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A simple approach to nondecreasing paths. (English) Zbl 07256098
Summary: We present a simple reduction of the problem of nondecreasing paths (with minimal last edge weight) in a directed edge-weighted graph to a reachability problem in a directed unweighted graph. The reduction yields an alternative simple method of solving the single-source nondecreasing paths problem in almost linear time. If the edge weights are integers then our algorithm can be implemented in \(O((n+m)\sqrt{\log\log n})\) time on the word RAM, where \(n\) and \(m\) stand for the number of vertices and edges in the input graph, respectively. By using the reduction, we obtain also a simple algorithm for the all-pairs nondecreasing paths problem. It runs in \(\widetilde{O}(n^\omega\min\{a w_G^\omega,W_G\})\) time, where \(aw_G\) denotes the average number of distinct weights of edges incident to the same vertex while \(W_G\) stands for the total number of distinct edge weights in the input graph \(G\), and \(\omega\) is the exponent of fast matrix multiplication.
68Q Theory of computing
Full Text: DOI
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