A Linear Time Algorithm for the k Maximal Sums Problem 1st Edition by Fredrik Bengtsson, Jingsen Chen ISBN 9783540305514

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Finding the sub-vector with the largest sum in a sequence of n numbers is known as the maximum sum problem. Finding the k sub-vectors with the largest sums is a natural extension of this, and is known as the k maximal sums problem. In this paper we design an optimal O(n+k) time algorithm for the k maximal sums problem. We use this algorithm to obtain algorithms solving the two-dimensional k maximal sums problem in O(m2 · n+ k) time, where the input is an m × n matrix with m ≤ n. We generalize this algorithm to solve the d-dimensional problem in O(n 2d−1 +k) time. The space usage of all the algorithms can be reduced to O(n d−1 + k). This leads to the first algorithm for the k maximal sums problem in one dimension using O(n + k) time and O(k) space.

A Linear Time Algorithm for the k Maximal Sums Problem 1st Table of contents:

• Problem Definition and Background
• Previous Approaches to Maximal Sums
• The Linear Time Algorithm
• Algorithm Design and Implementation
• Complexity Analysis
• Experimental Results and Evaluation
• Applications and Use Cases
• Challenges and Limitations
• Future Research Directions

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