Faster Algorithms for All Pairs Non-decreasing Paths Problem
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In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time Õ(n^3 + ω/2) = Õ(n^2.686). Here n is the number of vertices, and ω < 2.373 is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for (, )-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for (, )-matrix product. The previous best upper bound for APNP on weighted digraphs was Õ(n^1/2(3 + 3 - ω/ω + 1 + ω)) = Õ(n^2.78) [Duan, Gu, Zhang 2018]. We also show an Õ(n^2) time algorithm for APNP in undirected graphs which also reaches optimal within logarithmic factors.
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