Algorithms for the Maximum Subarray Problem Based on Matrix Multiplication
Hisao Tamaki,Takeshi Tokuyama +1 more
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TL;DR: The first subcubic algorithm is given, by reducing the problem to “funny matrix multiplication”, where the scalar product and addition in usual matrix multiplication are replaced by addition and max operations, respectively.
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Abstract: Given an M×N array of reals, we want to find a rectangular contiguous subarray such that the sum of the entries in the subarray is maximized. Since Bentley posed this problem in his Programming Pearls column in 1984 with an O (NM 2) time solution, no progress on the sequential complexity has been reported to date. We give the first subcubic algorithm, by reducing the problem to “funny matrix multiplication”, where the scalar product and addition in usual matrix multiplication are replaced by addition and max operations, respectively. We also give a faster e-approximation algorithm via the same reduction.
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Citations
•Proceedings Article
Learning To Count Objects in Images
Victor Lempitsky,Andrew Zisserman +1 more
- 06 Dec 2010
TL;DR: This work focuses on the practically-attractive case when the training images are annotated with dots, and introduces a new loss function, which is well-suited for visual object counting tasks and at the same time can be computed efficiently via a maximum subarray algorithm.
Subcubic Equivalences between Path, Matrix and Triangle Problems
Virginia Vassilevska Williams,Ryan Williams +1 more
- 23 Oct 2010
TL;DR: Generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure are shown.
Subcubic Equivalences Between Path, Matrix, and Triangle Problems
TL;DR: Generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure are shown.
Efficient Algorithms for the Maximum Subarray Problem by Distance Matrix Multiplication
TL;DR: An efficient algorithm that maximizes the sum of array elements of a subarray of a two-dimensional array that correlates two parameters involved in data, such as ages and income for the amount of sales per some period is designed.
106
Efficient algorithms for subwindow search in object detection and localization
Senjian An,Patrick Peursum,Wanquan Liu,Svetha Venkatesh +3 more
- 20 Jun 2009
TL;DR: Two significantly faster methods based on the linear-time Kadane's Algorithm for 1D maximum subarray search are presented, one of which is a novel, computationally superior branch-and-bound method and the other an approximate algorithm based on alternating search, whose computational complexity is typically O(n2).
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