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Efficient heuristic algorithms for the maximum subarray problem

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

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Efficient heuristic algorithms for the maximum subarray problem

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  1. Efficient heuristic algorithms for the maximum subarray problem Rung-Ren Lin and Kun-Mao Chao

  2. Preview • Trying to guess the answer intelligently. • Preliminary experiments show that these approaches are very promising for locating the maximum subarray in a given two-dimensional array.

  3. 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.

  4. 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.

  5. Applications

  6. Cont’d

  7. 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]. A TL

  8. 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] A A’

  9. Cont’d • for i = 2 to n do for j = 1 to n do A’[j][i] = A’[j][i] + A’[j][i-1] A’ TL

  10. Computing an Arbitrary Rectangle

  11. How to guess？ • Each rectangle can be computed by TL matrix, and the answer is MAX( + - - ). • the larger the better. • the smaller the better.

  12. 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).

  13. 4-Corner

  14. Happy New Year!!

  15. Interesting Questions • 128 gold. • The way to heaven and hell. • 10 smart prisoners.

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