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

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in n3−o(1) time, and it is widely open whether it has an O(n3−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) time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS’12] runs in O(M0.7519n2.5286) time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved O(M0.8043n2.4957) 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 n2.5−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(Mn) time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges. 2012 ACM Subject Classification Theory of computation → Shortest paths

[1]  Yinzhan Xu,et al.  Faster Dynamic Range Mode , 2020, ICALP.

[2]  Shiri Chechik,et al.  Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem , 2020, ICALP.

[3]  Virginia Vassilevska Williams,et al.  Monochromatic Triangles, Intermediate Matrix Products, and Convolutions , 2020, ITCS.

[4]  Grazia Lotti,et al.  On the Asymptotic Complexity of Rectangular Matrix Multiplication , 1983, Theor. Comput. Sci..

[5]  Gideon Yuval,et al.  An Algorithm for Finding All Shortest Paths Using N^(2.81) Infinite-Precision Multiplications , 1976, Inf. Process. Lett..

[6]  Mark H. Overmars,et al.  The Design of Dynamic Data Structures , 1987, Lecture Notes in Computer Science.

[7]  Noga Alon,et al.  On the Exponent of the All Pairs Shortest Path Problem , 1991, J. Comput. Syst. Sci..

[8]  Timothy M. Chan,et al.  Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths , 2021, ICALP.

[9]  Uri Zwick,et al.  All pairs shortest paths using bridging sets and rectangular matrix multiplication , 2000, JACM.

[10]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[11]  Fabrizio Grandoni,et al.  Truly Sub-cubic Algorithms for Language Edit Distance and RNA-Folding via Fast Bounded-Difference Min-Plus Product , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[12]  Timothy M. Chan,et al.  Linear-Space Data Structures for Range Mode Query in Arrays , 2011, Theory of Computing Systems.

[13]  François Le Gall,et al.  Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor , 2017, SODA.

[14]  Timothy M. Chan More algorithms for all-pairs shortest paths in weighted graphs , 2007, STOC '07.

[15]  J. Ian Munro,et al.  Improved Time and Space Bounds for Dynamic Range Mode , 2018, ESA.

[16]  Fabrizio Grandoni,et al.  All-Pairs LCA in DAGs: Breaking through the O(n2.5) barrier , 2020, ArXiv.

[17]  Michael J. Fischer,et al.  Boolean Matrix Multiplication and Transitive Closure , 1971, SWAT.

[18]  Raphael Yuster,et al.  Efficient algorithms on sets of permutations, dominance, and real-weighted APSP , 2009, SODA.

[19]  Raphael Yuster,et al.  Answering distance queries in directed graphs using fast matrix multiplication , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[20]  Josh Alman,et al.  A Refined Laser Method and Faster Matrix Multiplication , 2020, SODA.

[21]  Shiri Chechik,et al.  Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs , 2020, ICALP.

[22]  Victor Y. Pan,et al.  Fast Rectangular Matrix Multiplication and Applications , 1998, J. Complex..

[23]  Fabrizio Grandoni,et al.  Faster Replacement Paths and Distance Sensitivity Oracles , 2019, ACM Trans. Algorithms.

[24]  Yinzhan Xu,et al.  Truly Subcubic Min-Plus Product for Less Structured Matrices, with Applications , 2019, SODA.

[25]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[26]  Shiri Chechik,et al.  Near Optimal Algorithms For The Single Source Replacement Paths Problem , 2019, SODA.

[27]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[28]  V. V. Williams ON SOME FINE-GRAINED QUESTIONS IN ALGORITHMS AND COMPLEXITY , 2019, Proceedings of the International Congress of Mathematicians (ICM 2018).

[29]  Ryan Williams,et al.  Faster all-pairs shortest paths via circuit complexity , 2013, STOC.

[30]  Noga Alon,et al.  1 Derandomizing the Replacment Paths Algorithm of Roditty and Zwick [ 37 , 2019 .