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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.
##### MSC:
 68Q Theory of computing
##### Keywords:
graph algorithms; nondecreasing paths; time complexity
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##### References:
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