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Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths

Authors Yuzhou Gu, Adam Polak , Virginia Vassilevska Williams, Yinzhan Xu

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  • Theory of computation → Shortest paths
Keywords
  • APSP
  • Min-Plus Product
  • Range Mode
  • Single-Source Replacement Paths

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Abstract

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in n^{3-o(1)} time, and it is widely open whether it has an O(n^{3-ε}) time algorithm for ε > 0. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product.
A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an n-vertex graph with potentially negative edge weights in {-M, …, M}.
SSRP with positive integer edge weights bounded by M can be solved in Õ(Mn^ω) time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in O(M^{0.7519} n^{2.5286}) time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved O(M^{0.8043} n^{2.4957}) time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require n^{2.5-o(1)} time. 
Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an Õ(M n^{ω}) time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges.

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Yuzhou Gu, Adam Polak, Virginia Vassilevska Williams, and Yinzhan Xu. Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.75

Author Details

Yuzhou Gu
  • MIT, Cambridge, MA, USA
Adam Polak
  • École Polytechnique Fédérale de Lausanne, Switzerland
Virginia Vassilevska Williams
  • MIT, Cambridge, MA, USA
Yinzhan Xu
  • MIT, Cambridge, MA, USA

Funding

  • Polak, Adam: Supported by the Swiss National Science Foundation within the project Lattice Algorithms and Integer Programming (185030). Part of this work was done at Jagiellonian University, supported by Polish National Science Center grant 2017/27/N/ST6/01334.
  • Vassilevska Williams, Virginia: Supported by an NSF CAREER Award, NSF Grants CCF-1528078, CCF-1514339 and CCF-1909429, a BSF Grant BSF:2012338, a Google Research Fellowship and a Sloan Research Fellowship.
  • Xu, Yinzhan: Supported by NSF Grant CCF-1528078.

Acknowledgements

This project was started as part of an open problem session held alongside MIT subject 6.890 taught by the third author. The authors would like to acknowledge the rest of the participants in the open problems session including but not limited to Angelos Pelecanos, Nicole Wein and Yuancheng Yu.

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