Subcubic Min-Plus Product of Structured Matrices
Name
Xu-xyzhan-SM-EECS-2021-thesis.pdf
Description
Thesis PDF
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478.06 KB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Xu, Yinzhan
Advisor(s)
Vassilevska Williams, Virginia
Date Issued
June 2021
Publisher
Massachusetts Institute of Technology
Abstract
The All-Pairs Shortest Paths (APSP) problem is one of the most basic problems in computer science. The fastest known algorithms for APSP in ๐-node graphs run in ๐ยณโปโฐโฝยนโพ time, and it is a big open problem whether a truly subcubic, ๐(๐ยณโป superscript ๐) for ๐ > 0 time algorithm exists for APSP. The Min-Plus product of two ๐ ร ๐ matrices is known to be equivalent to APSP, where the optimal running times of the two problems differ by at most a constant factor. A natural way to approach understanding the complexity of APSP is thus understanding what structure (if any) is needed to solve Min-Plus Product in truly subcubic time. The goal of this thesis is to get truly subcubic algorithms for Min-Plus products for less structured inputs than what was previously known, and to apply them to versions of APSP and other problems. This thesis gives sub-cubic algorithms for two interesting cases of structured Min-Plus Products: Min-Plus product between matrices with a constant additive approximate rank and Min-Plus product between monotone matrices, whose definitions are deferred to the main text. These faster algorithms have a wide range of applications, including Geometric APSP, Maximum Subarray, Range Mode and Single Source Replacement Paths.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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