Algorithms for the Maximum Subarray Problem Based on Matrix Multiplication

Algorithms for the Maximum Algorithms for the Maximum Subarray Problem Based on Subarray Problem Based on

Matrix MultiplicationMatrix Multiplication

Authours ： Hisao Tamaki & Takeshi Tokuyama

Speaker ： Rung-Ren Lin

OutlineOutline

IntroductionIntroduction ½-approximation½-approximation Funny matrix multiplicationFunny matrix multiplication ReductionReduction Two little programsTwo little programs

IntroductionIntroduction ½-approximation½-approximation Funny matrix multiplicationFunny matrix multiplication ReductionReduction Two little programsTwo little programs

DefinitionDefinition

Given an Given an mm by by nn matrix, output the matrix, output the maximum subarray.maximum subarray.

HistoryHistory

Bentley proposes this problem in Bentley proposes this problem in 1984.1984.

Kadane’s algorithm solves 1-D Kadane’s algorithm solves 1-D problem in linear time.problem in linear time.

Kadane’s idea does not work for the Kadane’s idea does not work for the 2-D case.2-D case.

There’s an O(There’s an O(mm22nn) algorithm by using ) algorithm by using Kadane’s idea.Kadane’s idea.

Kadane’s algorithmKadane’s algorithm

S(i) = A(i) + max{S(i-1), 0}

i

PreprocessingPreprocessing

Given a matrix Given a matrix AA[1…[1…mm][1…][1…nn], we can ], we can compute compute BB[1…[1…mm][1…][1…nn] in O(] in O(mnmn) time ) time such that such that BB[[ii][][jj] represents the sum of ] represents the sum of AA[1…[1…ii][1…][1…jj].].

IntroductionIntroduction½-approximation½-approximation Funny matrix multiplicationFunny matrix multiplication ReductionReduction Two little programsTwo little programs

n

m

Time = mn/2 * 2 = mn

n

m

Time = mn/4 * 4= mn

Time ComplexityTime Complexity

OO((mnmn*log*logmm))

IntroductionIntroduction ½-approximation½-approximationFunny matrix multiplicationFunny matrix multiplication ReductionReduction Two little programsTwo little programs

DefinitionDefinition

Given two Given two nn by by nn matrices, matrices, AAijij & & BBijij

CCijij = max = maxk=1 to nk=1 to n{{AAikik + + BBkjkj}}

A B C

jj

i i

HistoryHistory

Funny matrix multiplication is well-Funny matrix multiplication is well-studied, since its computational studied, since its computational complexity is known to be equivalent complexity is known to be equivalent to that of “all-pairs shortest paths”.to that of “all-pairs shortest paths”.

Fredman constructs a subcubic Fredman constructs a subcubic algorithm with running timealgorithm with running time ：：

3

13

log

loglog*

n

nn

Cont’dCont’d

Takaoka improved toTakaoka improved to

in 1992.in 1992.

2

13

log

loglog*

n

nn

IntroductionIntroduction ½-approximation½-approximation Funny matrix multiplicationFunny matrix multiplicationReductionReduction Two little programsTwo little programs

DefinitionDefinition

TT((mm, , nn)) ：： computing time for the computing time for the mm by by nn matrix matrix

TTrowrow((mm, , nn)) ：： row-centeredrow-centered

TTcolcol((mm, , nn)) ：： column-centeredcolumn-centered

TT((mm, , nn) = ) = TTrowrow((mm, , nn) + ) + TTcolcol((mm, , nn) +) +

44TT((mm/2, /2, nn/2)/2)

Cont’dCont’d

TTcentercenter((mm, , nn)) ：： row & column-centeredrow & column-centered

TTrowrow((mm, , nn) = ) = TTcentercenter((mm, , nn) + 2) + 2TTrowrow((mm, , nn/2)/2)

TTcolcol((mm, , nn) = ) = TTcentercenter((mm, , nn) + 2) + 2TTcolcol((mm/2, /2, nn))

ReductionReduction

A B

C D

Cont’dCont’d

Let Let AA, , BB, , CC, , DD[1…[1…mm/2][1…/2][1…nn/2] be the /2] be the sum of the given area.sum of the given area.

XXijij = max = maxkk=1 to =1 to mm/2/2{{AAikik + + CCjkjk}}

YYijij = max = maxkk=1 to =1 to mm/2/2{{BBikik + + DDjkjk}}

output = max{output = max{XXij + ij + YYijij}}

IntroductionIntroduction ½-approximation½-approximation Funny matrix multiplicationFunny matrix multiplication ReductionReductionTwo little programsTwo little programs

13-card13-card

PacmanPacman

ChallengesChallenges

It’s difficult to search 3-D models.It’s difficult to search 3-D models.

The EndThe End