;
yield(out[:upb out]);
k1 := empty # empty: ending containing loop #
END
ESAC;
continue k1: k1 := k1 SHR 1
OD
OD
END;
main:(
[]STRING seqs = ("a","e","i","o","u");
- FOR SEQ seq IN # gen ncs(seqs, # ) DO ( #
- (SEQ seq)VOID:
print((seq, new line))
- OD # )
)</lang>
- Output:
iu eu eo eou eiu au ao aou ai aiu aio aiou aeu aeo aeou aeiu
AutoHotkey
using filtered templates ahk forum: discussion
<lang AutoHotkey>MsgBox % noncontinuous("a,b,c,d,e", ",") MsgBox % noncontinuous("1,2,3,4", ",")
noncontinuous(list, delimiter) { stringsplit, seq, list, %delimiter% n := seq0 ; sequence length Loop % x := (1<<n) - 1 { ; try all 0-1 candidate sequences
If !RegExMatch(b:=ToBin(A_Index,n),"^0*1*0*$") { ; drop continuous subsequences
Loop Parse, b
t .= A_LoopField ? seq%A_Index% " " : "" ; position -> number
t .= "`n" ; new sequences in new lines
}
} return t }
ToBin(n,W=16) { ; LS W-bits of Binary representation of n
Return W=1 ? n&1 : ToBin(n>>1,W-1) . n&1
}</lang>
BBC BASIC
<lang bbcbasic> DIM list1$(3)
list1$() = "1", "2", "3", "4"
PRINT "For [1, 2, 3, 4] non-continuous subsequences are:"
PROCnon_continuous_subsequences(list1$())
DIM list2$(4)
list2$() = "1", "2", "3", "4", "5"
PRINT "For [1, 2, 3, 4, 5] non-continuous subsequences are:"
PROCnon_continuous_subsequences(list2$())
END
DEF PROCnon_continuous_subsequences(l$())
LOCAL i%, j%, g%, n%, r%, s%, w%, a$, b$
n% = DIM(l$(),1)
FOR s% = 0 TO n%-2
FOR g% = s%+1 TO n%-1
a$ = "["
FOR i% = s% TO g%-1
a$ += l$(i%) + ", "
NEXT
FOR w% = 1 TO n%-g%
r% = n%+1-g%-w%
FOR i% = 1 TO 2^r%-1 STEP 2
b$ = a$
FOR j% = 0 TO r%-1
IF i% AND 2^j% b$ += l$(g%+w%+j%) + ", "
NEXT
PRINT LEFT$(LEFT$(b$)) + "]"
NEXT i%
NEXT w%
NEXT g%
NEXT s%
ENDPROC</lang>
- Output:
For [1, 2, 3, 4] non-continuous subsequences are: [1, 3] [1, 3, 4] [1, 4] [1, 2, 4] [2, 4] For [1, 2, 3, 4, 5] non-continuous subsequences are: [1, 3] [1, 3, 4] [1, 3, 5] [1, 3, 4, 5] [1, 4] [1, 4, 5] [1, 5] [1, 2, 4] [1, 2, 4, 5] [1, 2, 5] [1, 2, 3, 5] [2, 4] [2, 4, 5] [2, 5] [2, 3, 5] [3, 5]
Bracmat
<lang Bracmat>( ( noncontinuous
= sub
. ( sub
= su a nc
. !arg:(?su.?nc)
& !su
: %
%?a
( %:[%(sub$(!sjt.!nc !a))
| ?
& !nc:~
& out$(!nc !a)
& ~
)
)
& sub$(dummy !arg.)
|
)
& noncontinuous$(e r n i t) ); </lang>
- Output:
e n t e n e n i e n i t e i e i t e t e r i e r i t e r t e r n t r i r i t r t r n t n t
C
Note: This specimen can only handle lists of length less than the number of bits in an int. <lang C>#include <assert.h>
- include <stdio.h>
int main(int c, char **v) { unsigned int n = 1 << (c - 1), i = n, j, k; assert(n);
while (i--) { if (!(i & (i + (i & -(int)i)))) // consecutive 1s continue;
for (j = n, k = 1; j >>= 1; k++) if (i & j) printf("%s ", v[k]);
putchar('\n'); }
return 0; }</lang> Example use:
$ ./noncont 1 2 3 4 1 2 4 1 3 4 1 3 2 4 1 4 $ ./noncont 1 2 3 4 5 6 7 8 9 0 | wc -l 968
Using "consecutive + gap + any subsequence" to produce disjointed sequences: <lang c>#include <assert.h>
- include <stdio.h>
- include <stdlib.h>
void binprint(unsigned int n, unsigned int m) { char c[sizeof(n) * 8 + 1]; int i = 0; while (m >>= 1) c[i++] = n & m ? '#' : '-'; c[i] = 0; puts(c); }
int main(int c, char **v) { unsigned int n, gap, left, right; if (c < 2 || ! (n = 1 << atoi(v[1]))) n = 16;
for (gap = 2; gap < n; gap <<= 1) for (left = gap << 1; left < n; left |= left << 1) for (right = 1; right < gap; right++) binprint(left | right, n);
return 0; }</lang>
Recursive method
Using recursion and a state transition table. <lang c>#include <stdio.h>
typedef unsigned char sint; enum states { s_blnk = 0, s_tran, s_cont, s_disj };
/* Recursively look at each item in list, taking both choices of
picking the item or not. The state at each step depends on prvious pickings, with the state transition table:
blank + no pick -> blank blank + pick -> contiguous transitional + no pick -> transitional transitional + pick -> disjoint contiguous + no pick -> transitional contiguous + pick -> contiguous disjoint + pick -> disjoint disjoint + no pick -> disjoint
At first step, before looking at any item, state is blank. Because state is known at each step and needs not be calculated, it can be quite fast.
- /
unsigned char tbl[][2] = { { s_blnk, s_cont }, { s_tran, s_disj }, { s_tran, s_cont }, { s_disj, s_disj }, };
void pick(sint n, sint step, sint state, char **v, unsigned long bits) { int i, b; if (step == n) { if (state != s_disj) return; for (i = 0, b = 1; i < n; i++, b <<= 1) if ((b & bits)) printf("%s ", v[i]); putchar('\n'); return; }
bits <<= 1; pick(n, step + 1, tbl[state][0], v, bits); /* no pick */ pick(n, step + 1, tbl[state][1], v, bits | 1); /* pick */ }
int main(int c, char **v) { if (c - 1 >= sizeof(unsigned long) * 4) printf("Too many items"); else pick(c - 1, 0, s_blnk, v + 1, 0); return 0; }</lang>running it:
% ./a.out 1 2 3 4 1 3 1 4 2 4 1 2 4 1 3 4 % ./a.out 1 2 3 4 5 6 7 8 9 0 | wc -l 968
C++
<lang cpp>/* Not best code, wrote it really quick. Will add updated code using more C++11 features soon!
- /
- include <iostream>
- include <vector>
- include <algorithm>
- include <iterator>
int main() { std::vector<int> sequence = { 0, 1, 2, 3, 4 }; std::vector<int> work = { 0 }; std::vector<std::vector<int>> results; while (work != sequence) { bool zeroed = false; size_t index_zero_started = 0; for (size_t i = 0; i < work.size(); ++i) { if (work[i] >= sequence.size()) { if (i == 0) { work[i] = 0; work.push_back(0); index_zero_started = i; } else { ++work[i - 1]; for (size_t j = i; j < work.size(); ++j) { work[j] = 0; } index_zero_started = i - 1; } zeroed = true; break; } } if (zeroed) { for (size_t i = index_zero_started + 1; i < work.size(); ++i) { work[i] = work[i - 1] + 1; } } else { std::vector<int> temp_differences; std::adjacent_difference(std::begin(work), std::end(work), std::back_inserter(temp_differences)); if (std::find_if(std::begin(temp_differences) + 1, std::end(temp_differences), [](const int& n) { return n > 1; }) != std::end(temp_differences)) { results.push_back(work); } ++work.back(); } } std::cout << "Non-continuous subsequences of "; std::copy(std::begin(sequence), std::end(sequence), std::ostream_iterator<int>(std::cout, " ")); std::cout << std::endl; for (auto& e: results) { std::cout << "- "; std::copy(std::begin(e), std::end(e), std::ostream_iterator<int>(std::cout, " ")); std::cout << std::endl; } return 0; }</lang>
- Output:
Non-continuous subsequences of 0 1 2 3 4 - 0 2 - 0 3 - 0 4 - 1 3 - 1 4 - 2 4 - 0 1 3 - 0 1 4 - 0 2 3 - 0 2 4 - 0 3 4 - 1 2 4 - 1 3 4 - 0 1 2 4 - 0 1 3 4 - 0 2 3 4
Clojure
Here's a simple approach that uses the clojure.contrib.combinatorics library to generate subsequences, and then filters out the continuous subsequences using a naïve subseq test:
<lang lisp> (use '[clojure.contrib.combinatorics :only (subsets)])
(defn of-min-length [min-length]
(fn [s] (>= (count s) min-length)))
(defn runs [c l]
(map (partial take l) (take-while not-empty (iterate rest c))))
(defn is-subseq? [c sub]
(some identity (map = (runs c (count sub)) (repeat sub))))
(defn non-continuous-subsequences [s]
(filter (complement (partial is-subseq? s)) (subsets s)))
(filter (of-min-length 2) (non-continuous-subsequences [:a :b :c :d]))
</lang>
CoffeeScript
Use binary bitmasks to enumerate our sequences. <lang coffeescript> is_contigous_binary = (n) ->
# return true if binary representation of n is # of the form 1+0+ # examples: # 0 true # 1 true # 100 true # 110 true # 1001 false # 1010 false
# special case zero, or you'll get an infinite loop later return true if n == 0
# first remove 0s from end while n % 2 == 0 n = n / 2 # next, take advantage of the fact that a continuous # run of 1s would be of the form 2^n - 1 is_power_of_two(n + 1)
is_power_of_two = (m) ->
while m % 2 == 0 m = m / 2 m == 1
seq_from_bitmap = (arr, n) ->
# grabs elements from array according to a bitmap
# e.g. if n == 13 (1101), and arr = ['a', 'b', 'c', 'd'],
# then return ['a', 'c', 'd'] (flipping bits to 1011, so
# that least significant bit comes first)
i = 0
new_arr = []
while n > 0
if n % 2 == 1
new_arr.push arr[i]
n -= 1
n /= 2
i += 1
new_arr
non_contig_subsequences = (arr) ->
# Return all subsqeuences from an array that have a "hole" in # them. The order of the subsequences is not specified here. # This algorithm uses binary counting, so it is limited to # small lists, but large lists would be unwieldy regardless. bitmasks = [0...Math.pow(2, arr.length)] (seq_from_bitmap arr, n for n in bitmasks when !is_contigous_binary n)
arr = [1,2,3,4] console.log non_contig_subsequences arr for n in [1..10]
arr = [1..n]
num_solutions = non_contig_subsequences(arr).length
console.log "for n=#{n} there are #{num_solutions} solutions"
</lang>
- Output:
> coffee non_contig_subseq.coffee [ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 1, 2, 4 ], [ 1, 3, 4 ] ] for n=1 there are 0 solutions for n=2 there are 0 solutions for n=3 there are 1 solutions for n=4 there are 5 solutions for n=5 there are 16 solutions for n=6 there are 42 solutions for n=7 there are 99 solutions for n=8 there are 219 solutions for n=9 there are 466 solutions for n=10 there are 968 solutions
Common Lisp
<lang lisp>(defun all-subsequences (list)
(labels ((subsequences (tail &optional (acc '()) (result '()))
"Return a list of the subsequence designators of the
subsequences of tail. Each subsequence designator is a
list of tails of tail, the subsequence being the first
element of each tail."
(if (endp tail)
(list* (reverse acc) result)
(subsequences (rest tail) (list* tail acc)
(append (subsequences (rest tail) acc) result))))
(continuous-p (subsequence-d)
"True if the designated subsequence is continuous."
(loop for i in subsequence-d
for j on (first subsequence-d)
always (eq i j)))
(designated-sequence (subsequence-d)
"Destructively transforms a subsequence designator into
the designated subsequence."
(map-into subsequence-d 'first subsequence-d)))
(let ((nc-subsequences (delete-if #'continuous-p (subsequences list))))
(map-into nc-subsequences #'designated-sequence nc-subsequences))))</lang>
<lang lisp>(defun all-subsequences2 (list)
(labels ((recurse (s list)
(if (endp list)
(if (>= s 3)
'(())
'())
(let ((x (car list))
(xs (cdr list)))
(if (evenp s)
(append (mapcar (lambda (ys) (cons x ys))
(recurse (+ s 1) xs))
(recurse s xs))
(append (mapcar (lambda (ys) (cons x ys))
(recurse s xs))
(recurse (+ s 1) xs)))))))
(recurse 0 list)))</lang>
D
Recursive Version
<lang d>T[][] ncsub(T)(in T[] seq, in uint s=0) pure nothrow @safe {
if (seq.length) {
typeof(return) aux;
foreach (ys; ncsub(seq[1 .. $], s + !(s % 2)))
aux ~= seq[0] ~ ys;
return aux ~ ncsub(seq[1 .. $], s + s % 2);
} else
return new typeof(return)(s >= 3, 0);
}
void main() @safe {
import std.stdio;
[1, 2, 3].ncsub.writeln;
[1, 2, 3, 4].ncsub.writeln;
foreach (const nc; [1, 2, 3, 4, 5].ncsub)
nc.writeln;
}</lang>
- Output:
[[1, 3]] [[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]] [1, 2, 3, 5] [1, 2, 4, 5] [1, 2, 4] [1, 2, 5] [1, 3, 4, 5] [1, 3, 4] [1, 3, 5] [1, 3] [1, 4, 5] [1, 4] [1, 5] [2, 3, 5] [2, 4, 5] [2, 4] [2, 5] [3, 5]
Faster Lazy Version
This version doesn't copy the sub-arrays. <lang d>struct Ncsub(T) {
T[] seq;
int opApply(int delegate(ref T[]) dg) const {
immutable n = seq.length;
int result;
auto S = new T[n];
OUTER: foreach (immutable i; 1 .. 1 << n) {
uint lenS;
bool nc = false;
foreach (immutable j; 0 .. n + 1) {
immutable k = i >> j;
if (k == 0) {
if (nc) {
auto auxS = S[0 .. lenS];
result = dg(auxS);
if (result)
break OUTER;
}
break;
} else if (k % 2) {
S[lenS] = seq[j];
lenS++;
} else if (lenS)
nc = true;
}
}
return result; }
}
void main() {
import std.array, std.range;
//assert(24.iota.array.Ncsub!int.walkLength == 16_776_915);
auto r = 24.iota.array;
uint counter = 0;
foreach (s; Ncsub!int(r))
counter++;
assert(counter == 16_776_915);
}</lang>
Generator Version
This version doesn't copy the sub-arrays, and it's a little slower than the opApply-based version. <lang d>import std.stdio, std.array, std.range, std.concurrency;
Generator!(T[]) ncsub(T)(in T[] seq) {
return new typeof(return)({
immutable n = seq.length;
auto S = new T[n];
foreach (immutable i; 1 .. 1 << n) {
uint lenS = 0;
bool nc = false;
foreach (immutable j; 0 .. n + 1) {
immutable k = i >> j;
if (k == 0) {
if (nc)
yield(S[0 .. lenS]);
break;
} else if (k % 2) {
S[lenS] = seq[j];
lenS++;
} else if (lenS)
nc = true;
}
}
});
}
void main() {
assert(24.iota.array.ncsub.walkLength == 16_776_915);
[1, 2, 3].ncsub.writeln;
[1, 2, 3, 4].ncsub.writeln;
foreach (const nc; [1, 2, 3, 4, 5].ncsub)
nc.writeln;
}</lang>
Elixir
<lang elixir>defmodule RC do
defp masks(n) do
maxmask = trunc(:math.pow(2, n)) - 1
Enum.map(3..maxmask, &Integer.to_string(&1, 2))
|> Enum.filter_map(&contains_noncont(&1), &String.rjust(&1, n, ?0)) # padding
end
defp contains_noncont(n) do
Regex.match?(~r/10+1/, n)
end
defp apply_mask_to_list(mask, list) do
Enum.zip(to_char_list(mask), list)
|> Enum.filter_map(fn {include, _} -> include > ?0 end, fn {_, value} -> value end)
end
def ncs(list) do Enum.map(masks(length(list)), fn mask -> apply_mask_to_list(mask, list) end) end
end
IO.inspect RC.ncs([1,2,3]) IO.inspect RC.ncs([1,2,3,4]) IO.inspect RC.ncs('abcd')</lang>
- Output:
[[1, 3]] [[2, 4], [1, 4], [1, 3], [1, 3, 4], [1, 2, 4]] ['bd', 'ad', 'ac', 'acd', 'abd']
Erlang
Erlang's not optimized for strings or math, so this is pretty inefficient. Nonetheless, it works by generating the set of all possible "bitmasks" (represented as strings), filters for those with non-continuous subsequences, and maps from that set over the list. One immediate point for optimization that would complicate the code a bit would be to compile the regular expression, the problem being where you'd put it.
<lang erlang>-module(rosetta). -export([ncs/1]).
masks(N) ->
MaxMask = trunc(math:pow(2, N)),
Total = lists:map(fun(X) -> integer_to_list(X, 2) end,
lists:seq(3, MaxMask)),
Filtered = lists:filter(fun(X) -> contains_noncont(X) end, Total),
lists:map(fun(X) -> string:right(X, N, $0) end, Filtered). % padding
contains_noncont(N) ->
case re:run(N, "10+1") of
{match, _} -> true;
nomatch -> false
end.
apply_mask_to_list(Mask, List) ->
Zipped = lists:zip(Mask, List),
Filtered = lists:filter(fun({Include, _}) -> Include > 48 end, Zipped),
lists:map(fun({_, Value}) -> Value end, Filtered).
ncs(List) ->
lists:map(fun(Mask) -> apply_mask_to_list(Mask, List) end,
masks(length(List))).</lang>
- Output:
Eshell V5.10.1 (abort with ^G)
1> c(rosetta).
{ok,rosetta}
2> rosetta:ncs([1,2,3,4]).
[[2,4],[1,4],[1,3],[1,3,4],[1,2,4]]
Go
Generate the power set (power sequence, actually) with a recursive function, but keep track of the state of the subsequence on the way down. When you get to the bottom, if state == non-continuous, then include the subsequence. It's just filtering merged in with generation. <lang go>package main
import "fmt"
const ( // state:
m = iota // missing: all elements missing so far c // continuous: all elements included so far are continuous cm // one or more continuous followed by one or more missing cmc // non-continuous subsequence
)
func ncs(s []int) [][]int {
if len(s) < 3 {
return nil
}
return append(n2(nil, s[1:], m), n2([]int{s[0]}, s[1:], c)...)
}
var skip = []int{m, cm, cm, cmc} var incl = []int{c, c, cmc, cmc}
func n2(ss, tail []int, seq int) [][]int {
if len(tail) == 0 {
if seq != cmc {
return nil
}
return [][]int{ss}
}
return append(n2(append([]int{}, ss...), tail[1:], skip[seq]),
n2(append(ss, tail[0]), tail[1:], incl[seq])...)
}
func main() {
ss := ncs([]int{1, 2, 3, 4})
fmt.Println(len(ss), "non-continuous subsequences:")
for _, s := range ss {
fmt.Println(" ", s)
}
}</lang>
- Output:
5 non-continuous subsequences: [2 4] [1 4] [1 3] [1 3 4] [1 2 4]
Haskell
Generalized monadic filter
<lang haskell>action p x = if p x then succ x else x
fenceM p q s [] = guard (q s) >> return [] fenceM p q s (x:xs) = do
(f,g) <- p ys <- fenceM p q (g s) xs return $ f x ys
ncsubseq = fenceM [((:), action even), (flip const, action odd)] (>= 3) 0</lang>
- Output:
*Main> ncsubseq [1..3] [[1,3]] *Main> ncsubseq [1..4] [[1,2,4],[1,3,4],[1,3],[1,4],[2,4]] *Main> ncsubseq [1..5] [[1,2,3,5],[1,2,4,5],[1,2,4],[1,2,5],[1,3,4,5],[1,3,4],[1,3,5],[1,3],[1,4,5],[1,4],[1,5],[2,3,5],[2,4,5],[2,4],[2,5],[3,5]]
Filtered templates
This implementation works by computing templates of all possible subsequences of the given length of sequence, discarding the continuous ones, then applying the remaining templates to the input list.
<lang haskell>continuous = null . dropWhile not . dropWhile id . dropWhile not ncs xs = map (map fst . filter snd . zip xs) $
filter (not . continuous) $
mapM (const [True,False]) xs</lang>
Recursive
Recursive method with powerset as helper function.
<lang haskell>import Data.List
poset = foldr (\x p -> p ++ map (x:) p) [[]]
ncsubs [] = [[]] ncsubs (x:xs) = tail $ nc [x] xs
where nc [_] [] = [[]] nc (_:x:xs) [] = nc [x] xs nc xs (y:ys) = (nc (xs++[y]) ys) ++ map (xs++) (tail $ poset ys)</lang>
- Output:
*Main> ncsubs "aaa" ["aa"] (0.00 secs, 0 bytes) *Main> ncsubs [9..12] [[10,12],[9,10,12],[9,12],[9,11],[9,11,12]] (0.00 secs, 522544 bytes) *Main> ncsubs [] [[]] (0.00 secs, 0 bytes) *Main> ncsubs [1] [] (0.00 secs, 0 bytes)
A disjointed subsequence is a consecutive subsequence followed by a gap, then by any nonempty subsequence to its right: <lang haskell>import Data.List (subsequences, tails, delete)
disjoint a = concatMap (cutAt a) [1..length a - 2] where cutAt s n = [a ++ b | b <- delete [] (subsequences right), a <- init (tails left) ] where (left, _:right) = splitAt n s
main = print $ length $ disjoint [1..20]</lang>
Build a lexicographic list of consecutive subsequences, and a list of all subsequences, then subtract one from the other: <lang haskell>import Data.List (inits, tails)
subseqs = foldr (\x s -> [x] : map (x:) s ++ s) []
consecs = concatMap (tail.inits) . tails
minus [] [] = [] minus (a:as) bb@(b:bs) | a == b = minus as bs | otherwise = a:minus as bb
disjoint s = (subseqs s) `minus` (consecs s)
main = mapM_ print $ disjoint [1..4]</lang>
J
We select those combinations where the end of the first continuous subsequence appears before the start of the last continuous subsequence:
<lang J>allmasks=: 2 #:@i.@^ # firstend=:1 0 i.&1@E."1 ] laststart=: 0 1 {:@I.@E."1 ] noncont=: <@#~ (#~ firstend < laststart)@allmasks</lang>
Example use: <lang J> noncont 1+i.4 ┌───┬───┬───┬─────┬─────┐ │2 4│1 4│1 3│1 3 4│1 2 4│ └───┴───┴───┴─────┴─────┘
noncont 'aeiou'
┌──┬──┬──┬───┬───┬──┬──┬───┬──┬───┬───┬────┬───┬───┬────┬────┐ │iu│eu│eo│eou│eiu│au│ao│aou│ai│aiu│aio│aiou│aeu│aeo│aeou│aeiu│ └──┴──┴──┴───┴───┴──┴──┴───┴──┴───┴───┴────┴───┴───┴────┴────┘
#noncont i.10
968</lang>
Alternatively, since there are relatively few continuous sequences, we could specifically exclude them:
<lang J>contmasks=: a: ;@, 1 <:/~@i.&.>@i.@+ # noncont=: <@#~ (allmasks -. contmasks)</lang>
(we get the same behavior from this implementation)
Java
<lang java>public class NonContinuousSubsequences {
public static void main(String args[]) {
seqR("1234", "", 0, 0);
}
private static void seqR(String s, String c, int i, int added) {
if (i == s.length()) {
if (c.trim().length() > added)
System.out.println(c);
} else {
seqR(s, c + s.charAt(i), i + 1, added + 1);
seqR(s, c + ' ', i + 1, added);
}
}
}</lang>
12 4 1 34 1 3 1 4 2 4
JavaScript
Uses powerset() function from here. Uses a JSON stringifier from http://www.json.org/js.html
<lang javascript>function non_continuous_subsequences(ary) {
var non_continuous = new Array();
for (var i = 0; i < ary.length; i++) {
if (! is_array_continuous(ary[i])) {
non_continuous.push(ary[i]);
}
}
return non_continuous;
}
function is_array_continuous(ary) {
if (ary.length < 2)
return true;
for (var j = 1; j < ary.length; j++) {
if (ary[j] - ary[j-1] != 1) {
return false;
}
}
return true;
}
load('json2.js'); /* http://www.json.org/js.html */
print(JSON.stringify( non_continuous_subsequences( powerset([1,2,3,4]))));</lang>
- Output:
[[1,3],[1,4],[2,4],[1,2,4],[1,3,4]]
jq
In order to handle arrays of more than a handful of elements, we define non_continuous_subsequences/0 as a generator; that is, it produces a stream of arrays, each of which is a non-continuous subsequence of the given sequence.
Since the non-continuous subsequences are dense in the set of all subsets, we will use the powerset approach, and accordingly begin by defining subsets/0 as a generator. <lang jq># Generate a stream of subsets of the input array def subsets:
if length == 0 then [] else .[0] as $first | (.[1:] | subsets) | ., ([$first] + .) end ;
- Generate a stream of non-continuous indices in the range 0 <= i < .
def non_continuous_indices:
[range(0;.)] | subsets | select(length > 1 and length != 1 + .[length-1] - .[0]) ;
def non_continuous_subsequences:
(length | non_continuous_indices) as $ix | [.[ $ix[] ]] ;</lang>
Example: To show that the above approach can be used for relatively large n, let us count the number of non-continuous subsequences of [0, 1, ..., 19]. <lang jq>def count(f): reduce f as $i (0; . + 1);
count( [range(0;20)] | non_continuous_subsequences) </lang>
- Output:
$ jq -n -f powerset_generator.jq 1048365
Julia
This solution uses an iterator over non-contiguous sub-sequences, NCSubSeq. In the spirit of Julia's permutations and combinations built-ins, NCSubSeq provides an array of indices that can be used to create each subsequence from the full sequence. Sub-sequences are indexed by integers whose bit patterns indicate which members are included.
NCSubSeq works by filtering indices according to whether all 1s in these indices have bit pattern that are contiguous (using the iscontseq functions). This is an easy to implement approach. Greater efficiency might be achieved by exploiting the property that a sequence is contiguous if and only if its index is a difference of two powers of 2. This property is used to create the length(NCSubSeq(n)) function, which gives the number of non-contiguous sub-sequences of a sequence of length n.
NCSubSeq works transparently for sequence lengths up to WORD_SIZE-1 (typically 63). It can be extended to work for longer sequences by casting n to a larger integer, e.g. using Big(n). A more polished implementation would handle this extension behind the scenes.
Iterator and Functions <lang Julia> iscontseq(n::Integer) = count_zeros(n) == leading_zeros(n) + trailing_zeros(n) iscontseq(n::BigInt) = !ismatch(r"0", rstrip(bin(n), '0'))
function makeint2seq(n::Integer)
const idex = collect(1:n)
function int2seq(m::Integer)
d = digits(m, 2, n)
idex[d .== 1]
end
return int2seq
end
immutable NCSubSeq{T<:Integer}
n::T
end
type NCSubState{T<:Integer}
m::T m2s::Function
end
Base.length(a::NCSubSeq) = 2^a.n - div(a.n*(a.n+1), 2) - 1 Base.start(a::NCSubSeq) = NCSubState(5, makeint2seq(a.n)) Base.done(a::NCSubSeq, as::NCSubState) = 2^a.n-3 < as.m
function Base.next(a::NCSubSeq, as::NCSubState)
s = as.m2s(as.m)
as.m += 1
while iscontseq(as.m)
as.m += 1
end
return (s, as)
end </lang>
Main <lang Julia> n = 4 print("Testing NCSubSeq for ", n, " items:\n ") for a in NCSubSeq(n)
print(" ", a)
end println()
s = "Rosetta" cs = split(s, "") m = 10 n = length(NCSubSeq(length(s))) - m println() println("The first and last ", m, " NC sub-sequences of \"", s, "\":") for (i,a) in enumerate(NCSubSeq(length(cs)))
i <= m || n < i || continue
println(@sprintf "%6d %s" i join(cs[a], ""))
i == m || continue
println(" .. ......")
end
t = {} append!(t, collect(1:10)) append!(t, collect(20:10:40)) append!(t, big(50):50:200) println() println("Numbers of NC sub-sequences of a given length:") for i in t
println(@sprintf("%7d => ", i), length(NCSubSeq(i)))
end </lang>
- Output:
Testing NCSubSeq for 4 items:
[1,3] [1,4] [2,4] [1,2,4] [1,3,4]
The first and last 10 NC sub-sequences of "Rosetta":
1 Rs
2 Re
3 oe
4 Roe
5 Rse
6 Rt
7 ot
8 Rot
9 st
10 Rst
.. ......
90 otta
91 Rotta
92 stta
93 Rstta
94 ostta
95 Rostta
96 Retta
97 oetta
98 Roetta
99 Rsetta
Numbers of NC sub-sequences of a given length:
1 => 0
2 => 0
3 => 1
4 => 5
5 => 16
6 => 42
7 => 99
8 => 219
9 => 466
10 => 968
20 => 1048365
30 => 1073741358
40 => 1099511626955
50 => 1125899906841348
100 => 1267650600228229401496703200325
150 => 1427247692705959881058285969449495136382735298
200 => 1606938044258990275541962092341162602522202993782792835281275
Kotlin
<lang scala>// version 1.1.2
fun <T> ncs(a: Array<T>) {
fun generate(m: Int, k: Int, c: IntArray) {
if (k == m) {
if (c[m - 1] != c[0] + m - 1) {
for (i in 0 until m) print("${a[c[i]]} ")
println()
}
}
else {
for (j in 0 until a.size) {
if (k == 0 || j > c[k - 1]) {
c[k] = j
generate(m, k + 1, c)
}
}
}
}
for (m in 2 until a.size) {
val c = IntArray(m)
generate(m, 0, c)
}
}
fun main(args: Array<String>) {
val a = arrayOf(1, 2, 3, 4)
ncs(a)
println()
val ca = arrayOf('a', 'b', 'c', 'd', 'e')
ncs(ca)
}</lang>
- Output:
1 3 1 4 2 4 1 2 4 1 3 4 a c a d a e b d b e c e a b d a b e a c d a c e a d e b c e b d e a b c e a b d e a c d e
Mathematica
We make all the subsets then filter out the continuous ones:
<lang Mathematica>GoodBad[i_List]:=Not[MatchQ[Differences[i],{1..}|{}]] n=5 Select[Subsets[Range[n]],GoodBad]</lang>
gives back:
<lang Mathematica> {{1,3},{1,4},{1,5},{2,4},{2,5},{3,5},{1,2,4},{1,2,5},{1,3,4},{1,3,5},{1,4,5},{2,3,5},{2,4,5},{1,2,3,5},{1,2,4,5},{1,3,4,5}}</lang>
Nim
<lang nim>import sequtils
proc ncsub[T](se: seq[T], s = 0): seq[seq[T]] =
result = @[]
if se.len > 0:
let
x = se[0..0]
xs = se[1 .. -1]
p2 = s mod 2
p1 = (s + 1) mod 2
for ys in ncsub(xs, s + p1):
result.add(x & ys)
result.add(ncsub(xs, s + p2))
elif s >= 3:
result.add(@[])
echo "ncsub(", toSeq 1.. 3, ") = ", ncsub(toSeq 1..3) echo "ncsub(", toSeq 1.. 4, ") = ", ncsub(toSeq 1..4) echo "ncsub(", toSeq 1.. 5, ") = ", ncsub(toSeq 1..5)</lang>
- Output:
ncsub(@[1, 2, 3]) = @[@[1, 3]] ncsub(@[1, 2, 3, 4]) = @[@[1, 2, 4], @[1, 3, 4], @[1, 3], @[1, 4], @[2, 4]] ncsub(@[1, 2, 3, 4, 5]) = @[@[1, 2, 3, 5], @[1, 2, 4, 5], @[1, 2, 4], @[1, 2, 5], @[1, 3, 4, 5], @[1, 3, 4], @[1, 3, 5], @[1, 3], @[1, 4, 5], @[1, 4], @[1, 5], @[2, 3, 5], @[2, 4, 5], @[2, 4], @[2, 5], @[3, 5]]
OCaml
<lang ocaml>let rec fence s = function
[] ->
if s >= 3 then
[[]]
else
[]
| x :: xs ->
if s mod 2 = 0 then
List.map
(fun ys -> x :: ys)
(fence (s + 1) xs)
@
fence s xs
else
List.map
(fun ys -> x :: ys)
(fence s xs)
@
fence (s + 1) xs
let ncsubseq = fence 0</lang>
- Output:
# ncsubseq [1;2;3];; - : int list list = [[1; 3]] # ncsubseq [1;2;3;4];; - : int list list = [[1; 2; 4]; [1; 3; 4]; [1; 3]; [1; 4]; [2; 4]] # ncsubseq [1;2;3;4;5];; - : int list list = [[1; 2; 3; 5]; [1; 2; 4; 5]; [1; 2; 4]; [1; 2; 5]; [1; 3; 4; 5]; [1; 3; 4]; [1; 3; 5]; [1; 3]; [1; 4; 5]; [1; 4]; [1; 5]; [2; 3; 5]; [2; 4; 5]; [2; 4]; [2; 5]; [3; 5]]
Oz
A nice application of finite set constraints. We just describe what we want and the constraint system will deliver it: <lang oz>declare
fun {NCSubseq SeqList}
Seq = {FS.value.make SeqList}
proc {Script Result}
%% the result is a subset of Seq
{FS.subset Result Seq}
%% at least one element of Seq is missing
local Gap in
{FS.include Gap Seq}
{FS.exclude Gap Result}
%% and this element is between the smallest
%% and the largest elements of the subsequence
Gap >: {FS.int.min Result}
Gap <: {FS.int.max Result}
end
%% enumerate all such sets
{FS.distribute naive [Result]}
end
in
{Map {SearchAll Script} FS.reflect.lowerBoundList}
end
in
{Inspect {NCSubseq [1 2 3 4]}}</lang>
PARI/GP
Just a simple script, but it's I/O bound so efficiency isn't a concern. (Almost all subsequences are non-contiguous so looping over all possibilities isn't that bad. For length 20 about 99.98% of subsequences are non-contiguous.) <lang parigp>noncontig(n)=n>>=valuation(n,2);n++;n>>=valuation(n,2);n>1; nonContigSubseq(v)={
for(i=5,2^#v-1,
if(noncontig(i),
print(vecextract(v,i))
)
)
}; nonContigSubseq([1,2,3]) nonContigSubseq(["a","b","c","d","e"])</lang>
- Output:
[1, 3] ["a", "c"] ["a", "d"] ["b", "d"] ["a", "b", "d"] ["a", "c", "d"] ["a", "e"] ["b", "e"] ["a", "b", "e"] ["c", "e"] ["a", "c", "e"] ["b", "c", "e"] ["a", "b", "c", "e"] ["a", "d", "e"] ["b", "d", "e"] ["a", "b", "d", "e"] ["a", "c", "d", "e"]
Perl
<lang perl>my ($max, @current); sub non_continuous {
my ($idx, $has_gap, $found) = @_;
for ($idx .. $max) {
push @current, $_;
# print "@current\n" if $has_gap; # uncomment for huge output
$found ++ if $has_gap;
$found += non_continuous($_ + 1, $has_gap) if $_ < $max;
pop @current;
$has_gap = @current; # don't set gap flag if it's empty still
}
$found;
}
$max = 20; # 1048365 sequences, 10 seconds-ish print "found ", non_continuous(1), " sequences\n";</lang>
Perl 6
<lang perl6>sub non_continuous_subsequences ( *@list ) {
@list.combinations.grep: { 1 != all( .[ 0 ^.. .end] Z- .[0 ..^ .end] ) }
}
say non_continuous_subsequences( 1..3 )».gist; say non_continuous_subsequences( 1..4 )».gist; say non_continuous_subsequences( ^4 ).map: {[<a b c d>[.list]].gist};</lang>
- Output:
((1 3)) ((1 3) (1 4) (2 4) (1 2 4) (1 3 4)) ([a c] [a d] [b d] [a b d] [a c d])
Phix
Straightforward recursive implementation, the only minor trick is that a gap does not
mean non-contiguous until you actually take something later.
Counts non-contiguous subsequences of sequences 1..20 in just over 0.5 seconds
<lang Phix>bool countonly = false
integer count = 0
procedure ncs(sequence rest, integer ri=0, sequence taken={}, bool contig=false, bool gap=false)
if ri>=length(rest) then
if contig then
if countonly then
count += 1
else
?taken
end if
end if
else
ri += 1
ncs(rest,ri,taken&rest[ri],gap,gap)
ncs(rest,ri,taken,contig,length(taken)!=0)
end if
end procedure
ncs({1,2,3}) ?"===" ncs({1,2,3,4}) ?"===" countonly = true atom t0 = time() sequence s = {} for i=1 to 20 do
count = 0 ncs(tagset(i)) s = append(s,count)
end for ?time()-t0 ?s</lang>
- Output:
{1,3}
"==="
{1,2,4}
{1,3,4}
{1,3}
{1,4}
{2,4}
"==="
0.515
{0,0,1,5,16,42,99,219,466,968,1981,4017,8100,16278,32647,65399,130918,261972,524097,1048365}
PicoLisp
<lang PicoLisp>(de ncsubseq (Lst)
(let S 0
(recur (S Lst)
(ifn Lst
(and (>= S 3) '(NIL))
(let (X (car Lst) XS (cdr Lst))
(ifn (bit? 1 S) # even
(conc
(mapcar '((YS) (cons X YS))
(recurse (inc S) XS) )
(recurse S XS) )
(conc
(mapcar '((YS) (cons X YS))
(recurse S XS) )
(recurse (inc S) XS) ) ) ) ) ) ) )</lang>
Pop11
We modify classical recursive generation of subsets, using variables to keep track if subsequence is continuous.
<lang pop11>define ncsubseq(l);
lvars acc = [], gap_started = false, is_continuous = true;
define do_it(l1, l2);
dlocal gap_started;
lvars el, save_is_continuous = is_continuous;
if l2 = [] then
if not(is_continuous) then
cons(l1, acc) -> acc;
endif;
else
front(l2) -> el;
back(l2) -> l2;
not(gap_started) and is_continuous -> is_continuous;
do_it(cons(el, l1), l2);
save_is_continuous -> is_continuous;
not(l1 = []) or gap_started -> gap_started;
do_it(l1, l2);
endif;
enddefine;
do_it([], rev(l));
acc;
enddefine;
ncsubseq([1 2 3 4 5]) =></lang>
- Output:
[[1 3] [1 4] [2 4] [1 2 4] [1 3 4] [1 5] [2 5] [1 2 5] [3 5] [1 3 5]
[2 3 5] [1 2 3 5] [1 4 5] [2 4 5] [1 2 4 5] [1 3 4 5]]
PowerShell
<lang PowerShell>Function SubSequence ( [Array] $S, [Boolean] $all=$false ) {
$sc = $S.count
if( $sc -gt ( 2 - [Int32] $all ) ) {
[void] $sc--
0..$sc | ForEach-Object {
$gap = $_
"$( $S[ $_ ] )"
if( $gap -lt $sc )
{
SubSequence ( ( $gap + 1 )..$sc | Where-Object { $_ -ne $gap } ) ( ( $gap -ne 0 ) -or $all ) | ForEach-Object {
[String]::Join( ',', ( ( [String]$_ ).Split(',') | ForEach-Object {
$lt = $true
} {
if( $lt -and ( $_ -gt $gap ) )
{
$S[ $gap ]
$lt = $false
}
$S[ $_ ]
} {
if( $lt )
{
$S[ $gap ]
}
}
) )
}
}
}
#[String]::Join( ',', $S)
} else {
$S | ForEach-Object { [String] $_ }
}
}
Function NonContinuous-SubSequence ( [Array] $S ) {
$sc = $S.count
if( $sc -eq 3 )
{
[String]::Join( ',', $S[ ( 0,2 ) ] )
} elseif ( $sc -gt 3 ) {
[void] $sc--
$gaps = @()
$gaps += ( ( NonContinuous-SubSequence ( 1..$sc ) ) | ForEach-Object {
$gap1 = ",$_,"
"0,{0}" -f ( [String]::Join( ',', ( 1..$sc | Where-Object { $gap1 -notmatch "$_," } ) ) )
} )
$gaps += 1..( $sc - 1 )
2..( $sc - 1 ) | ForEach-Object {
$gap2 = $_ - 1
$gaps += ( ( SubSequence ( $_..$sc ) ) | ForEach-Object {
"$gap2,$_"
} )
}
#Write-Host "S $S gaps $gaps"
$gaps | ForEach-Object {
$gap3 = ",$_,"
"$( 0..$sc | Where-Object { $gap3 -notmatch ",$_," } | ForEach-Object {
$S[$_]
} )" -replace ' ', ','
}
} else {
$null
}
}
( NonContinuous-SubSequence 'a','b','c','d','e' ) | Select-Object length, @{Name='value';Expression={ $_ } } | Sort-Object length, value | ForEach-Object { $_.value }</lang>
Prolog
Works with SWI-Prolog.
We explain to Prolog how to build a non continuous subsequence of a list L, then we ask Prolog to fetch all the subsequences.
<lang Prolog> % fetch all the subsequences ncsubs(L, LNCSL) :- setof(NCSL, one_ncsubs(L, NCSL), LNCSL).
% how to build one subsequence one_ncsubs(L, NCSL) :- extract_elem(L, NCSL); ( sublist(L, L1), one_ncsubs(L1, NCSL)).
% extract one element of the list % this element is neither the first nor the last. extract_elem(L, NCSL) :- length(L, Len), Len1 is Len - 2, between(1, Len1, I), nth0(I, L, Elem), select(Elem, L, NCS1), ( NCSL = NCS1; extract_elem(NCS1, NCSL)).
% extract the first or the last element of the list sublist(L, SL) :- (L = [_|SL]; reverse(L, [_|SL1]), reverse(SL1, SL)). </lang> Example : <lang Prolog>?- ncsubs([a,e,i,o,u], L). L = [[a,e,i,u],[a,e,o],[a,e,o,u],[a,e,u],[a,i],[a,i,o],[a,i,o,u],[a,i,u],[a,o],[a,o,u],[a,u],[e,i,u],[e,o],[e,o,u],[e,u],[i,u]]</lang>
Python
<lang python>def ncsub(seq, s=0):
if seq:
x = seq[:1]
xs = seq[1:]
p2 = s % 2
p1 = not p2
return [x + ys for ys in ncsub(xs, s + p1)] + ncsub(xs, s + p2)
else:
return [[]] if s >= 3 else []</lang>
- Output:
>>> ncsub(range(1, 4)) [[1, 3]] >>> ncsub(range(1, 5)) [[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]] >>> ncsub(range(1, 6)) [[1, 2, 3, 5], [1, 2, 4, 5], [1, 2, 4], [1, 2, 5], [1, 3, 4, 5], [1, 3, 4], [1, 3, 5], [1, 3], [1, 4, 5], [1, 4], [1, 5], [2, 3, 5], [2, 4, 5], [2, 4], [2, 5], [3, 5]]
A faster Python + Psyco JIT version:
<lang python>from sys import argv import psyco
def C(n, k):
result = 1
for d in xrange(1, k+1):
result *= n
n -= 1
result /= d
return result
nsubs = lambda n: sum(C(n, k) for k in xrange(3, n+1))
def ncsub(seq):
n = len(seq) result = [None] * nsubs(n) pos = 0
for i in xrange(1, 2 ** n):
S = []
nc = False
for j in xrange(n + 1):
k = i >> j
if k == 0:
if nc:
result[pos] = S
pos += 1
break
elif k % 2:
S.append(seq[j])
elif S:
nc = True
return result
from sys import argv import psyco psyco.full() n = 10 if len(argv) < 2 else int(argv[1]) print len( ncsub(range(1, n)) )</lang>
R
The idea behind this is to loop over the possible lengths of subsequence, finding all subsequences then discarding those which are continuous.
<lang r>ncsub <- function(x) {
n <- length(x)
a <- seq_len(n)
seqlist <- list()
for(i in 2:(n-1))
{
seqs <- combn(a, i) # Get all subseqs
ok <- apply(seqs, 2, function(x) any(diff(x)!=1)) # Find noncts ones
newseqs <- unlist(apply(seqs[,ok], 2, function(x) list(x)), recursive=FALSE) # Convert matrix to list of its columns
seqlist <- c(seqlist, newseqs) # Append to existing list
}
lapply(seqlist, function(index) x[index])
}
- Example usage
ncsub(1:4) ncsub(letters[1:5])</lang>
Racket
Take a simple subsets definition: <lang racket> (define (subsets l)
(if (null? l) '(())
(append (for/list ([l2 (subsets (cdr l))]) (cons (car l) l2))
(subsets (cdr l)))))
</lang> since the subsets are returned in their original order, it is also a sub-sequences function.
Now add to it a "state" counter which count one for each chunk of items included or excluded. It's always even when we're in an excluded chunk (including the beginning) and odd when we're including items -- increment it whenever we switch from one kind of chunk to the other. This means that we should only include subsequences where the state is 3 (included->excluded->included) or more. Note that this results in code that is similar to the "Generalized monadic filter" entry, except a little simpler.
<lang racket>
- lang racket
(define (non-continuous-subseqs l)
(let loop ([l l] [x 0])
(if (null? l) (if (>= x 3) '(()) '())
(append (for/list ([l2 (loop (cdr l) (if (even? x) (add1 x) x))])
(cons (car l) l2))
(loop (cdr l) (if (odd? x) (add1 x) x))))))
(non-continuous-subseqs '(1 2 3 4))
- => '((1 2 4) (1 3 4) (1 3) (1 4) (2 4))
</lang>
REXX
This REXX version also works with non-numeric (alphabetic) items (as well as numbers). <lang rexx>/*REXX program lists all the non─continuous subsequences (NCS), given a sequence. */ parse arg list /*obtain the arguments from the C. L. */ if list= | list==',' then list=1 2 3 4 5 /*Not specified? Then use the default.*/ say 'list=' space(list); say /*display the list to the terminal. */ w=words(list) /*W: is the number of items in list. */ $=left(123456789, w) /*build a string of decimal digits. */ tail=right($, max(0, w-2)) /*construct a fast tail for comparisons*/
- =0 /* [↓] L: length of Jth item. */
do j=13 to left($,1) || tail; L=length(j) /*step through list (using smart start)*/
if verify(j, $)\==0 then iterate /*Not one of the chosen (sequences) ? */
f=left(j,1) /*use the fist decimal digit of J. */
NCS=0 /*there isn't a non─continuous subseq. */
do k=2 to L; _=substr(j, k, 1) /*extract a single decimal digit of J.*/
if _ <= f then iterate j /*if next digit ≤, then skip this digit*/
if _ \== f+1 then NCS=1 /*it's OK as of now (that is, so far).*/
f=_ /*now have a new next decimal digit. */
end /*k*/
if \NCS then iterate /*not OK? Then skip this number (item)*/
#=#+1 /*Eureka! We found a number (or item).*/
@=; do m=1 for L /*build a sequence string to display. */
@=@ word(list, substr(j, m, 1)) /*pick off a number (item) to display. */
end /*m*/
say 'a non─continuous subsequence: ' @ /*show the non─continuous subsequence. */ end /*j*/
say if #==0 then #='no' /*make it look more gooder Angleshy. */ say # "non─continuous subsequence"s(#) 'were found.' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ s: if arg(1)==1 then return ; return word(arg(2) 's', 1) /*pluralizer.*/</lang> output when using the input: 1 2 3 4
list= 1 2 3 4 a non-continuous subsequence: 1 3 a non-continuous subsequence: 1 4 a non-continuous subsequence: 2 4 a non-continuous subsequence: 1 2 4 a non-continuous subsequence: 1 3 4 5 non-continuous subsequences were found.
output when using the following input: a e I o u
list= a e I o u a non-continuous subsequence: a I a non-continuous subsequence: a o a non-continuous subsequence: a u a non-continuous subsequence: e o a non-continuous subsequence: e u a non-continuous subsequence: I u a non-continuous subsequence: a e o a non-continuous subsequence: a e u a non-continuous subsequence: a I o a non-continuous subsequence: a I u a non-continuous subsequence: a o u a non-continuous subsequence: e I u a non-continuous subsequence: e o u a non-continuous subsequence: a e I u a non-continuous subsequence: a e o u a non-continuous subsequence: a I o u 16 non-continuous subsequences were found.
output when using the following [channel Islands (Great Britain)] as input: Alderney Guernsey Herm Jersey Sark
list= Alderney Guernsey Herm Jersey Sark a non-continuous subsequence: Alderney Herm a non-continuous subsequence: Alderney Jersey a non-continuous subsequence: Alderney Sark a non-continuous subsequence: Guernsey Jersey a non-continuous subsequence: Guernsey Sark a non-continuous subsequence: Herm Sark a non-continuous subsequence: Alderney Guernsey Jersey a non-continuous subsequence: Alderney Guernsey Sark a non-continuous subsequence: Alderney Herm Jersey a non-continuous subsequence: Alderney Herm Sark a non-continuous subsequence: Alderney Jersey Sark a non-continuous subsequence: Guernsey Herm Sark a non-continuous subsequence: Guernsey Jersey Sark a non-continuous subsequence: Alderney Guernsey Herm Sark a non-continuous subsequence: Alderney Guernsey Jersey Sark a non-continuous subsequence: Alderney Herm Jersey Sark 16 non-continuous subsequences were found.
output when using the following [six noble gases] as input: helium neon argon krypton xenon radon
list= helium neon argon krypton xenon radon a non-continuous subsequence: helium argon a non-continuous subsequence: helium krypton a non-continuous subsequence: helium xenon a non-continuous subsequence: helium radon a non-continuous subsequence: neon krypton a non-continuous subsequence: neon xenon a non-continuous subsequence: neon radon a non-continuous subsequence: argon xenon a non-continuous subsequence: argon radon a non-continuous subsequence: krypton radon a non-continuous subsequence: helium neon krypton a non-continuous subsequence: helium neon xenon a non-continuous subsequence: helium neon radon a non-continuous subsequence: helium argon krypton a non-continuous subsequence: helium argon xenon a non-continuous subsequence: helium argon radon a non-continuous subsequence: helium krypton xenon a non-continuous subsequence: helium krypton radon a non-continuous subsequence: helium xenon radon a non-continuous subsequence: neon argon xenon a non-continuous subsequence: neon argon radon a non-continuous subsequence: neon krypton xenon a non-continuous subsequence: neon krypton radon a non-continuous subsequence: neon xenon radon a non-continuous subsequence: argon krypton radon a non-continuous subsequence: argon xenon radon a non-continuous subsequence: helium neon argon xenon a non-continuous subsequence: helium neon argon radon a non-continuous subsequence: helium neon krypton xenon a non-continuous subsequence: helium neon krypton radon a non-continuous subsequence: helium neon xenon radon a non-continuous subsequence: helium argon krypton xenon a non-continuous subsequence: helium argon krypton radon a non-continuous subsequence: helium argon xenon radon a non-continuous subsequence: helium krypton xenon radon a non-continuous subsequence: neon argon krypton radon a non-continuous subsequence: neon argon xenon radon a non-continuous subsequence: neon krypton xenon radon a non-continuous subsequence: helium neon argon krypton radon a non-continuous subsequence: helium neon argon xenon radon a non-continuous subsequence: helium neon krypton xenon radon a non-continuous subsequence: helium argon krypton xenon radon 42 non-continuous subsequences were found.
Ruby
Uses code from Power Set.
<lang ruby>class Array
def func_power_set
inject([[]]) { |ps,item| # for each item in the Array
ps + # take the powerset up to now and add
ps.map { |e| e + [item] } # it again, with the item appended to each element
}
end
def non_continuous_subsequences
func_power_set.reject {|seq| continuous?(seq)}
end
def continuous?(seq)
seq.each_cons(2) {|a, b| return false if a.succ != b}
true
end
end
p (1..3).to_a.non_continuous_subsequences p (1..4).to_a.non_continuous_subsequences p (1..5).to_a.non_continuous_subsequences p ("a".."d").to_a.non_continuous_subsequences</lang>
- Output:
[[1, 3]] [[1, 3], [1, 4], [2, 4], [1, 2, 4], [1, 3, 4]] [[1, 3], [1, 4], [2, 4], [1, 2, 4], [1, 3, 4], [1, 5], [2, 5], [1, 2, 5], [3, 5], [1, 3, 5], [2, 3, 5], [1, 2, 3, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5], [1, 3, 4, 5]] [["a", "c"], ["a", "d"], ["b", "d"], ["a", "b", "d"], ["a", "c", "d"]]
It is not the value of the array element and when judging continuation in the position, it changes as follows. <lang ruby>class Array
def continuous?(seq)
seq.each_cons(2) {|a, b| return false if index(a)+1 != index(b)}
true
end
end
p %w(a e i o u).non_continuous_subsequences</lang>
- Output:
[["a", "i"], ["a", "o"], ["e", "o"], ["a", "e", "o"], ["a", "i", "o"], ["a", "u"], ["e", "u"], ["a", "e", "u"], ["i", "u"], ["a", "i", "u"], ["e", "i", "u"], ["a", "e", "i", "u"], ["a", "o", "u"], ["e", "o", "u"], ["a", "e", "o", "u"], ["a", "i", "o", "u"]]
Scheme
<lang scheme>(define (ncsubseq lst)
(let recurse ((s 0)
(lst lst))
(if (null? lst)
(if (>= s 3)
'(())
'())
(let ((x (car lst))
(xs (cdr lst)))
(if (even? s)
(append
(map (lambda (ys) (cons x ys))
(recurse (+ s 1) xs))
(recurse s xs))
(append
(map (lambda (ys) (cons x ys))
(recurse s xs))
(recurse (+ s 1) xs)))))))</lang>
- Output:
> (ncsubseq '(1 2 3)) ((1 3)) > (ncsubseq '(1 2 3 4)) ((1 2 4) (1 3 4) (1 3) (1 4) (2 4)) > (ncsubseq '(1 2 3 4 5)) ((1 2 3 5) (1 2 4 5) (1 2 4) (1 2 5) (1 3 4 5) (1 3 4) (1 3 5) (1 3) (1 4 5) (1 4) (1 5) (2 3 5) (2 4 5) (2 4) (2 5) (3 5))
Seed7
<lang seed7>$ include "seed7_05.s7i";
const func array bitset: ncsub (in bitset: seq, in integer: s) is func
result
var array bitset: subseq is 0 times {};
local
var bitset: x is {};
var bitset: xs is {};
var bitset: ys is {};
begin
if seq <> {} then
x := {min(seq)};
xs := seq - x;
for ys range ncsub(xs, s + 1 - s rem 2) do
subseq &:= x | ys;
end for;
subseq &:= ncsub(xs, s + s rem 2);
elsif s >= 3 then
subseq &:= {};
end if;
end func;
const proc: main is func
local
var bitset: seq is {};
begin
for seq range ncsub({1, 2, 3, 4}, 0) do
writeln(seq);
end for;
end func;</lang>
- Output:
{1, 2, 4}
{1, 3, 4}
{1, 3}
{1, 4}
{2, 4}
Sidef
<lang ruby>func non_continuous(min, max, subseq=[], has_gap=false) {
static current = [];
range(min, max).each { |i|
current.push(i);
has_gap && subseq.append([current...]);
i < max && non_continuous(i.inc, max, subseq, has_gap);
current.pop;
has_gap = current.len;
}
subseq;
}
say non_continuous(1, 3); say non_continuous(1, 4); say non_continuous("a", "d");</lang>
- Output:
[[1, 3]] [[1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2, 4]] [["a", "b", "d"], ["a", "c"], ["a", "c", "d"], ["a", "d"], ["b", "d"]]
Standard ML
<lang sml>fun fence s [] =
if s >= 3 then
[[]]
else
[]
| fence s (x :: xs) =
if s mod 2 = 0 then
map
(fn ys => x :: ys)
(fence (s + 1) xs)
@
fence s xs
else
map
(fn ys => x :: ys)
(fence s xs)
@
fence (s + 1) xs
fun ncsubseq xs = fence 0 xs</lang>
- Output:
- ncsubseq [1,2,3]; val it = [[1,3]] : int list list - ncsubseq [1,2,3,4]; val it = [[1,2,4],[1,3,4],[1,3],[1,4],[2,4]] : int list list - ncsubseq [1,2,3,4,5]; val it = [[1,2,3,5],[1,2,4,5],[1,2,4],[1,2,5],[1,3,4,5],[1,3,4],[1,3,5],[1,3], [1,4,5],[1,4],[1,5],[2,3,5],...] : int list list
Tcl
This Tcl implementation uses the subsets function from Power Set, which is acceptable as that conserves the ordering, as well as a problem-specific test function is_not_continuous and a generic list filter lfilter:
<lang Tcl> proc subsets l {
set res [list [list]]
foreach e $l {
foreach subset $res {lappend res [lappend subset $e]}
}
return $res
}
proc is_not_continuous seq {
set last [lindex $seq 0]
foreach e [lrange $seq 1 end] {
if {$e-1 != $last} {return 1}
set last $e
}
return 0
}
proc lfilter {f list} {
set res {}
foreach i $list {if [$f $i] {lappend res $i}}
return $res
}
% lfilter is_not_continuous [subsets {1 2 3 4}] {1 3} {1 4} {2 4} {1 2 4} {1 3 4}</lang>
Ursala
To do it the lazy programmer way, apply the powerset library function to the list, which will generate all continuous and non-continuous subsequences of it, and then delete the subsequences that are also substrings (hence continuous) using a judicious combination of the built in substring predicate (K3), negation (Z), and distributing filter (K17) operator suffixes. This function will work on lists of any type. To meet the requirement for structural equivalence, the list items are first uniquely numbered (num), and the numbers are removed afterwards (rSS).
<lang Ursala>#import std
noncontinuous = num; ^rlK3ZK17rSS/~& powerset
- show+
examples = noncontinuous 'abcde'</lang>
- Output:
abce abd abde abe ac acd acde ace ad ade ae bce bd bde be ce
zkl
<lang zkl>fcn non_continuous_subsequences(ary){
pwerSet(ary).filter(fcn(list){(not isContinuous(list)) })
} fcn isContinuous(ary){
if(ary.len()<2) return(True);
foreach n in (ary.len()-1){ if(1+ary[n]!=ary[n+1]) return(False); }
return(True);
} non_continuous_subsequences(T(1,2,3,4)).println();</lang> <lang>fcn pwerSet(list){
(0).pump(list.len(),List,List,Utils.Helpers.pickNFrom.fp1(list),
T(T,Void.Write,Void.Write) ) .append(list)
}</lang> <lang zkl>fcn brokenSubsequences(str){
pwerSet(str.split("")).apply("concat")
.filter('wrap(substr){ (not str.holds(substr)) })
} brokenSubsequences("1234").println();</lang>
- Output:
L(L(1,3),L(1,4),L(2,4),L(1,2,4),L(1,3,4))
L("13","14","24","124","134")