2026-09-19T16:59:44Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1392342022-01-15T03:01:45Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Vassilevska Williams, Virginia
Xu, Yinzhan
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2022-01-14T14:58:26Z
2022-01-14T14:58:26Z
2021-06
2021-06-24T19:42:21.535Z
https://hdl.handle.net/1721.1/139234
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.
S.M.
Massachusetts Institute of Technology
In Copyright - Educational Use Permitted
Copyright MIT
http://rightsstatements.org/page/InC-EDU/1.0/
Subcubic Min-Plus Product of Structured Matrices
Thesis
application/pdf
Master
Master of Science in Electrical Engineering and Computer Science
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="f8f02cda-44d5-41c1-ad62-cf19f47b1ba9">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Title>Subcubic Min-Plus Product of Structured Matrices</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2021-06</PublicationDate>
<Authors>
<Author>
<DisplayName>Xu, Yinzhan</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>http://rightsstatements.org/page/InC-EDU/1.0/</License>
<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.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>