Randomized Algorithm for the Sum Selection 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 $s(i, j) = \sum_{t = i}^{j}{a_{t}}$ is k over all ${n(n-1)} \over {2}$ segments. We will give a randomized algorithm for this problem that runs in expected O(n log n) time. Applying this algorithm we can obtain an algorithm for the kMaximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in expected O(n log n + k) time. The previously best known algorithm for the kMaximum Sums Problem runs in O(n log2n + k) time in the worst case.

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

[2]  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..

[3]  Jirí Matousek,et al.  Randomized Optimal Algorithm for Slope Selection , 1991, Inf. Process. Lett..

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

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

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

[7]  Fredrik Bengtsson,et al.  Efficient Algorithms for k Maximum Sums , 2004, ISAAC.

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

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

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

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

[12]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

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

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

[15]  David M. Mount,et al.  A randomized algorithm for slope selection , 1992, Int. J. Comput. Geom. Appl..

[16]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[17]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.