Efficient Algorithms for the Sum Selection Problem and K Maximum Sums Problem

Given a sequence of n real numbers A = a1, a2,..., an and a positive integer k, the Sum Selection Problem is to find the segment A(i,j) = ai , ai+1,..., aj such that the rank of the sum s(i, j) = ∑t=ijat is k over all $\frac{n(n-1)}{2}$ segments. We present a deterministic algorithm for this problem that runs in O(n logn) time. The previously best known randomized algorithm for this problem runs in expected O(n logn) time. Applying this algorithm we can obtain a deterministic algorithm for the k Maximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in O(n logn + k) time. The previously best known randomized and deterministic algorithms for the k Maximum Sums Problem run respectively in expected O(n logn + k) and O(n log2n + k) time in the worst case.

[1]  Narsingh Deo,et al.  Parallel Processing Letters C World Scientiic Publishing Company Parallel Algorithms for Maximum Subsequence and Maximum Subarray , 2022 .

[2]  E. Szemerédi,et al.  Sorting inc logn parallel steps , 1983 .

[3]  Richard Cole,et al.  Slowing down sorting networks to obtain faster sorting algorithms , 2015, JACM.

[4]  Selim G. Akl,et al.  Application of Broadcasting with Selective Reduction to the Maximal Sum Subsegment Problem , 1991, Int. J. High Speed Comput..

[5]  János Komlós,et al.  An 0(n log n) sorting network , 1983, STOC.

[6]  Douglas R. Smith.  Applications of a Strategy for Designing Divide-and-Conquer Algorithms , 1987, Sci. Comput. Program..

[7]  Jon Louis Bentley.  Programming pearls: perspective on performance , 1984, CACM.

[8]  Tomasz Imielinski,et al.  Mining association rules between sets of items in large databases , 1993, SIGMOD Conference.

[9]  Yasuhiko Morimoto,et al.  Data mining using two-dimensional optimized association rules: scheme, algorithms, and visualization , 1996, SIGMOD '96.

[10]  Tadao Takaoka,et al.  Algorithms for the problem of K maximum sums and a VLSI algorithm for the K maximum subarrays problem , 2004, 7th International Symposium on Parallel Architectures, Algorithms and Networks, 2004. Proceedings..

[11]  David Gries,et al.  A Note on a Standard Strategy for Developing Loop Invariants and Loops , 1982, Sci. Comput. Program..

[12]  Fredrik Bengtsson,et al.  Efficient Algorithms for k Maximum Sums , 2006, Algorithmica.

[13]  Kuan-Yu Chen,et al.  Improved algorithms for the k maximum-sums problems , 2006, Theor. Comput. Sci..

[14]  Philip S. Yu.  Review - Mining Association Rules between Sets of Items in Large Databases , 1999, ACM SIGMOD Digit. Rev..

[15]  Endre Szemerédi,et al.  An Optimal-Time Algorithm for Slope Selection , 1989, SIAM J. Comput..

[16]  Jon Bentley,et al.  Programming pearls: algorithm design techniques , 1984, CACM.

[17]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[18]  Hisao Tamaki,et al.  Algorithms for the maximum subarray problem based on matrix multiplication , 1998, SODA '98.

[19]  D. T. Lee,et al.  Randomized algorithm for the sum selection problem , 2005, Theor. Comput. Sci..

[20]  Bernard Chazelle,et al.  Optimal Slope Selection Via Cuttings , 1994, CCCG.

[21]  Selim G. Akl,et al.  Parallel Maximum Sum Algorithms on Interconnection Networks , 1999 .

[22]  Tadao Takaoka,et al.  Efficient Algorithms for the Maximum Subarray Problem by Distance Matrix Multiplication , 2002, CATS.