{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/139234"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/139234","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Subcubic Min-Plus Product of Structured Matrices","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 𝑛-node graphs run in 𝑛³⁻⁰⁽¹⁾ time, and it is a big open problem whether a truly subcubic, 𝑂(𝑛³⁻ superscript 𝜀) for 𝜀 > 0 time algorithm exists for APSP. The Min-Plus product of two 𝑛 × 𝑛 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_html":"The All-Pairs Shortest Paths (APSP) problem is one of the most basic problems in computer science. The fastest known algorithms for APSP in 𝑛-node graphs run in 𝑛³⁻⁰⁽¹⁾ time, and it is a big open problem whether a truly subcubic, 𝑂(𝑛³⁻ superscript 𝜀) for 𝜀 > 0 time algorithm exists for APSP. The Min-Plus product of two 𝑛 × 𝑛 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_has_math":false,"creators":["Xu, Yinzhan"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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The fastest known algorithms for APSP in 𝑛-node graphs run in 𝑛³⁻⁰⁽¹⁾ time, and it is a big open problem whether a truly subcubic, 𝑂(𝑛³⁻ superscript 𝜀) for 𝜀 > 0 time algorithm exists for APSP. The Min-Plus product of two 𝑛 × 𝑛 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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["Subcubic Min-Plus Product of Structured Matrices"]}]}],"canonical_facts":{"dc:contributor.advisor":["Vassilevska Williams, Virginia"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"],"dc:creator":["Xu, Yinzhan"],"dc:date.accessioned":["2022-01-14T14:58:26Z"],"dc:date.available":["2022-01-14T14:58:26Z"],"dc:date.issued":["2021-06"],"dc:description.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 𝑛-node graphs run in 𝑛³⁻⁰⁽¹⁾ time, and it is a big open problem whether a truly subcubic, 𝑂(𝑛³⁻ superscript 𝜀) for 𝜀 > 0 time algorithm exists for APSP. The Min-Plus product of two 𝑛 × 𝑛 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."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/139234"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Subcubic Min-Plus Product of Structured Matrices"],"dc:type":["Thesis"],"thesis:degree_name":["Master","Master of Science in Electrical Engineering and Computer Science"]},"updated_at":"2026-07-22T22:22:29Z"}