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Improved Time Bounds for All Pairs Non-decreasing Paths in General Digraphs

Authors Ran Duan, Yong Gu, Le Zhang

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Author Details

Ran Duan
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Yong Gu
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Le Zhang
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

Funding

This work was partially supported by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61361136003.
  • Duan, Ran: Supported by a China Youth 1000-Talent grant.

Cite As Get BibTex

Ran Duan, Yong Gu, and Le Zhang. Improved Time Bounds for All Pairs Non-decreasing Paths in General Digraphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.44

Abstract

We present improved algorithms for solving the All Pairs Non-decreasing Paths (APNP) problem on weighted digraphs. Currently, the best upper bound on APNP is O~(n^{(9+omega)/4})=O(n^{2.844}), obtained by Vassilevska Williams [TALG 2010 and SODA'08], where omega<2.373 is the usual exponent of matrix multiplication. Our first algorithm improves the time bound to O~(n^{2+omega/3})=O(n^{2.791}). The algorithm determines, for every pair of vertices s, t, the minimum last edge weight on a non-decreasing path from s to t, where a non-decreasing path is a path on which the edge weights form a non-decreasing sequence. The algorithm proposed uses the combinatorial properties of non-decreasing paths. Also a slightly improved algorithm with running time O(n^{2.78}) is presented.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Graph algorithms
  • Matrix multiplication
  • Non-decreasing paths

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