Maximum subarray problem: Difference between revisions

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The maximum subarray problem was proposed by [[Ulf Grenander]] in 1977 as a simplified model for [[maximum likelihood estimation]] of patterns in digitized images.{{sfn|Bentley|1984|p=868-869}}

Grenander was looking to find a rectangular subarray with maximum sum, in a two-dimensional array of real numbers. A brute-force algorithm for the two-dimensional problem runs in ''O''(''n''⁶) time; because this was prohibitively slow, Grenander proposed the one-dimensional problem to gain insight into its structure. Grenander derived an algorithm that solves the one-dimensional problem in ''O''(''n''²) time, using [[prefix sum]]{{NoteTag

|By using a precomputed table of cumulative sums <math>S[k] = \sum_{x=1}^k A[x]</math> to compute the subarray sum <math>\sum_{x=i}^j A[x] = S[j] - S[i-1]</math> in constant time<!--sentence fragment-->

}}, improving the brute force running time of ''O''(''n''³). When [[Michael Shamos]] heard about the problem, he overnight devised an ''O''(''n'' log ''n'') [[divide-and-conquer algorithm]] for it.

}}

improving the brute force running time of ''O''(''n''³). When [[Michael Shamos]] heard about the problem, he overnight devised an ''O''(''n'' log ''n'') [[divide-and-conquer algorithm]] for it.

Soon after, Shamos described the one-dimensional problem and its history at a [[Carnegie Mellon University]] seminar attended by [[Jay Kadane]], who designed within a minute an ''O''(''n'')-time algorithm,{{sfn|Bentley|1984|p=868-869}}{{sfn|Bentley|1989|p=76-77}}{{sfn|Gries|1982|p=211}} which is as fast as possible.<ref group=note>since every algorithm must at least scan the array once which already takes ''O''(''n'') time</ref> In 1982, [[David Gries]] obtained the same ''O''(''n'')-time algorithm by applying [[Edsger W. Dijkstra |Dijkstra]]'s "standard strategy";{{sfn|Gries|1982|p=209-211}} in 1989, [[Richard Bird (computer scientist)|Richard Bird]] derived it by purely algebraic manipulation of the brute-force algorithm using the [[Bird–Meertens formalism]].{{sfn|Bird|1989|loc=Sect.8, p.126}}

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== Applications ==

{{expert needed|Computational biology|section|reason=fix inline tags|date=September 2019}}

Maximum subarray problems arise in many fields, such as genomic [[sequence analysis]] and [[computer vision]].

Genomic sequence analysis employs maximum subarray algorithms to identify important biological segments of protein sequences.{{citation needed|date=Marchthat 2020}}have unusual properties, by assigning scores to points within the sequence that are positive when a motif to be recognized is present, and negative when it is not, and then seeking the maximum subarray among these scores. These problems include conserved segments, GC-rich regions, tandem repeats, low-complexity filter, DNA binding domains, and regions of high charge.<ref>{{citation neededharvtxt|date=OctoberRuzzo|Tompa|1999}}; 2017{{harvtxt|Alves|Cáceres|Song|2004}}</ref>

In [[computer vision]], bitmap images generally consist only of positive values, for which the maximum subarray problem is trivial: the result is always the whole array. However, after subtracting a threshold value (such as the average pixel value) from each pixel, so that above-average pixels will be positive and below-average pixels will be negative, the maximum subarray problem can be applied to the modified image to detect bright areas within it.<ref>{{harvtxt|Bae|Takaoka|2006}}; {{harvtxt|Weddell|Read|Thaher|Takaoka|2013}}</ref>

In [[computer vision]], maximum-subarray algorithms are used on bitmap images to detect the brightest area in an image.

==Kadane's algorithm==

===EmptyNo empty subarrays admitted===

{| align="right" class="wikitable collapsible collapsed"

[[Joseph Born Kadane|Kadane's]] algorithm scans the given array <math>A[1\ldots n]</math> from left to right.

! Example run

|-

| [[File:Kadane run −2,1,−3,4,−1,2,1,−5,4.gif|thumb|500px|Execution of Kadane's algorithm on the [[#top|above]] example array. ''{{color|#0000c0|Blue}}:'' subarray with largest sum ending at ''i''; ''{{color|#00c000|green}}:'' subarray with largest sum encountered so far; a lower case letter indicates an empty array; variable ''i'' is left implicit in Python code.]]

|}

[[Joseph Born Kadane|Kadane's]] original algorithm solves the problem version when empty subarrays are admitted. It scans the given array <math>A[1\ldots n]</math> from left to right.

In the <math>j</math>th step, it computes the subarray with the largest sum ending at <math>j</math>; this sum is maintained in variable <code>current_sum</code>.{{NoteTag

|named <code>MaxEndingHere</code> in {{harvtxt|Bentley|1989}}, and <code>c</code> in {{harvtxt|Gries|1982}}

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and easily obtained as the maximum of all values of <code>current_sum</code> seen so far, cf. line 7 of the algorithm.

As a [[loop invariant]], in the <math>j</math>th step, the old value of <code>current_sum</code> holds the maximum over all <math>i \in \{ 1,\ldots, j-1 \}</math> of the sum <math>A[i]+\cdots+A[j-1]</math>.{{NoteTag

|This sum is <math>0</math> when <math>i=j</math>, corresponding to the empty subarray <math>A[j\ldots j-1]</math>.

}}

Therefore, <code>current_sum</code><math>+A[j]</math>{{NoteTag|

In the Python code below, <math>A[j]</math> is expressed as <code>x</code>, with the index <math>j</math> left implicit.

}}

is the maximum over all <math>i \in \{ 1,\ldots, j-1 \}</math> of the sum <math>A[i]+\cdots+A[j]</math>. To extend the latter maximum to cover also the case <math>i=j+1</math>, it is sufficient to consider also the emptysingleton subarray <math>A[j+1 \; \ldots \; j]</math>. This is done in line 6 by assigning <math>\max(0A[j],</math><code>current_sum</code><math>+A[j])</math> as the new value of <code>current_sum</code>, which after that holds the maximum over all <math>i \in \{ 1, \ldots, j+1 \}</math> of the sum <math>A[i]+\cdots+A[j]</math>.

Thus, the problem can be solved with the following code,{{sfn|Bentley|1989|p=74}}{{sfn78,171|ps=. Bentley, like Gries, first introduces the variant admitting empty subarrays, see [[#Empty subarrays admitted|1982|p=211below]], and describes only the changes.}} expressed here in [[Python (programming language)|Python]]:.

<syntaxhighlight lang="python" line>

def max_subarray(numbers):

"""Find the largest sum of any contiguous subarray."""

best_sum = 0float('-inf')

current_sum = 0

for x in numbers:

current_sum = max(0x, current_sum + x)

best_sum = max(best_sum, current_sum)

return best_sum

</syntaxhighlight>

If the input contains no positive element, the returned value is that of the largest element (i.e., the value closest to 0), or negative infinity if the input was empty. For correctness, an exception should be raised when the input array is empty, since an empty array has no maximum nonempty subarray. If the array is nonempty, its first element could be used in place of negative infinity, if needed to avoid mixing numeric and non-numeric values.

This version of the algorithm will return 0 if the input contains no positive elements (including when the input is empty).

The algorithm can be adapted to the case which disallowsallows empty subarrays or to keep track of the starting and ending indices of the maximum subarray.

This algorithm calculates the maximum subarray ending at each position from the maximum subarray ending at the previous position, so it can be viewed as a trivial case of [[dynamic programming]].

===No emptyEmpty subarrays admitted===

{| align="right" class="wikitable collapsible collapsed"

For the variant of the problem which disallows empty subarrays, <code>best_sum</code> should be initialized to negative infinity instead{{sfn|Bentley|1989|p=78,171}}

! Example run

|-

| [[File:Kadane run −2,1,−3,4,−1,2,1,−5,4.gif|thumb|500px|Execution of Kadane's algorithm on the [[#top|above]] example array. ''{{color|#0000c0|Blue}}:'' subarray with largest sum ending at ''i''; ''{{color|#00c000|green}}:'' subarray with largest sum encountered so far; a lower case letter indicates an empty array; variable ''i'' is left implicit in Python code.]]

|}

Kadane's algorithm, as originally published, is for solving the problem variant which allows empty subarrays.{{sfn|Bentley|1989|p=74}}{{sfn|Gries|1982|p=211}}

In such variant, the answer is 0 when the input contains no positive elements (including when the input is empty).

The variant is obtained with two changes in code: in line 3, <code>best_sum</code> should be initialized to 0 to account for the empty subarray <math>A[0 \ldots -1]</math>

<syntaxhighlight lang="python" line start="3">

best_sum = - infinity0;

</syntaxhighlight>

and alsoline 6 in the for loop <code>current_sum</code> should be updated asto <code>max(x0, current_sum + x)</code>.{{NoteTag

|While the latter modification is not mentioned by {{harvtxt|Bentley|1989}} does not mention this difference, itusing achieves<code>x</code> maintaininginstead of <code>0</code> in the modified[[#No empty subarrays admitted|above]] version without empty subarrays achieves maintaining its loop invariant <code>current_sum</code><math>=\max_{i \in \{ 1, ..., j-1 \}} A[i]+...+A[j-1]</math> at the beginning of the <math>j</math>th step.

}}

<syntaxhighlight lang="python" line start="6">

current_sum = max(x0, current_sum + x)

</syntaxhighlight>

In that case, if the input contains no positive element, the returned value is that of the largest element (i.e., the value closest to 0), or negative infinity if the input was empty. For correctness, an exception should be raised when the input array is empty, since an empty array has no maximum nonempty subarray.

As a [[loop invariant]], in the <math>j</math>th step, the old value of <code>current_sum</code> holds the maximum over all <math>i \in \{ 1,\ldots, j \}</math> of the sum <math>A[i]+\cdots+A[j-1]</math>.{{NoteTag

|This sum is <math>0</math> when <math>i=j</math>, corresponding to the empty subarray <math>A[j\ldots j-1]</math>.

}}

Therefore, <code>current_sum</code><math>+A[j]</math>

is the maximum over all <math>i \in \{ 1,\ldots, j \}</math> of the sum <math>A[i]+\cdots+A[j]</math>. To extend the latter maximum to cover also the case <math>i=j+1</math>, it is sufficient to consider also the empty subarray <math>A[j+1 \; \ldots \; j]</math>. This is done in line 6 by assigning <math>\max(0,</math><code>current_sum</code><math>+A[j])</math> as the new value of <code>current_sum</code>, which after that holds the maximum over all <math>i \in \{ 1, \ldots, j+1 \}</math> of the sum <math>A[i]+\cdots+A[j]</math>. Machine-verified [[C (programming language)|C]] / [[Frama-C]] code of both variants can be found [[:commons:File:Kadane run −2,1,−3,4,−1,2,1,−5,4.gif#Source code|here]].

===Computing the best subarray's position===

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{{NoteFoot|30em}}

==ReferencesNotes==

{{Reflist}}

==References==

*{{citation

| last1 = Alves | first1 = Carlos E. R.

| last2 = Cáceres | first2 = Edson

| last3 = Song | first3 = Siang W.

| editor1-last = Kranzlmüller | editor1-first = Dieter

| editor2-last = Kacsuk | editor2-first = Péter

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| volume = 3241

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| doi-access = free

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}}

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}}.

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| journal = The Computer Journal

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}}

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}}

}}{{dead link|date=May 2021|bot=medic}}{{cbignore|bot=medic}}

*{{citation

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| contribution = A linear time algorithm for the ''k'' maximal sums problem

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| hdl=1813/6370

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| last1 = Ruzzo | first1 = Walter L.

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| editor4-last = Brutlag | editor4-first = Douglas L.

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| editor6-last = Mewes | editor6-first = Hans-Werner

| editor7-last = Zimmer | editor7-first = Ralf

| contribution = A Linear Time Algorithm for Finding All Maximal Scoring Subsequences

| contribution-url = https://www.aaai.org/Library/ISMB/1999/ismb99-027.php

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*{{citation

| first = Tadao | last = Takaoka

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| pages = 446–452

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| access-date = November 17, 2018

}}

*{{citation

| last1 = Weddell | first1 = Stephen John

| last2 = Read | first2 = Tristan

| last3 = Thaher | first3 = Mohammed

| last4 = Takaoka | first4 = Tadao

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| issue = 4

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| title = Maximum subarray algorithms for use in astronomical imaging

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| year = 2013| bibcode = 2013JEI....22d3011W

}}

== External links ==

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[[Category:Optimization algorithms and methods]]

[[Category:Dynamic programming]]

[[Category:Polynomial-time problems]]

[[Category:Articles with example Python (programming language) code]]