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A simple approach to nondecreasing paths
Information Processing Letters ( IF 0.7 ) Pub Date : 2020-07-01 , DOI: 10.1016/j.ipl.2020.105992
Mirosław Kowaluk , Andrzej Lingas

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)loglogn) 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 O˜(nωmin{awGω,WG}) time, where awG denotes the average number of distinct weights of edges incident to the same vertex while WG stands for the total number of distinct edge weights in the input graph G, and ω is the exponent of fast matrix multiplication.



中文翻译:

一种不减少路径的简单方法

我们将有向边加权图中的非递减路径问题(具有最小的最后边权)简化为有向未加权图中的可达性问题。减少产生了另一种简单的方法,可以在几乎线性的时间内解决单源非递减路径问题。如果边缘权重是整数,那么我们的算法可以在Øñ+日志日志ñ字RAM上的时间,其中nm分别代表输入图中顶点和边的数量。通过使用约简,我们还获得了用于所有对的非递减路径问题的简单算法。它在Øñω{一种wGωw ^G} 时间,地点 一种wG 表示入射到同一顶点的边的不同权重的平均数,而 w ^G代表输入图G中不同边缘权重的总数,并且ω是快速矩阵乘法的指数。

更新日期:2020-07-01
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