Improved Bounds for Rectangular Monotone Min-Plus Product
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In a recent breakthrough paper, Chi et al. (STOC'22) introduce an Õ(n^3 + ω/2) time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions n × n and entries bounded by O(n). This greatly improves upon the previous Õ(n^12 + ω/5) time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: M-bounded Single Source Replacement Path, Batch Range Mode, k-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al. since the improvement requires additional optimization.
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