Non-continuous subsequences
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.
- Metrics
- Counting
- Word frequency
- Letter frequency
- Jewels and stones
- I before E except after C
- Bioinformatics/base count
- Count occurrences of a substring
- Count how many vowels and consonants occur in a string
- Remove/replace
- XXXX redacted
- Conjugate a Latin verb
- Remove vowels from a string
- String interpolation (included)
- Strip block comments
- Strip comments from a string
- Strip a set of characters from a string
- Strip whitespace from a string -- top and tail
- Strip control codes and extended characters from a string
- Anagrams/Derangements/shuffling
- Word wheel
- ABC problem
- Sattolo cycle
- Knuth shuffle
- Ordered words
- Superpermutation minimisation
- Textonyms (using a phone text pad)
- Anagrams
- Anagrams/Deranged anagrams
- Permutations/Derangements
- Find/Search/Determine
- ABC words
- Odd words
- Word ladder
- Semordnilap
- Word search
- Wordiff (game)
- String matching
- Tea cup rim text
- Alternade words
- Changeable words
- State name puzzle
- String comparison
- Unique characters
- Unique characters in each string
- Extract file extension
- Levenshtein distance
- Palindrome detection
- Common list elements
- Longest common suffix
- Longest common prefix
- Compare a list of strings
- Longest common substring
- Find common directory path
- Words from neighbour ones
- Change e letters to i in words
- Non-continuous subsequences
- Longest common subsequence
- Longest palindromic substrings
- Longest increasing subsequence
- Words containing "the" substring
- Sum of the digits of n is substring of n
- Determine if a string is numeric
- Determine if a string is collapsible
- Determine if a string is squeezable
- Determine if a string has all unique characters
- Determine if a string has all the same characters
- Longest substrings without repeating characters
- Find words which contains all the vowels
- Find words which contains most consonants
- Find words which contains more than 3 vowels
- Find words which first and last three letters are equals
- Find words which odd letters are consonants and even letters are vowels or vice_versa
- Formatting
- Substring
- Rep-string
- Word wrap
- String case
- Align columns
- Literals/String
- Repeat a string
- Brace expansion
- Brace expansion using ranges
- Reverse a string
- Phrase reversals
- Comma quibbling
- Special characters
- String concatenation
- Substring/Top and tail
- Commatizing numbers
- Reverse words in a string
- Suffixation of decimal numbers
- Long literals, with continuations
- Numerical and alphabetical suffixes
- Abbreviations, easy
- Abbreviations, simple
- Abbreviations, automatic
- Song lyrics/poems/Mad Libs/phrases
- Mad Libs
- Magic 8-ball
- 99 Bottles of Beer
- The Name Game (a song)
- The Old lady swallowed a fly
- The Twelve Days of Christmas
- Tokenize
- Text between
- Tokenize a string
- Word break problem
- Tokenize a string with escaping
- Split a character string based on change of character
- Sequences
11l
<lang 11l>F ncsub(seq, s = 0)
I seq.empty
R I s >= 3 {[[Int]()]} E Int()
E
V x = seq[0.<1]
V xs = seq[1..]
V p2 = s % 2
V p1 = !p2
R ncsub(xs, s + p1).map(ys -> @x + ys) [+] ncsub(xs, s + p2)
print(ncsub(Array(1..3))) print(ncsub(Array(1..4))) print(ncsub(Array(1..5)))</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]]
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 csharp>using System; using System.Collections.Generic; using System.Linq;
class Program {
public static void Main() {
var sequence = new[] { "A", "B", "C", "D" };
foreach (var subset in Subsets(sequence.Length).Where(s => !IsContinuous(s))) {
Console.WriteLine(string.Join(" ", subset.Select(i => sequence[i])));
}
}
static IEnumerable<List<int>> Subsets(int length) {
int[] values = Enumerable.Range(0, length).ToArray();
var stack = new Stack<int>(length);
for (int i = 0; stack.Count > 0 || i < length; ) {
if (i < length) {
stack.Push(i++);
yield return (from index in stack.Reverse() select values[index]).ToList();
} else {
i = stack.Pop() + 1;
if (stack.Count > 0) i = stack.Pop() + 1;
}
}
}
static bool IsContinuous(List<int> list) => list[list.Count - 1] - list[0] + 1 == list.Count;
}</lang>
- Output:
A B D A C A C D A D B D
C++
<lang cpp> /*
* Nigel Galloway, July 19th., 2017 - Yes well is this any better? */
class N{
uint n,i,g,e,l;
public:
N(uint n): n(n-1),i{},g{},e(1),l(n-1){}
bool hasNext(){
g=(1<<n)+e;for(i=l;i<n;++i) g+=1<<i;
if (l==2) {l=--n; e=1; return true;}
if (e<((1<<(l-1))-1)) {++e; return true;}
e=1; --l; return (l>0);
}
uint next() {return g;}
}; </lang> Which may be used as follows: <lang cpp> int main(){
N n(4); while (n.hasNext()) std::cout << n.next() << "\t* " << std::bitset<4>(n.next()) << std::endl;
} </lang>
- Output:
9 * 1001 10 * 1010 11 * 1011 13 * 1101 5 * 0101
I can count the length of the sequence: <lang cpp> int main(){
N n(31);
int z{};for (;n.hasNext();++z); std::cout << z << std::endl;
} </lang>
- Output:
2147483151
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]]
F#
Generate only the non-continuous subsequences
<lang fsharp> (*
A function to generate only the non-continuous subsequences. Nigel Galloway July 20th., 2017
- )
let N n =
let fn n = Seq.map (fun g->(2<<<n)+g)
let rec fg n = seq{if n>0 then yield! seq{1..((1<<<n)-1)}|>fn n; yield! fg (n-1)|>fn n}
Seq.collect fg ({1..(n-2)})
</lang> This may be used as follows: <lang fsharp> let Ng ng = N ng |> Seq.iter(fun n->printf "%2d -> " n; {0..(ng-1)}|>Seq.iter (fun g->if (n&&&(1<<<g))>0 then printf "%d " (g+1));printfn "") Ng 4 </lang>
- Output:
5 -> 1 3 9 -> 1 4 10 -> 2 4 11 -> 1 2 4 13 -> 1 3 4
Counting the number of non-continuous subsequences is interesting:
> Seq.length (N 20);; Real: 00:00:00.169, CPU: 00:00:00.169, GC gen0: 0, gen1: 0 val it : int = 1048365 > Seq.length (N 23);; Real: 00:00:01.238, CPU: 00:00:01.239, GC gen0: 0, gen1: 0 val it : int = 8388331 > Seq.length (N 24);; Real: 00:00:02.520, CPU: 00:00:02.523, GC gen0: 0, gen1: 0 val it : int = 16776915 > Seq.length (N 25);; Real: 00:00:04.926, CPU: 00:00:04.930, GC gen0: 0, gen1: 0 val it : int = 33554106
Generate all subsequences and filter out the continuous
<lang fsharp> (*
A function to filter out continuous subsequences. Nigel Galloway July 24th., 2017
- )
let Nonseq n=
let fn = function
|((n,0),true )->(n+1,1)
|((n,_),false)->(n,0)
|(n,_) ->n
{5..(1<<<n)-1}|>Seq.choose(fun i->if fst({0..n-1}|>Seq.takeWhile(fun n->(1<<<(n-1))<i)|>Seq.fold(fun n g->fn (n,(i&&&(1<<<g)>0)))(0,0)) > 1 then Some(i) else None)
</lang> Again counting the number of non-continuous subsequences
> Seq.length (Nonseq 20);; Real: 00:00:02.356, CPU: 00:00:02.389, GC gen0: 183, gen1: 0 val it : int = 1048365 > Seq.length (Nonseq 23);; Real: 00:00:20.714, CPU: 00:00:20.950, GC gen0: 1571, gen1: 0 val it : int = 8388331 > Seq.length (Nonseq 24);; Real: 00:00:43.129, CPU: 00:00:43.601, GC gen0: 3216, gen1: 0 val it : int = 16776915 > Seq.length (Nonseq 25);; Real: 00:01:28.853, CPU: 00:01:29.869, GC gen0: 6577, gen1: 0 val it : int = 33554106
Conclusion
Find a better filter or use the generator.
FreeBASIC
<lang freebasic>Sub Subsecuencias_no_continuas(l() As String)
Dim As Integer i, j, g, n, r, s, w
Dim As String a, b, c
n = Ubound(l)
For s = 0 To n-2
For g = s+1 To n-1
a = "["
For i = s To g-1
a += l(i) + ", "
Next i
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 Then b += l(g+w+j) + ", "
Next j
'Print Left(Left(b)) + "]"
c = (Left(b, Len (b)-1))
Print Left(c, Len(c)-1) + "]"
Next i
Next w
Next g
Next s
End Sub
Dim lista1(3) As String = {"1", "2", "3", "4"} Print "Para [1, 2, 3, 4] las subsecuencias no continuas son:" Subsecuencias_no_continuas(lista1()) Dim lista2(4) As String = {"e", "r", "n", "i", "t"} Print "Para [e, r, n, i, t] las subsecuencias no continuas son:" Subsecuencias_no_continuas(lista2()) Sleep</lang>
- Output:
Para [1, 2, 3, 4] las subsecuencias no continuas son: [1, 3] [1, 3, 4] [1, 4] [1, 2, 4] [2, 4] Para [e, r, n, i, t] las subsecuencias no continuas son: [e, n] [e, n, i] [e, n, t] [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]
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>using Printf, IterTools
import Base.IteratorSize, Base.iterate, Base.length
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)
idex = collect(1:n)
function int2seq(m::Integer)
d = digits(m, base=2, pad=n)
idex[d .== 1]
end
return int2seq
end
struct NCSubSeq{T<:Integer}
n::T
end
mutable struct NCSubState{T<:Integer}
m::T m2s::Function
end
Base.IteratorSize(::NCSubSeq) = Base.HasLength() Base.length(a::NCSubSeq) = 2 ^ a.n - a.n * (a.n + 1) ÷ 2 - 1 function Base.iterate(a::NCSubSeq, as::NCSubState=NCSubState(5, makeint2seq(a.n)))
if 2 ^ a.n - 3 < as.m
return nothing
end
s = as.m2s(as.m)
as.m += 1
while iscontseq(as.m)
as.m += 1
end
return (s, as)
end
n = 4 println("Testing NCSubSeq for ", n, " items:\n ", join(NCSubSeq(n), " "))
s = "Rosetta" cs = split(s, "") m = 10 n = length(NCSubSeq(length(cs))) - m println("\nThe first and last ", m, " NC sub-sequences of \"", s, "\":") for (i, a) in enumerate(NCSubSeq(length(s)))
i <= m || n < i || continue
println(@sprintf "%6d %s" i join(cs[a], ""))
i == m || continue
println(" .. ......")
end
println("\nThe first and last ", m, " NC sub-sequences of \"", s, "\"") for x in IterTools.Iterators.flatten([1:10; 20:10:40; big.(50:50:200)])
@printf "%7d → %d\n" x length(NCSubSeq(x))
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
The first and last 10 NC sub-sequences of "Rosetta"
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("${a[c[i]]} ")
println()
}
}
else {
for (j in 0 until a.size) {
if (k == 0 || j > c[k - 1]) {
c[k] = j
generate(m, k + 1, c)
}
}
}
}
for (m in 2 until a.size) {
val c = IntArray(m)
generate(m, 0, c)
}
}
fun main(args: Array<String>) {
val a = arrayOf(1, 2, 3, 4)
ncs(a)
println()
val ca = arrayOf('a', 'b', 'c', 'd', 'e')
ncs(ca)
}</lang>
- Output:
1 3 1 4 2 4 1 2 4 1 3 4 a c a d a e b d b e c e a b d a b e a c d a c e a d e b c e b d e a b c e a b d e a c d e
M2000 Interpreter
<lang M2000 Interpreter> Module Non_continuous_subsequences (item$(), display){ Function positions(n) { function onebit { =lambda b=false (&c)-> { =b :if c then b~:c=not b } } dim k(n)=onebit(), p(n) m=true flush for i=1 to 2^n { for j=0 to n-1 :p(j)= k(j)(&m) :next m1=p(0) m2=0 for j=1 to n-1 if m2 then if m1>p(j) then m2=2:exit for if m1 < p(j) then m2++ m1=p(j) next if m2=2 then data cons(p())' push a copy of p() to end of stack m=true } =array([]) }
a=positions(len(item$())) if display then For i=0 to len(a)-1 b=array(a,i) line$=format$("{0::-5})",i+1,) for j=0 to len(b)-1 if array(b,j) then line$+=" "+item$(j) next print line$ doc$<=line$+{ } next end if line$="Non continuous subsequences:"+str$(len(a)) Print line$ doc$<=line$+{ } } global doc$ document doc$ ' change string to document object Non_continuous_subsequences ("1","2","3","4"), true Non_continuous_subsequences ("a","e","i","o","u"), true Non_continuous_subsequences ("R","o","s","e","t","t","a"), true Non_continuous_subsequences ("1","2","3","4","5","6","7","8","9","0"), false clipboard doc$ </lang>
- Output:
1) 1 3
2) 1 4
3) 2 4
4) 1 2 4
5) 1 3 4
Non continuous subsequences: 5
1) a i
2) a o
3) e o
4) a e o
5) a i o
6) a u
7) e u
8) a e u
9) i u
10) a i u
11) e i u
12) a e i u
13) a o u
14) e o u
15) a e o u
16) a i o u
Non continuous subsequences: 16
1) R s
2) R e
3) o e
4) R o e
5) R s e
6) R t
7) o t
8) R o t
9) s t
10) R s t
11) o s t
12) R o s t
13) R e t
14) o e t
15) R o e t
16) R s e t
17) R t
18) o t
19) R o t
20) s t
21) R s t
22) o s t
23) R o s t
24) e t
25) R e t
26) o e t
27) R o e t
28) s e t
29) R s e t
30) o s e t
31) R o s e t
32) R t t
33) o t t
34) R o t t
35) s t t
36) R s t t
37) o s t t
38) R o s t t
39) R e t t
40) o e t t
41) R o e t t
42) R s e t t
43) R a
44) o a
45) R o a
46) s a
47) R s a
48) o s a
49) R o s a
50) e a
51) R e a
52) o e a
53) R o e a
54) s e a
55) R s e a
56) o s e a
57) R o s e a
58) t a
59) R t a
60) o t a
61) R o t a
62) s t a
63) R s t a
64) o s t a
65) R o s t a
66) e t a
67) R e t a
68) o e t a
69) R o e t a
70) s e t a
71) R s e t a
72) o s e t a
73) R o s e t a
74) R t a
75) o t a
76) R o t a
77) s t a
78) R s t a
79) o s t a
80) R o s t a
81) e t a
82) R e t a
83) o e t a
84) R o e t a
85) s e t a
86) R s e t a
87) o s e t a
88) R o s e t a
89) R t t a
90) o t t a
91) R o t t a
92) s t t a
93) R s t t a
94) o s t t a
95) R o s t t a
96) R e t t a
97) o e t t a
98) R o e t t a
99) R s e t t a
Non continuous subsequences: 99
Non continuous subsequences: 968
Mathematica/Wolfram Language
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>
- Output:
{{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}}
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) = @_;
my $found;
for ($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; print "found ", non_continuous(1), " sequences\n";</lang>
- Output:
found 1048365 sequences
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 of length 1..20 in under half a second
with javascript_semantics integer count = 0 procedure ncs(sequence rest, object taken, integer ri=0, bool contig=false, bool gap=false) if ri>=length(rest) then if contig then if integer(taken) then count += 1 else ?taken end if end if else ri += 1 ncs(rest,iff(integer(taken)?taken+1:deep_copy(taken)&rest[ri]),ri,gap,gap) ncs(rest,taken,ri,contig,iff(integer(taken)?taken!=0:length(taken)!=0)) end if end procedure ncs({1,2,3},{}) ?"===" ncs({1,2,3,4},{}) ?"===" atom t0 = time() sequence s = {} for i=1 to 20 do count = 0 ncs(tagset(i),0) s = append(s,count) end for ?elapsed(time()-t0) pp(s)
- Output:
{1,3}
"==="
{1,2,4}
{1,3,4}
{1,3}
{1,4}
{2,4}
"==="
"0.3s"
{0,0,1,5,16,42,99,219,466,968,1981,4017,8100,16278,32647,65399,130918,261972,524097,1048365}
Picat
This approach uses power_set/1 (from the util module) to get the proper indices.
<lang Picat>import util.
go =>
println(1..4=non_cont(1..4)),
L = "abcde".reverse(),
println(L=non_cont(L)),
println(ernit=non_cont("ernit")),
println(aaa=non_cont("aaa")),
println(aeiou=non_cont("aeiou")),
nl,
println("Printing just the lengths for 1..N for N = 1..20:"),
foreach(N in 1..20)
println(1..N=non_cont(1..N).length) % just the length
end,
nl.
% get all the non-continuous subsequences non_cont(L) = [ [L[I] : I in S] : S in non_cont_ixs(L.length)].
% get all the index positions that are non-continuous non_cont_ixs(N) = [ P: P in power_set(1..N), length(P) > 1, P.last() - P.first() != P.length-1].</lang>
- Output:
[1,2,3,4] = [[2,4],[1,4],[1,3],[1,3,4],[1,2,4]] edcba = [ca,da,db,dba,dca,ea,eb,eba,ec,eca,ecb,ecba,eda,edb,edba,edca] ernit = [nt,rt,ri,rit,rnt,et,ei,eit,en,ent,eni,enit,ert,eri,erit,ernt] aaa = [aa] aeiou = [iu,eu,eo,eou,eiu,au,ao,aou,ai,aiu,aio,aiou,aeu,aeo,aeou,aeiu] Printing just the lengths for 1..N for N = 1..20: [1] = 0 [1,2] = 0 [1,2,3] = 1 [1,2,3,4] = 5 [1,2,3,4,5] = 16 [1,2,3,4,5,6] = 42 [1,2,3,4,5,6,7] = 99 [1,2,3,4,5,6,7,8] = 219 [1,2,3,4,5,6,7,8,9] = 466 [1,2,3,4,5,6,7,8,9,10] = 968 [1,2,3,4,5,6,7,8,9,10,11] = 1981 [1,2,3,4,5,6,7,8,9,10,11,12] = 4017 [1,2,3,4,5,6,7,8,9,10,11,12,13] = 8100 [1,2,3,4,5,6,7,8,9,10,11,12,13,14] = 16278 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] = 32647 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] = 65399 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] = 130918 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] = 261972 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] = 524097 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] = 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>
Quackery
A sequence of n items has 2^n possible subsequences, including the empty sequence. These correspond to the numbers 0 to 2^n-1, where a one in the binary expansion of the number indicates inclusion in the subsequence of the corresponding item in the sequence. Non-continuous subsequences correspond to numbers where the binary expansion of the number has a one, followed by one or more zeroes, followed by a one.
<lang Quackery> [ dup 1 & dip [ 1 >> ] ] is 2/mod ( n --> n n )
[ 0 swap
[ dup 0 != while
2/mod iff
[ dip 1+ ] done
again ]
[ dup 0 != while
2/mod not iff
[ dip 1+ ] done
again ]
[ dup 0 != while
2/mod iff
[ dip 1+ ] done
again ]
drop 3 = ] is noncontinuous ( n --> b )
[ [] unrot
[ dup 0 != while
dip behead tuck
1 & iff
[ nested dip rot
join unrot ]
else drop
1 >> again ]
2drop ] is bitems ( [ n --> [ )
[ [] swap
dup size bit times
[ i^ noncontinuous if
[ dup i^ bitems
nested rot
join swap ] ]
drop ] is ncsubs ( [ --> [ )
' [ 1 2 3 4 ] ncsubs echo cr $ "quackery" ncsubs 72 wrap$</lang>
- Output:
[ [ 1 3 4 ] [ 1 2 4 ] [ 2 4 ] [ 1 4 ] [ 1 3 ] ] qackery quckery uckery qckery quakery uakery qakery akery qukery ukery qkery quacery uacery qacery acery qucery ucery qcery cery quaery uaery qaery aery query uery qery quackry uackry qackry ackry quckry uckry qckry ckry quakry uakry qakry akry qukry ukry qkry kry quacry uacry qacry acry qucry ucry qcry cry quary uary qary ary qury ury qry quackey uackey qackey ackey quckey uckey qckey ckey quakey uakey qakey akey qukey ukey qkey key quacey uacey qacey acey qucey ucey qcey cey quaey uaey qaey aey quey uey qey ey quacky uacky qacky acky qucky ucky qcky cky quaky uaky qaky aky quky uky qky ky quacy uacy qacy acy qucy ucy qcy cy quay uay qay ay quy uy qy qacker qucker ucker qcker quaker uaker qaker aker quker uker qker quacer uacer qacer acer qucer ucer qcer cer quaer uaer qaer aer quer uer qer quackr uackr qackr ackr quckr uckr qckr ckr quakr uakr qakr akr qukr ukr qkr kr quacr uacr qacr acr qucr ucr qcr cr quar uar qar ar qur ur qr qacke qucke ucke qcke quake uake qake ake quke uke qke quace uace qace ace quce uce qce ce quae uae qae ae que ue qe qack quck uck qck quak uak qak ak quk uk qk qac quc uc qc qa
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>
Raku
(formerly 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])
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 optional argument from the CL.*/ 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. */ nums= strip( left(123456789, w) ) /*build a string of decimal digits. */ tail= right(nums, max(0, w-2) ) /*construct a fast tail for comparisons*/
- = 0 /*#: number of non─continuous subseq. */
do j=13 to left(nums,1) || tail /*step through list (using smart start)*/
if verify(j, nums) \== 0 then iterate /*Not one of the chosen (sequences) ? */
f= left(j, 1) /*use the fist decimal digit of J. */
NCS= 0 /*so far, no non─continuous subsequence*/
do k=2 for length(j)-1 /*search for " " " */
x= substr(j, k, 1) /*extract a single decimal digit of J.*/
if x <= f then iterate j /*if next digit ≤, then skip this digit*/
if x \== f+1 then NCS= 1 /*it's OK as of now (that is, so far).*/
f= x /*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 length(j) /*build a sequence string to display. */
$= $ word(list, substr(j, m, 1) ) /*obtain a number (or item) to display.*/
end /*m*/
say 'a non─continuous subsequence: ' $ /*show the non─continuous subsequence. */
end /*j*/
say /*help ensure visual fidelity in output*/ if #==0 then #= 'no' /*make it look more gooder Angleshy. */ say # "non─continuous subsequence"s(#) 'were found.' /*handle plurals.*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ s: if arg(1)==1 then return ; return word( arg(2) 's', 1) /*simple pluralizer.*/</lang>
- output when using the input of: 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 input of: 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 [five 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.
Ring
<lang ring>
- Project : Non-continuous subsequences
load "stdlib.ring" list = [1,2,3,4] items = newlist(pow(2,len(list))-1,len(list)) see "For [1, 2, 3, 4] non-continuous subsequences are:" + nl powerset(list,4) showarray(items,4) see nl
list = [1,2,3,4,5] items = newlist(pow(2,len(list))-1,len(list)) see "For [1, 2, 3, 4, 5] non-continuous subsequences are:" + nl powerset(list,5) showarray(items,5)
func showarray(items,ind)
for n = 1 to len(items)
flag = 0
for m = 1 to ind - 1
if items[n][m] = 0 or items[n][m+1] = 0
exit
ok
if (items[n][m] + 1) != items[n][m+1]
flag = 1
exit
ok
next
if flag = 1
see "["
str = ""
for x = 1 to len(items[n])
if items[n][x] != 0
str = str + items[n][x] + " "
ok
next
str = left(str, len(str) - 1)
see str + "]" + nl
ok
next
func powerset(list,ind)
num = 0
num2 = 0
items = newlist(pow(2,len(list))-1,ind)
for i = 2 to (2 << len(list)) - 1 step 2
num2 = 0
num = num + 1
for j = 1 to len(list)
if i & (1 << j)
num2 = num2 + 1
if list[j] != 0
items[num][num2] = list[j]
ok
ok
next
next
return items
</lang> Output:
For [1, 2, 3, 4] non-continuous subsequences are: [1 3] [1 4] [2 4] [1 2 4] [1 3 4] For [1, 2, 3, 4, 5] non-continuous subsequences are: [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]
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"]]
Scala
<lang Scala>object NonContinuousSubSequences extends App {
private def seqR(s: String, c: String, i: Int, added: Int): Unit = {
if (i == s.length) {
if (c.trim.length > added) println(c)
} else {
seqR(s, c + s(i), i + 1, added + 1)
seqR(s, c + " ", i + 1, added)
}
}
seqR("1234", "", 0, 0)
}</lang>
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 $res
}
% lfilter is_not_continuous [subsets {1 2 3 4}] {1 3} {1 4} {2 4} {1 2 4} {1 3 4}</lang>
Ursala
To do it the lazy programmer way, apply the powerset library function to the list, which will generate all continuous and non-continuous subsequences of it, and then delete the subsequences that are also substrings (hence continuous) using a judicious combination of the built in substring predicate (K3), negation (Z), and distributing filter (K17) operator suffixes. This function will work on lists of any type. To meet the requirement for structural equivalence, the list items are first uniquely numbered (num), and the numbers are removed afterwards (rSS).
<lang Ursala>#import std
noncontinuous = num; ^rlK3ZK17rSS/~& powerset
- show+
examples = noncontinuous 'abcde'</lang>
- Output:
abce abd abde abe ac acd acde ace ad ade ae bce bd bde be ce
VBScript
<lang vb>'Non-continuous subsequences - VBScript - 03/02/2021
Function noncontsubseq(l)
Dim i, j, g, n, r, s, w, m
Dim a, b, c
n = Ubound(l)
For s = 0 To n-2
For g = s+1 To n-1
a = "["
For i = s To g-1
a = a & l(i) & ", "
Next 'i
For w = 1 To n-g
r = n+1-g-w
For i = 1 To 2^r-1 Step 2
b = a
For j = 0 To r-1
If i And 2^j Then b=b & l(g+w+j) & ", "
Next 'j
c = (Left(b, Len(b)-1))
WScript.Echo Left(c, Len(c)-1) & "]"
m = m+1
Next 'i
Next 'w
Next 'g
Next 's
noncontsubseq = m
End Function 'noncontsubseq
list = Array("1", "2", "3", "4") WScript.Echo "List: [" & Join(list, ", ") & "]" nn = noncontsubseq(list) WScript.Echo nn & " non-continuous subsequences" </lang>
- Output:
List: [1, 2, 3, 4] [1, 3] [1, 3, 4] [1, 4] [1, 2, 4] [2, 4] 5 non-continuous subsequences
Wren
Needed a bit of doctoring to do the character example as Wren only has strings. <lang ecmascript>import "/fmt" for Fmt
var ncs = Fn.new { |a|
var f = "$d "
if (a[0] is String) {
for (i in 0...a.count) a[i] = a[i].bytes[0]
f = "$c "
}
var generate // recursive
generate = Fn.new { |m, k, c|
if (k == m) {
if (c[m - 1] != c[0] + m - 1) {
for (i in 0...m) Fmt.write(f, a[c[i]])
System.print()
}
} else {
for (j in 0...a.count) {
if (k == 0 || j > c[k - 1]) {
c[k] = j
generate.call(m, k + 1, c)
}
}
}
}
for (m in 2...a.count) {
var c = List.filled(m, 0)
generate.call(m, 0, c)
}
}
var a = [1, 2, 3, 4] ncs.call(a) System.print() var ca = ["a", "b", "c", "d", "e"] ncs.call(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
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 zkl>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")
- Programming Tasks
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