Greatest subsequential sum
You are encouraged to solve this task according to the task description, using any language you may know.
Given a sequence of integers, find a continuous subsequence which maximizes the sum of its elements, that is, the elements of no other single subsequence add up to a value larger than this one. An empty subsequence is considered to have the sum 0; thus if all elements are negative, the result must be the empty sequence.
Ada
<lang ada>with Ada.Text_Io; use Ada.Text_Io;
procedure Max_Subarray is
type Int_Array is array (Positive range <>) of Integer; Empty_Error : Exception; function Max(Item : Int_Array) return Int_Array is Start : Positive; Finis : Positive; Max_Sum : Integer := Integer'First; Sum : Integer; begin if Item'Length = 0 then raise Empty_Error; end if; for I in Item'range loop Sum := 0; for J in I..Item'Last loop Sum := Sum + Item(J); if Sum > Max_Sum then Max_Sum := Sum; Start := I; Finis := J; end if; end loop; end loop; return Item(Start..Finis); end Max; A : Int_Array := (-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1); B : Int_Array := Max(A);
begin
for I in B'range loop Put_Line(Integer'Image(B(I))); end loop;
exception
when Empty_Error => Put_Line("Array being analyzed has no elements.");
end Max_Subarray;</lang>
ALGOL 68
<lang algol68>main: (
[]INT a = (-1 , -2 , 3 , 5 , 6 , -2 , -1 , 4 , -4 , 2 , -1);
INT begin max, end max, max sum, sum;
sum := 0; begin max := 0; end max := -1; max sum := 0;
FOR begin FROM LWB a TO UPB a DO sum := 0; FOR end FROM begin TO UPB a DO sum +:= a[end]; IF sum > max sum THEN max sum := sum; begin max := begin; end max := end FI OD OD;
FOR i FROM begin max TO end max DO print(a[i]) OD
)</lang> Output:
+3 +5 +6 -2 -1 +4
AutoHotkey
classic algorithm: <lang AutoHotkey>seq = -1,-2,3,5,6,-2,-1,4,-4,2,-1 max := sum := start := 0 Loop Parse, seq, `,
If (max < sum+=A_LoopField) max := sum, a := start, b := A_Index Else If sum <= 0 sum := 0, start := A_Index
- read out the best subsequence
Loop Parse, seq, `,
s .= A_Index > a && A_Index <= b ? A_LoopField "," : ""
MsgBox % "Max = " max "`n[" SubStr(s,1,-1) "]"</lang>
AWK
<lang awk># Finds the subsequence of ary[1] to ary[len] with the greatest sum.
- Sets subseq[1] to subseq[n] and returns n. Also sets subseq["sum"].
- An empty subsequence has sum 0.
function maxsubseq(subseq, ary, len, b, bp, bs, c, cp, i) { b = 0 # best sum c = 0 # current sum bp = 0 # position of best subsequence bn = 0 # length of best subsequence cp = 1 # position of current subsequence
for (i = 1; i <= len; i++) { c += ary[i] if (c < 0) { c = 0 cp = i + 1 } if (c > b) { b = c bp = cp bn = i + 1 - cp } }
for (i = 1; i <= bn; i++) subseq[i] = ary[bp + i - 1] subseq["sum"] = b return bn }</lang>
Demonstration: <lang awk># Joins the elements ary[1] to ary[len] in a string. function join(ary, len, i, s) { s = "[" for (i = 1; i <= len; i++) { s = s ary[i] if (i < len) s = s ", " } s = s "]" return s }
- Demonstrates maxsubseq().
function try(str, ary, len, max, maxlen) { len = split(str, ary) print "Array: " join(ary, len) maxlen = maxsubseq(max, ary, len) print " Maximal subsequence: " \ join(max, maxlen) ", sum " max["sum"] }
BEGIN { try("-1 -2 -3 -4 -5") try("0 1 2 -3 3 -1 0 -4 0 -1 -4 2") try("-1 -2 3 5 6 -2 -1 4 -4 2 -1") }</lang>
Output:
Array: [-1, -2, -3, -4, -5] Maximal subsequence: [], sum 0 Array: [0, 1, 2, -3, 3, -1, 0, -4, 0, -1, -4, 2] Maximal subsequence: [0, 1, 2], sum 3 Array: [-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1] Maximal subsequence: [3, 5, 6, -2, -1, 4], sum 15
Bracmat
This program iterates over all subsequences by forced backtracking, caused by the failing node ~
at the end of the middle part of the pattern. The combination of flags [%
on an expression creates a pattern that succeeds if and only if the expression is successfully evaluated. sjt
is an extra argument to any function that is part of a pattern and it contains the current subexpression candidate. Inside the pattern the function sum
sums over all elements in sjt
. The currently longest maximal subsequence is kept in seq
.
<lang bracmat> ( 0:?max
& :?seq & -1 -2 3 5 6 -2 -1 4 -4 2 -1 : ? [%( ( = s sum . ( sum = A . !arg:%?A ?arg&!A+sum$!arg | 0 ) & ( sum$!sjt:>!max:?max & !sjt:?seq | ) ) $ & ~ ) ?
| !seq ) </lang>
<lang>3 5 6 -2 -1 4</lang>
C
<lang c>#include "stdio.h"
typedef struct Range {
int start, end, sum;
} Range;
Range maxSubseq(const int sequence[], const int len) {
int maxSum = 0, thisSum = 0, i = 0; int start = 0, end = -1, j;
for (j = 0; j < len; j++) { thisSum += sequence[j]; if (thisSum < 0) { i = j + 1; thisSum = 0; } else if (thisSum > maxSum) { maxSum = thisSum; start = i; end = j; } }
Range r; if (start <= end && start >= 0 && end >= 0) { r.start = start; r.end = end + 1; r.sum = maxSum; } else { r.start = 0; r.end = 0; r.sum = 0; } return r;
}
int main(int argc, char **argv) {
int a[] = {-1 , -2 , 3 , 5 , 6 , -2 , -1 , 4 , -4 , 2 , -1}; int alength = sizeof(a)/sizeof(a[0]);
Range r = maxSubseq(a, alength); printf("Max sum = %d\n", r.sum); int i; for (i = r.start; i < r.end; i++) printf("%d ", a[i]); printf("\n");
return 0;
}</lang> Output:
Max sum = 15 3 5 6 -2 -1 4
C++
<lang cpp>#include <utility> // for std::pair
- include <iterator> // for std::iterator_traits
- include <iostream> // for std::cout
- include <ostream> // for output operator and std::endl
- include <algorithm> // for std::copy
- include <iterator> // for std::output_iterator
// Function template max_subseq // // Given a sequence of integers, find a subsequence which maximizes // the sum of its elements, that is, the elements of no other single // subsequence add up to a value larger than this one. // // Requirements: // * ForwardIterator is a forward iterator // * ForwardIterator's value_type is less-than comparable and addable // * default-construction of value_type gives the neutral element // (zero) // * operator+ and operator< are compatible (i.e. if a>zero and // b>zero, then a+b>zero, and if a<zero and b<zero, then a+b<zero) // * [begin,end) is a valid range // // Returns: // a pair of iterators describing the begin and end of the // subsequence template<typename ForwardIterator>
std::pair<ForwardIterator, ForwardIterator> max_subseq(ForwardIterator begin, ForwardIterator end)
{
typedef typename std::iterator_traits<ForwardIterator>::value_type value_type;
ForwardIterator seq_begin = begin, seq_end = seq_begin; value_type seq_sum = value_type(); ForwardIterator current_begin = begin; value_type current_sum = value_type();
value_type zero = value_type();
for (ForwardIterator iter = begin; iter != end; ++iter) { value_type value = *iter; if (zero < value) { if (current_sum < zero) { current_sum = zero; current_begin = iter; } } else { if (seq_sum < current_sum) { seq_begin = current_begin; seq_end = iter; seq_sum = current_sum; } } current_sum += value; }
if (seq_sum < current_sum) { seq_begin = current_begin; seq_end = end; seq_sum = current_sum; }
return std::make_pair(seq_begin, seq_end);
}
// the test array int array[] = { -1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1 };
// function template to find the one-past-end pointer to the array template<typename T, int N> int* end(T (&arr)[N]) { return arr+N; }
int main() {
// find the subsequence std::pair<int*, int*> seq = max_subseq(array, end(array));
// output it std::copy(seq.first, seq.second, std::ostream_iterator<int>(std::cout, " ")); std::cout << std::endl;
return 0;
}</lang>
C#
The challange <lang csharp>using System;
namespace Tests_With_Framework_4 {
class Program { static void Main(string[] args) { int[] integers = { -1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1 }; int length = integers.Length; int maxsum, beginmax, endmax, sum; maxsum = beginmax = sum = 0; endmax = -1;
for (int i = 0; i < length; i++) { sum = 0; for (int k = i; k < length; k++) { sum += integers[k]; if (sum > maxsum) { maxsum = sum; beginmax = i; endmax = k; } } }
for (int i = beginmax; i <= endmax; i++) Console.WriteLine(integers[i]);
Console.ReadKey(); } }
}</lang>
Clojure
Naive algorithm: <lang clojure>(defn max-subseq-sum [coll]
(->> (take-while seq (iterate rest coll)) ; tails (mapcat #(reductions conj [] %)) ; inits (apply max-key #(reduce + %)))) ; max sum</lang>
Sample output: <lang clojure>user> (max-subseq-sum [-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1]) [3 5 6 -2 -1 4]</lang>
CoffeeScript
<lang coffeescript> max_sum_seq = (sequence) ->
# This runs in linear time. [sum_start, sum, max_sum, max_start, max_end] = [0, 0, 0, 0, 0] for n, i in sequence sum += n if sum > max_sum max_sum = sum max_start = sum_start max_end = i + 1 if sum < 0 # start new sequence sum = 0 sum_start = i + 1 sequence[max_start...max_end]
- tests
console.log max_sum_seq [-1, 0, 15, 3, -9, 12, -4] console.log max_sum_seq [-1] console.log max_sum_seq [4, -10, 3] </lang>
Common Lisp
Linear version returns the maximum subsequence sum, the subsequence with the maximum sum, and start and end indices for the subsequence within the original sequence. Based on the implementation at Word Aligned. Leading zeroes aren't trimmed from the subsequence.
<lang lisp>(defun max-subseq (list)
(let ((best-sum 0) (current-sum 0) (end 0)) ;; determine the best sum, and the end of the max subsequence (do ((list list (rest list)) (i 0 (1+ i))) ((endp list)) (setf current-sum (max 0 (+ current-sum (first list)))) (when (> current-sum best-sum) (setf end i best-sum current-sum))) ;; take the subsequence of list ending at end, and remove elements ;; from the beginning until the subsequence sums to best-sum. (let* ((sublist (subseq list 0 (1+ end))) (sum (reduce #'+ sublist))) (do ((start 0 (1+ start)) (sublist sublist (rest sublist)) (sum sum (- sum (first sublist)))) ((or (endp sublist) (eql sum best-sum)) (values best-sum sublist start (1+ end)))))))</lang>
For example,
> (max-subseq '(-1 -2 -3 -4 -5)) 0 NIL 1 1
> (max-subseq '(0 1 2 -3 3 -1 0 -4 0 -1 -4 2)) 3 (0 1 2) 0 3
D
Adapted from the Python version: <lang d>import std.stdio: writeln;
pure nothrow inout(T[]) maxSubseq(T)(inout T[] sequence) {
int maxSum, thisSum, i, start, end = -1;
foreach (j, x; sequence) { thisSum += x; if (thisSum < 0) { i = j + 1; thisSum = 0; } else if (thisSum > maxSum) { maxSum = thisSum; start = i; end = j; } }
if (start <= end && start >= 0 && end >= 0) return sequence[start .. end + 1]; else return [];
}
void main() {
auto a1 = [-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1]; writeln("Maximal subsequence: ", maxSubseq(a1));
auto a2 = [-1, -2, -3, -5, -6, -2, -1, -4, -4, -2, -1]; writeln("Maximal subsequence: ", maxSubseq(a2));
}</lang> Output:
Maximal subsequence: [3, 5, 6, -2, -1, 4] Maximal subsequence:
E
This implementation tests every combination, but it first examines the sequence to reduce the number of combinations tried: We need not consider beginning the subsequence at any point which is not the beginning, or a change from negative to positive. We need not consider ending the subsequence at any point which is not the end, or a change from positive to negative. (Zero is moot and treated as negative.)
This algorithm is therefore where is the size of the sequence and is the number of sign changes in the sequence. I think it could be improved to by recording the positive and negative intervals' sums during the initial pass and accumulating the sum of those sums in the inner for loop.
maxSubseq
returns both the maximum sum found, and the interval of indexes which produces it.
<lang e>pragma.enable("accumulator")
def maxSubseq(seq) {
def size := seq.size()
# Collect all intervals of indexes whose values are positive def intervals := { var intervals := [] var first := 0 while (first < size) { var next := first def seeing := seq[first] > 0 while (next < size && (seq[next] > 0) == seeing) { next += 1 } if (seeing) { # record every positive interval intervals with= first..!next } first := next } intervals } # For recording the best result found var maxValue := 0 var maxInterval := 0..!0 # Try all subsequences beginning and ending with such intervals. for firstIntervalIx => firstInterval in intervals { for lastInterval in intervals(firstIntervalIx) { def interval := (firstInterval.getOptStart())..!(lastInterval.getOptBound()) def value := accum 0 for i in interval { _ + seq[i] } if (value > maxValue) { maxValue := value maxInterval := interval } } } return ["value" => maxValue, "indexes" => maxInterval]
}</lang>
<lang e>def seq := [-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1] def [=> value, => indexes] := maxSubseq(seq) println(`$\ Sequence: $seq Maximum subsequence sum: $value Indexes: ${indexes.getOptStart()}..${indexes.getOptBound().previous()} Subsequence: ${seq(indexes.getOptStart(), indexes.getOptBound())} `)</lang>
Euphoria
<lang euphoria>function maxSubseq(sequence s)
integer sum, maxsum, first, last maxsum = 0 first = 1 last = 0 for i = 1 to length(s) do sum = 0 for j = i to length(s) do sum += s[j] if sum > maxsum then maxsum = sum first = i last = j end if end for end for return s[first..last]
end function
? maxSubseq({-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1}) ? maxSubseq({}) ? maxSubseq({-1, -5, -3})</lang>
Output:
{3,5,6,-2,-1,4} {} {}
Factor
<lang factor>USING: kernel locals math math.order sequences ;
- max-with-index ( elt0 ind0 elt1 ind1 -- elt ind )
elt0 elt1 < [ elt1 ind1 ] [ elt0 ind0 ] if ;
- last-of-max ( accseq -- ind ) -1 swap -1 [ max-with-index ] reduce-index nip ;
- max-subseq ( seq -- subseq )
dup 0 [ + 0 max ] accumulate swap suffix last-of-max head dup 0 [ + ] accumulate swap suffix [ neg ] map last-of-max tail ;</lang> <lang factor>( scratchpad ) { -1 -2 3 5 6 -2 -1 4 -4 2 -1 } max-subseq dup sum swap . . { 3 5 6 -2 -1 4 } 15</lang>
Forth
<lang forth>2variable best variable best-sum
- sum ( array len -- sum )
0 -rot cells over + swap do i @ + cell +loop ;
- max-sub ( array len -- sub len )
over 0 best 2! 0 best-sum ! dup 1 do \ foreach length 2dup i - cells over + swap do \ foreach start i j sum dup best-sum @ > if best-sum ! i j best 2! else drop then cell +loop loop 2drop best 2@ ;
- .array ." [" dup 0 ?do over i cells + @ . loop ." ] = " sum . ;
create test -1 , -2 , 3 , 5 , 6 , -2 , -1 , 4 , -4 , 2 , -1 ,
test 11 max-sub .array \ [3 5 6 -2 -1 4 ] = 15</lang>
Fortran
<lang fortran>program MaxSubSeq
implicit none
integer, parameter :: an = 11 integer, dimension(an) :: a = (/ -1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1 /)
integer, dimension(an,an) :: mix integer :: i, j integer, dimension(2) :: m
forall(i=1:an,j=1:an) mix(i,j) = sum(a(i:j)) m = maxloc(mix) ! a(m(1):m(2)) is the wanted subsequence print *, a(m(1):m(2))
end program MaxSubSeq</lang>
Go
<lang go>package main
import "fmt"
func gss(s []int) ([]int, int) {
var best, start, end, sum, sumStart int for i, x := range s { sum += x switch { case sum > best: best = sum start = sumStart end = i + 1 case sum < 0: sum = 0 sumStart = i + 1 } } return s[start:end], best
}
var testCases = [][]int{
{-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1}, {-1, 1, 2, -5, -6}, {}, {-1, -2, -1},
}
func main() {
for _, c := range testCases { fmt.Println("Input: ", c) subSeq, sum := gss(c) fmt.Println("Sub seq:", subSeq) fmt.Println("Sum: ", sum, "\n") }
}</lang> Output:
Input: [-1 -2 3 5 6 -2 -1 4 -4 2 -1] Sub seq: [3 5 6 -2 -1 4] Sum: 15 Input: [-1 1 2 -5 -6] Sub seq: [1 2] Sum: 3 Input: [] Sub seq: [] Sum: 0 Input: [-1 -2 -1] Sub seq: [] Sum: 0
Haskell
Naive approach which tests all possible subsequences, as in a few of the other examples. For fun, this is in point-free style and doesn't use loops: <lang haskell>import Data.List (inits, tails, maximumBy) import Data.Ord (comparing)
subseqs :: [a] -> a subseqs = concatMap inits . tails
maxsubseq :: (Ord a, Num a) => [a] -> [a] maxsubseq = maximumBy (comparing sum) . subseqs
main = print $ maxsubseq [-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1]</lang> Secondly, the linear time constant space approach: <lang haskell>maxsubseq xs = loop (0,0,xs) (0,0,xs) xs where
loop (s,n,xs) _ [] = take n xs loop best@(bsum,_,_) (s,n,ys) (x:xs) | s' < 0 = loop best (0,0,xs) xs | s' > bsum = loop next next xs | otherwise = loop best next xs where s' = x + s n' = n + 1 next = n' `seq` (s',n',ys)
main = print $ maxsubseq [-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1]</lang>
Icon and Unicon
<lang Icon>procedure main() L1 := [-1,-2,3,5,6,-2,-1,4,-4,2,-1] # sample list L := [-1,1,2,3,4,-11]|||L1 # prepend a local maximum into the mix write(ximage(maxsubseq(L))) end
link ximage # to show lists
procedure maxsubseq(L) #: return the subsequence of L with maximum positive sum local i,maxglobal,maxglobalI,maxlocal,maxlocalI
maxglobal := maxlocal := 0 # global and local maxima
every i := 1 to *L do {
if (maxlocal := max(maxlocal +L[i],0)) > 0 then if /maxlocalI then maxlocalI := [i,i] else maxlocalI[2] := i # local maxima subscripts else maxlocalI := &null # reset subsequence if maxglobal <:= maxlocal then # global maxima maxglobalI := copy(maxlocalI) }
return L[(\maxglobalI)[1]:maxglobalI[2]] | [] # return sub-sequence or empty list end</lang>
J
<lang j>maxss=: monad define
AS =. 0,; <:/~@i.&.> #\y MX =. (= >./) AS +/ . * y y #~ {. MX#AS
)</lang>
y is the input vector, an integer list.
AS means "all sub-sequences." It is a binary table. Each row indicates one sub-sequence; the count of columns equals the length of the input.
MX means "maxima." It is the first location in AS where the corresponding sum is largest.
Totals for the subsequences are calculated by the phrase 'AS +/ . * y' which is the inner product of the binary table with the input vector.
All solutions are found but only one is returned, to fit the output requirement. If zero is the maximal sum the empty array is always returned, although sub-sequences of positive length (comprised of zeros) might be more interesting.
Example use:
<lang j> maxss _1 _2 3 5 6 _2 _1 4 _4 2 _1
3 5 6 _2 _1 4</lang>
Note: if we just want the sum of the maximum subsequence,and are not concerned with the subsequence itself, we can use:
<lang j>maxs=: [:>./(0>.+)/\.</lang>
Example use: <lang j> maxs _1 _2 3 5 6 _2 _1 4 _4 2 _1 15</lang>
This suggests a variant: <lang j>maxSS=:monad define
sums=: (0>.+)/\. y start=: sums i. max=: >./ sums max (] {.~ #@] |&>: (= +/\) i. 1:) y}.~start
)</lang> or <lang j>maxSS2=:monad define
start=. (i. >./) (0>.+)/\. y ({.~ # |&>: [: (i.>./@,&0) +/\) y}.~start
)</lang>
These variants are considerably faster than the first implementation, on long sequences.
Java
This is not a particularly efficient solution, but it gets the job done.
The method nextChoices was modified from an RIT CS lab. <lang java>import java.util.Scanner; import java.util.ArrayList;
public class Sub{
private static int[] indices;
public static void main(String[] args){ ArrayList<Long> array= new ArrayList<Long>(); //the main set Scanner in = new Scanner(System.in); while(in.hasNextLong()) array.add(in.nextLong()); long highSum= Long.MIN_VALUE;//start the sum at the lowest possible value ArrayList<Long> highSet= new ArrayList<Long>(); //loop through all possible subarray sizes including 0 for(int subSize= 0;subSize<= array.size();subSize++){ indices= new int[subSize]; for(int i= 0;i< subSize;i++) indices[i]= i; do{ long sum= 0;//this subarray sum variable ArrayList<Long> temp= new ArrayList<Long>();//this subarray //sum it and save it for(long index:indices) {sum+= array.get(index); temp.add(array.get(index));} if(sum > highSum){//if we found a higher sum highSet= temp; //keep track of it highSum= sum; } }while(nextIndices(array));//while we haven't tested all subarrays } System.out.println("Sum: " + highSum + "\nSet: " + highSet); } /** * Computes the next set of choices from the previous. The * algorithm tries to increment the index of the final choice * first. Should that fail (index goes out of bounds), it * tries to increment the next-to-the-last index, and resets * the last index to one more than the next-to-the-last. * Should this fail the algorithm keeps starting at an earlier * choice until it runs off the start of the choice list without * Finding a legal set of indices for all the choices. * * @return true unless all choice sets have been exhausted. * @author James Heliotis */
private static boolean nextIndices(ArrayList<Long> a) { for(int i= indices.length-1;i >= 0;--i){ indices[i]++; for(int j=i+1;j < indices.length;++j){ indices[j]= indices[j - 1] + 1;//reset the last failed try } if(indices[indices.length - 1] < a.size()){//if this try went out of bounds return true; } } return false; }
}</lang>
This one runs in linear time, and isn't generalized. <lang java>private static int BiggestSubsum(int[] t) {
int sum = 0; int maxsum = 0;
for (int i : t) { sum += i; if (sum < 0) sum = 0; maxsum = sum > maxsum ? sum : maxsum; } return maxsum;
}</lang>
JavaScript
Simple brute force approach. <lang javascript> function MaximumSubsequence( population ) { var maxValue = 0; var subsequence = [];
for( var i=0, len=population.length; i < len; i++ ) { for( var j=i; j <= len; j++ ) { var subsequence = population.slice(i,j); var value = sumValues(subsequence); if( value > maxValue ) { maxValue = value; greatest = subsequence; }; } }
return greatest; }
function sumValues(arr) { var result = 0; for( var i=0, len=arr.length; i < len; i++) { result += arr[i]; } return result; }</lang>
Lua
<lang lua>function sumt(t, start, last) return start <= last and t[start] + sumt(t, start+1, last) or 0 end function maxsub(ary, idx)
local idx = idx or 1 if not ary[idx] then return {} end local maxsum, last = 0, idx for i = idx, #ary do if sumt(ary, idx, i) > maxsum then maxsum, last = sumt(ary, idx, i), i end end local v = maxsub(ary, idx + 1) if maxsum < sumt(v, 1, #v) then return v end local ret = {} for i = idx, last do ret[#ret+1] = ary[i] end return ret
end</lang>
M4
<lang M4>divert(-1) define(`setrange',`ifelse(`$3',`',$2,`define($1[$2],$3)`'setrange($1,
incr($2),shift(shift(shift($@))))')')
define(`asize',decr(setrange(`a',1,-1,-2,3,5,6,-2,-1,4,-4,2,-1))) define(`get',`defn(`$1[$2]')') define(`for',
`ifelse($#,0,``$0, `ifelse(eval($2<=$3),1, `pushdef(`$1',$2)$4`'popdef(`$1')$0(`$1',incr($2),$3,`$4')')')')
define(`maxsum',0) for(`x',1,asize,
`define(`sum',0)`'for(`y',x,asize, `define(`sum',eval(sum+get(`a',y)))`'ifelse(eval(sum>maxsum),1, `define(`maxsum',sum)`'define(`xmax',x)`'define(`ymax',y)')')')
divert for(`x',xmax,ymax,`get(`a',x) ')</lang>
Mathematica
Method 1
First we define 2 functions, one that gives all possibles subsequences (as a list of lists of indices) for a particular length. Then another extract those indices adds them up and looks for the largest sum. <lang Mathematica>Sequences[m_]:=Prepend[Flatten[Table[Partition[Range[m],n,1],{n,m}],1],{}] MaximumSubsequence[x_List]:=Module[{sums},
sums={x#,Total[x#]}&/@Sequences[Length[x]]; First[First[sums[[Ordering[sums,-1,#12<#22&]]]]]
]</lang>
Method 2
<lang Mathematica>MaximumSubsequence[x_List]:=Last@SortBy[Flatten[Table[xa;;b, {b,Length[x]}, {a,b}],1],Total]</lang>
Examples
<lang Mathematica>MaximumSubsequence[{-1,-2,3,5,6,-2,-1,4,-4,2,-1}] MaximumSubsequence[{2,4,5}] MaximumSubsequence[{2,-4,3}] MaximumSubsequence[{4}] MaximumSubsequence[{}]</lang> gives back:
{3,5,6,-2,-1,4} {2,4,5} {3} {4} {}
OCaml
<lang ocaml>let maxsubseq =
let rec loop sum seq maxsum maxseq = function | [] -> maxsum, List.rev maxseq | x::xs -> let sum = sum + x and seq = x :: seq in if sum < 0 then loop 0 [] maxsum maxseq xs else if sum > maxsum then loop sum seq sum seq xs else loop sum seq maxsum maxseq xs in loop 0 [] 0 []
let _ =
maxsubseq [-1 ; -2 ; 3 ; 5 ; 6 ; -2 ; -1 ; 4; -4 ; 2 ; -1]</lang>
This returns a pair of the maximum sum and (one of) the maximum subsequence(s).
Oz
<lang oz>declare
fun {MaxSubSeq Xs}
fun {Step [Sum0 Seq0 MaxSum MaxSeq] X} Sum = Sum0 + X Seq = X|Seq0 in if Sum > MaxSum then %% found new maximum [Sum Seq Sum Seq] elseif Sum < 0 then %% discard negative subseqs [0 nil MaxSum MaxSeq] else [Sum Seq MaxSum MaxSeq] end end
[_ _ _ MaxSeq] = {FoldL Xs Step [0 nil 0 nil]} in {Reverse MaxSeq} end
in
{Show {MaxSubSeq [~1 ~2 3 5 6 ~2 ~1 4 ~4 2 1]}}</lang>
PARI/GP
Naive quadratic solution (with end-trimming). <lang parigp>grsub(v)={
my(mn=1,mx=#v,r=0,at,c); if(vecmax(v)<=0,return([1,0])); while(v[mn]<=0,mn++); while(v[mx]<=0,mx--); for(a=mn,mx, c=0; for(b=a,mx, c+=v[b]; if(c>r,r=c;at=[a,b]) ) ); at
};</lang>
Pascal
<lang pascal>Program GreatestSubsequentialSum(output);
var
a: array[1..11] of integer = (-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1); i, j: integer; seqStart, seqEnd: integer; maxSum, seqSum: integer;
begin
maxSum := 0; seqStart := 0; seqEnd := -1; for i := low(a) to high(a) do begin seqSum := 0; for j := i to high(a) do begin seqSum := seqSum + a[j]; if seqSum > maxSum then begin maxSum := seqSum; seqStart := i; seqEnd := j; end; end; end;
writeln ('Sequence: '); for i := low(a) to high(a) do write (a[i]:3); writeln; writeln ('Subsequence with greatest sum: '); for i := low(a) to seqStart - 1 do write (' ':3); for i := seqStart to seqEnd do write (a[i]:3); writeln; writeln ('Sum:'); writeln (maxSum);
end.</lang> Output:
:> ./GreatestSubsequentialSum Sequence: -1 -2 3 5 6 -2 -1 4 -4 2 -1 Subsequence with greatest sum: 3 5 6 -2 -1 4 Sum: 15
Perl
O(n) running-sum method: <lang perl>use strict;
sub max_sub(\@) { my ($a, $maxs, $maxe, $s, $sum, $maxsum) = shift; foreach (0 .. $#$a) { my $t = $sum + $a->[$_]; ($s, $sum) = $t > 0 ? ($s, $t) : ($_ + 1, 0);
if ($maxsum < $sum) { $maxsum = $sum; ($maxs, $maxe) = ($s, $_ + 1) } } @$a[$maxs .. $maxe - 1] }
my @a = map { int(rand(20) - 10) } 1 .. 10; my @b = (-1) x 10;
print "seq: @a\nmax: [ @{[max_sub @a]} ]\n"; print "seq: @b\nmax: [ @{[max_sub @b]} ]\n";</lang>output
seq: -7 5 -3 0 5 -5 -1 -1 -5 1 max: [ 5 -3 0 5 ] seq: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 max: [ ]
Naive and potentionally very slow method: <lang perl>use strict;
my @a = (-1 , -2 , 3 , 5 , 6 , -2 , -1 , 4 , -4 , 2 , -1);
my @maxsubarray; my $maxsum = 0;
foreach my $begin (0..$#a) {
foreach my $end ($begin..$#a) { my $sum = 0; $sum += $_ foreach @a[$begin..$end]; if($sum > $maxsum) { $maxsum = $sum; @maxsubarray = @a[$begin..$end]; } }
}
print "@maxsubarray\n";</lang>
Perl 6
<lang perl6>sub maxsubseq (*@a) {
my ($start, $end, $sum, $maxsum) = -1, -1, 0, 0; for @a.kv -> $i, $x { $sum += $x; if $maxsum < $sum { ($maxsum, $end) = $sum, $i; } elsif $sum < 0 { ($sum, $start) = 0, $i; } } return @a[$start ^.. $end];
}</lang>
Another solution, not translated from any other language:
For each starting position, we calculate all the subsets starting at that position. They are combined with the best subset ($max_subset) from previous loops, to form (@subsets). The best of those @subsets is saved at the new $max_subset.
Consuming the array (.shift) allows us to skip tracking the starting point; it is always 0.
The empty sequence is used to initialize $max_subset, which fufills the "all negative" requirement of the problem.
Note that once the triangular comma bug is resolved, the inner-loop subset calculation line can be shortened to "my @subsets = [\,] @a;". <lang perl6>sub max_sub-seq ( *@a ) {
my $max_subset = []; while @a { my @subsets = @a.keys.map: { [ @a[0..$_] ] }; @subsets.push($max_subset); $max_subset = @subsets.max: { [+] .list }; @a.shift; }
return $max_subset;
}
max_sub-seq( -1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1 ).perl.say; max_sub-seq( -2, -2, -1, 3, 5, 6, -1, 4, -4, 2, -1 ).perl.say; max_sub-seq( -2, -2, -1, -3, -5, -6, -1, -4, -4, -2, -1 ).perl.say;</lang>
Output:
[3, 5, 6, -2, -1, 4] [3, 5, 6, -1, 4] []
PicoLisp
<lang PicoLisp>(maxi '((L) (apply + L))
(mapcon '((L) (maplist reverse (reverse L))) (-1 -2 3 5 6 -2 -1 4 -4 2 -1) ) )</lang>
Output:
-> (3 5 6 -2 -1 4)
Prolog
Constraint Handling Rules
CHR is a programming language created by Professor Thom Frühwirth.
Works with SWI-Prolog and module CHR written by Tom Schrijvers and Jan Wielemaker.
<lang Prolog>:- use_module(library(chr)).
- - chr_constraint
init_chr/2, seq/2, % gss(Deb, Len, TT) gss/3, % gsscur(Deb, Len, TT, IdCur) gsscur/4, memoseq/3, clean/0, greatest_subsequence/0.
greatest_subsequence <=>
L = [-1 , -2 , 3 , 5 , 6 , -2 , -1 , 4 , -4 , 2 , -1],
init_chr(1, L),
find_chr_constraint(gss(Deb, Len, V)),
clean,
writeln(L),
forall(between(1, Len, I),
( J is I+Deb-1, nth1(J, L, N), format('~w ', [N]))),
format('==> ~w ~n', [V]).
% destroy last constraint gss clean \ gss(_,_,_) <=> true. clean <=> true.
init_chr_end @ init_chr(_, []) <=> gss(0, 0, 0), gsscur(1,0,0,1).
init_chr_loop @ init_chr(N, [H|T]) <=> seq(N, H), N1 is N+1, init_chr(N1, T).
% here, we memorize the list gsscur_with_negative @ gsscur(Deb, Len, TT, N), seq(N, V) <=> V =< 0 |
memoseq(Deb, Len, TT),
TT1 is TT + V, N1 is N+1, % if TT1 becomes negative, % we begin a new subsequence ( TT1 < 0 -> gsscur(N1,0,0,N1) ; Len1 is Len + 1, gsscur(Deb, Len1, TT1, N1)).
gsscur_with_positive @ gsscur(Deb, Len, TT, N), seq(N, V) <=> V > 0 |
TT1 is TT + V,
N1 is N+1, Len1 is Len + 1, gsscur(Deb, Len1, TT1, N1).
gsscur_end @ gsscur(Deb, Len, TT, _N) <=> memoseq(Deb, Len, TT).
memoseq(_DC, _LC, TTC), gss(D, L, TT) <=> TTC =< TT | gss(D, L, TT).
memoseq(DC, LC, TTC), gss(_D, _L, TT) <=> TTC > TT | gss(DC, LC, TTC).</lang> OutPut:
?- greatest_subsequence. [-1,-2,3,5,6,-2,-1,4,-4,2,-1] 3 5 6 -2 -1 4 ==> 15 true ; false.
PureBasic
<lang PureBasic>If OpenConsole()
Define s$, a, b, p1, p2, sum, max, dm=(?EndOfMyData-?MyData) Dim Seq.i(dm/SizeOf(Integer)) CopyMemory(?MyData,@seq(),dm) For a=0 To ArraySize(seq()) sum=0 For b=a To ArraySize(seq()) sum+seq(b) If sum>max max=sum p1=a p2=b EndIf Next Next For a=p1 To p2 s$+str(seq(a)) If a<p2 s$+"+" EndIf Next PrintN(s$+" = "+str(max)) Print("Press ENTER to quit"): Input() CloseConsole()
EndIf
DataSection
MyData: Data.i -1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1 EndOfMyData:
EndDataSection</lang>
Python
Naive, inefficient but really simple solution which tests all possible subsequences, as in a few of the other examples: <lang python>def maxsubseq(seq):
return max((seq[begin:end] for begin in xrange(len(seq)+1) for end in xrange(begin, len(seq)+1)), key=sum)</lang>
Classic linear-time constant-space solution based on algorithm from "Programming Pearls" book. <lang python>def maxsum(sequence):
"""Return maximum sum.""" maxsofar, maxendinghere = 0, 0 for x in sequence: # invariant: ``maxendinghere`` and ``maxsofar`` are accurate for ``x[0..i-1]`` maxendinghere = max(maxendinghere + x, 0) maxsofar = max(maxsofar, maxendinghere) return maxsofar</lang>
Adapt the above-mentioned solution to return maximizing subsequence. See http://www.java-tips.org/java-se-tips/java.lang/finding-maximum-contiguous-subsequence-sum-using-divide-and-conquer-app.html <lang python>def maxsumseq(sequence):
start, end, sum_start = -1, -1, -1 maxsum_, sum_ = 0, 0 for i, x in enumerate(sequence): sum_ += x if maxsum_ < sum_: # found maximal subsequence so far maxsum_ = sum_ start, end = sum_start, i elif sum_ < 0: # start new sequence sum_ = 0 sum_start = i assert maxsum_ == maxsum(sequence) assert maxsum_ == sum(sequence[start + 1:end + 1]) return sequence[start + 1:end + 1]</lang>
Modify ``maxsumseq()`` to allow any iterable not just sequences. <lang python>def maxsumit(iterable):
maxseq = seq = [] start, end, sum_start = -1, -1, -1 maxsum_, sum_ = 0, 0 for i, x in enumerate(iterable): seq.append(x); sum_ += x if maxsum_ < sum_: maxseq = seq; maxsum_ = sum_ start, end = sum_start, i elif sum_ < 0: seq = []; sum_ = 0 sum_start = i assert maxsum_ == sum(maxseq[:end - start]) return maxseq[:end - start]</lang>
Elementary tests: <lang python>f = maxsumit assert f([]) == [] assert f([-1]) == [] assert f([0]) == [] assert f([1]) == [1] assert f([1, 0]) == [1] assert f([0, 1]) == [0, 1] assert f([0, 1, 0]) == [0, 1] assert f([2]) == [2] assert f([2, -1]) == [2] assert f([-1, 2]) == [2] assert f([-1, 2, -1]) == [2] assert f([2, -1, 3]) == [2, -1, 3] assert f([2, -1, 3, -1]) == [2, -1, 3] assert f([-1, 2, -1, 3]) == [2, -1, 3] assert f([-1, 2, -1, 3, -1]) == [2, -1, 3] assert f([-1, 1, 2, -5, -6]) == [1,2]</lang>
R
<lang R>max.subseq <- function(x) {
cumulative <- cumsum(x) min.cumulative.so.far <- Reduce(min, cumulative, accumulate=TRUE) end <- which.max(cumulative-min.cumulative.so.far) begin <- which.min(c(0, cumulative[1:end])) if (end >= begin) x[begin:end] else x[c()]
}</lang> Sample output: <lang r>> max.subseq(c(-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1)) [1] 3 5 6 -2 -1 4</lang>
Racket
Linear time version, returns the maximum subsequence and its sum. <lang scheme>(define (max-subseq l)
(define-values (_ result _1 max-sum) (for/fold ([seq '()] [max-seq '()] [sum 0] [max-sum 0]) ([i l]) (cond [(> (+ sum i) max-sum) (values (cons i seq) (cons i seq) (+ sum i) (+ sum i))] [(< (+ sum i) 0) (values '() max-seq 0 max-sum)] [else (values (cons i seq) max-seq (+ sum i) max-sum)]))) (values (reverse result) max-sum))</lang>
For example: <lang Racket>> (max-subseq '(-1 -2 3 5 6 -2 -1 4 -4 2 -1)) '(3 5 6 -2 -1 4) 15</lang>
REXX
version 1
This REXX version will find the shortest greatest continous subsequence sum. <lang rexx>/*REXX program to find sum of the greatest continous subsequence sum. */
arg @ /*get the arugment LIST (if any).*/ say 'words='words(@) 'list='@ /*show WORDS and LIST to console.*/ say sum=word(@,1) /*a "starter" sum (of a sequence)*/ w=words(@) /*number of words in the list. */ at=1 /*where the sequence starts at. */ L=0 /*the length of the sequence. */
/*───────────────────────────────────────process the list───────────────*/
do j=1 for w; f=word(@,j)
do k=j to w; s=f
do m=j+1 to k s=s+word(@,m) end /*m*/
if s>sum then do; sum=s; at=j; L=k-j+1; end end /*k*/
end /*j*/
/*──────────────────────────────────────────────────────────────────────*/
seq=subword(@,at,L) if seq== then seq="[NULL]" sum=word(sum 0,1) say 'sum='sum/1 "sequence="seq</lang> Output when the following was used for the list: -1 -2 3 5 6 -2 -1 4 -4 2 -1
words=11 list=-1 -2 3 5 6 -2 -1 4 -4 2 -1 sum=15 sequence=3 5 6 -2 -1 4
Output when the following was used for the list: 1 2 3 4 -777 1 2 3 4 0 0
words=12 list=1 2 3 4 0 -777 1 2 3 4 0 0 sum=10 sequence=1 2 3 4
version 2
This REXX version will find the longest greatest continous subsequence sum. <lang rexx>/*REXX program to find sum of the greatest continous subsequence sum. */
arg @ /*get the arugment LIST (if any).*/ say 'words='words(@) 'list='@ /*show WORDS and LIST to console.*/ say sum=word(@,1) /*a "starter" sum (of a sequence)*/ w=words(@) /*number of words in the list. */ at=1 /*where the sequence starts at. */ L=0 /*the length of the sequence. */
/*───────────────────────────────────────process the list───────────────*/
do j=1 for w; f=word(@,j)
do k=j to w; s=f
do m=j+1 to k s=s+word(@,m) end /*m*/
_=k-j+1 if (s==sum & _>L) | s>sum then do; sum=s; at=j; L=_; end end /*k*/
end /*j*/
/*──────────────────────────────────────────────────────────────────────*/
seq=subword(@,at,L) if seq== then seq="[NULL]" sum=word(sum 0,1) say 'sum='sum/1 "sequence="seq</lang> Output when the following was used for the list: 1 2 3 4 -777 1 2 3 4 0 0
words=12 list=1 2 3 4 0 -777 1 2 3 4 0 0 sum=10 sequence=1 2 3 4 0 0
Ruby
Answer is stored in "slice": <lang ruby>Infinity = 1.0/0 def subarray_sum(arr)
max, slice = -Infinity, [] arr.each_with_index do |n, i| (i...arr.length).each do |j| sum = arr[i..j].inject(0) { |x, sum| sum += x } if sum > max max = sum slice = arr[i..j] end end end [max, slice]
end</lang>
A better answer would run in O(n) instead of O(n^2) using numerical properties to remove the need for the inner loop.
<lang ruby># the trick is that at any point
- in the iteration if starting a new chain is
- better than your current score with this element
- added to it, then do so.
- the interesting part is proving the math behind it
Infinity = 1.0/0 def subarray_sum(arr)
curr,max = -Infinity,-Infinity first,last= 0,0 arr.each_with_index do |e,i| curr = e + curr if(e>curr) curr = e first=i end if(curr > max) max = curr last=i end end return max,arr[first...last+1]
end
input=[1,2,3,4,5,-8,-9,-20,40,25,-5] p subarray_sum(input) =>[65, [40, 25]]
input=[-3, -1] p subarray_sum(input) =>[-1, [-1]]</lang>
Scala
The first solution solves the problem as specified, the second gives preference to the longest subsequence in case of ties. They are both vulnerable to integer overflow.
The third solution accepts any type N for which there's a Numeric[N], which includes all standard numeric types, and can be extended to include user defined numeric classes.
The last solution keeps to linear time by increasing complexity slightly. <lang scala>def maxSubseq(l: List[Int]) = l.scanRight(Nil : List[Int]) {
case (el, acc) if acc.sum + el < 0 => Nil case (el, acc) => el :: acc
} max Ordering.by((_: List[Int]).sum)
def biggestMaxSubseq(l: List[Int]) = l.scanRight(Nil : List[Int]) {
case (el, acc) if acc.sum + el < 0 => Nil case (el, acc) => el :: acc
} max Ordering.by((ss: List[Int]) => (ss.sum, ss.length))
def biggestMaxSubseq[N](l: List[N])(implicit n: Numeric[N]) = {
import n._ l.scanRight(Nil : List[N]) { case (el, acc) if acc.sum + el < zero => Nil case (el, acc) => el :: acc } max Ordering.by((ss: List[N]) => (ss.sum, ss.length))
}
def linearBiggestMaxSubseq[N](l: List[N])(implicit n: Numeric[N]) = {
import n._ l.scanRight((zero, Nil : List[N])) { case (el, (acc, _)) if acc + el < zero => (zero, Nil) case (el, (acc, ss)) => (acc + el, el :: ss) } max Ordering.by((t: (N, List[N])) => (t._1, t._2.length)) _2
}</lang>
Scheme
<lang scheme>(define (maxsubseq in)
(let loop ((_sum 0) (_seq (list)) (maxsum 0) (maxseq (list)) (l in)) (if (null? l) (cons maxsum (reverse maxseq)) (let* ((x (car l)) (sum (+ _sum x)) (seq (cons x _seq))) (if (> sum 0) (if (> sum maxsum) (loop sum seq sum seq (cdr l)) (loop sum seq maxsum maxseq (cdr l))) (loop 0 (list) maxsum maxseq (cdr l)))))))</lang>
This returns a cons of the maximum sum and (one of) the maximum subsequence(s).
Seed7
<lang seed7>$ include "seed7_05.s7i";
const func array integer: maxSubseq (in array integer: sequence) is func
result var array integer: maxSequence is 0 times 0; local var integer: number is 0; var integer: index is 0; var integer: currentSum is 0; var integer: currentStart is 1; var integer: maxSum is 0; var integer: startPos is 0; var integer: endPos is 0; begin for number key index range sequence do currentSum +:= number; if currentSum < 0 then currentStart := succ(index); currentSum := 0; elsif currentSum > maxSum then maxSum := currentSum; startPos := currentStart; endPos := index; end if; end for; if startPos <= endPos and startPos >= 1 and endPos >= 1 then maxSequence := sequence[startPos .. endPos]; end if; end func;
const proc: main is func
local const array integer: a1 is [] (-1, -2, 3, 5, 6, -2, -1, 4, -4, 2, -1); const array integer: a2 is [] (-1, -2, -3, -5, -6, -2, -1, -4, -4, -2, -1); var integer: number is 0; begin write("Maximal subsequence:"); for number range maxSubseq(a1) do write(" " <& number); end for; writeln; write("Maximal subsequence:"); for number range maxSubseq(a2) do write(" " <& number); end for; writeln; end func;</lang>
Output:
Maximal subsequence: 3 5 6 -2 -1 4 Maximal subsequence:
Standard ML
<lang sml>val maxsubseq = let
fun loop (_, _, maxsum, maxseq) [] = (maxsum, rev maxseq) | loop (sum, seq, maxsum, maxseq) (x::xs) = let val sum = sum + x val seq = x :: seq in if sum < 0 then loop (0, [], maxsum, maxseq) xs else if sum > maxsum then loop (sum, seq, sum, seq) xs else loop (sum, seq, maxsum, maxseq) xs end
in
loop (0, [], 0, [])
end;
maxsubseq [~1, ~2, 3, 5, 6, ~2, ~1, 4, ~4, 2, ~1]</lang> This returns a pair of the maximum sum and (one of) the maximum subsequence(s).
Tcl
<lang tcl>package require Tcl 8.5 set a {-1 -2 3 5 6 -2 -1 4 -4 2 -1}
- from the Perl solution
proc maxsumseq1 {a} {
set len [llength $a] set maxsum 0 for {set start 0} {$start < $len} {incr start} { for {set end $start} {$end < $len} {incr end} { set sum 0 incr sum [expr [join [lrange $a $start $end] +]] if {$sum > $maxsum} { set maxsum $sum set maxsumseq [lrange $a $start $end] } } } return $maxsumseq
}
- from the Python solution
proc maxsumseq2 {sequence} {
set start -1 set end -1 set maxsum_ 0 set sum_ 0 for {set i 0} {$i < [llength $sequence]} {incr i} { set x [lindex $sequence $i] incr sum_ $x if {$maxsum_ < $sum_} { set maxsum_ $sum_ set end $i } elseif {$sum_ < 0} { set sum_ 0 set start $i } } assert {$maxsum_ == [maxsum $sequence]} assert {$maxsum_ == [sum [lrange $sequence [expr {$start + 1}] $end]]} return [lrange $sequence [expr {$start + 1}] $end]
}
proc maxsum {sequence} {
set maxsofar 0 set maxendinghere 0 foreach x $sequence { set maxendinghere [expr {max($maxendinghere + $x, 0)}] set maxsofar [expr {max($maxsofar, $maxendinghere)}] } return $maxsofar
}
proc assert {condition {message "Assertion failed!"}} {
if { ! [uplevel 1 [list expr $condition]]} { return -code error $message }
}
proc sum list {
expr [join $list +]
}
puts "sequence: $a"
puts "maxsumseq1: [maxsumseq1 $a]"
puts [time {maxsumseq1 $a} 1000]
puts "maxsumseq2: [maxsumseq2 $a]"
puts [time {maxsumseq2 $a} 1000]</lang>
outputs
sequence: -1 -2 3 5 6 -2 -1 4 -4 2 -1 maxsumseq1: 3 5 6 -2 -1 4 367.041 microseconds per iteration maxsumseq2: 3 5 6 -2 -1 4 74.623 microseconds per iteration
Ursala
This example solves the problem by the naive algorithm of testing all possible subsequences. <lang Ursala>#import std
- import int
max_subsequence = zleq$^l&r/&+ *aayK33PfatPRTaq ^/~& sum:-0
- cast %zL
example = max_subsequence <-1,-2,3,5,6,-2,-1,4,-4,2,-1></lang> The general theory of operation is as follows.
- The
max_subsequence
function is a composition of three functions, one to generate the sequences, one to sum all of them, and one to pick out the one with the maximum sum. - The function that sums all the sequences is
* ^/~& sum:-0
which applies to every member of a list (by the*
operator) and forms a pair (using the^
operator) of the identity function (~&
) of its argument, and the reduction (:-
) of the sum over a list with a vacuous case result of 0. - The function that picks out the maximum sum is
zleq$^l&r/&
, which uses the maximizing operator ($^
) over a list of pairs with respect to the integer ordering relation (zleq
) applied to the right sides of the pairs (&r
), after which the left side (l
) of the maximizing pair is extracted. The/&
inserts an extra pair(<>,0)
at the beginning of the list before searching it in case it's empty or has only negative sums. - The function that generates all the sequences is
~&aayK33PfatPRTaq
, which appears as a suffix of the*
operator rather than being used explicitly. - The sequence generating function is in the form of a recursive conditional (
q
) with predicatea
, inductive caseayK33PfatPRT
and base casea
, meaning that in the base case of an empty list argument, the argument itself is returned. - The inductive case,
ayK33PfatPRT
is a concatenation (T
) of two functionsayK33
andfatPR
- The latter function,
fatPR
is a recursive call (R
) of the enclosing recursive conditional (f
) with the tail of the argument (at
). - The remaining function,
ayK33
uses the triangle-squared combinatorK33
of the list-lead operatory
applied to the argumenta
. - The list lead operator
y
by itself takes a non-empty list as an argument and returns a copy with the last item deleted. - The triangle-squared combinator
K33
constructs a function that takes an input list of a length , constructs a list of copies of it, and applies its operand 0 times to the head, once to the head of tail, twice to the head of the tail of the tail, and so on. Hence, an operand ofy
will generate the list of all prefixes of a list.
output:
<3,5,6,-2,-1,4>
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