Efficient heuristic algorithms for the maximum subarray problem

Rung-Ren Lin and Kun-Mao Chao

Preview

Trying to guess the answer intelligently. Preliminary experiments show that these appr

oaches are very promising for locating the maximum subarray in a given two-dimensional array.

Review of Maximum Subarray

Bentley posed the maximum subarray problem in his book “Programming Pearls” in 1984.

He introduces Kadane's algorithm for the one-dimensional case, whose time is linear.

Cont’d

Given an m×n array of numbers, Bentley solved the problem in O(m2n) time.

An improvement O(m2n(loglogm/logm)0.5) was given by Tamaki et al. in 1998.This algorithm is heavily recursive and complicated.

Applications

20 25 30 35 40 45 50 55 60$

$$

$$$

$$$$

$$$$$

$$$$$$

$$$$$$$

Cont’d

Heuristic Methods

Given a 2-D array A[1..m][1..n], let TL[i][j] denote the sum of the rectangle A[1..i][1..j].

-3 1 2

2 -1 0

0 -2 1

-3 -2 0

-1 -1 1

-1 -3 0

A TL

Constructing TL Matrix

for i = 2 to n do

for j = 1 to n do

A[i][j] = A[i][j] + A[i-1][j]

-3 1 2

2 -1 0

0 -2 1

-3 -1 2

-1 0 2

-1 -2 3

A A’

Cont’d

for i = 2 to n do

for j = 1 to n do

A’[j][i] = A’[j][i] + A’[j][i-1]

-3 1 2

-1 0 2

-1 -2 3

-3 -2 0

-1 -1 1

-1 -3 0

A’ TL

Computing an Arbitrary Rectangle

How to guess ？

Each rectangle can be computed by TL matrix, and the answer is MAX( + - - ).

the larger the better. the smaller the better.

Cont’d

We try only those entries which are in the top k-th, or in the bottom k-th for a given k.

We test only O(k) times instead of O(n) times. Since there are in total O(m2) pairs, this step takes O(km2).

4-Corner

Happy New Year!!

Interesting Questions

128 gold. The way to heaven and hell. 10 smart prisoners.