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Quantum Algorithm for Shortest Path Search in Directed Acyclic Graph
Moscow University Computational Mathematics and Cybernetics Pub Date : 2019-05-02 , DOI: 10.3103/s0278641919010023
K. R. Khadiev , L. I. Safina

In this work, we consider a well-known “Single Source Shortest Path Search” problems for weighted directed acyclic graphs (DAGs). We suggest a quantum algorithm with time complexity \(O(\sqrt {nm} \,\log \;n)\) and O(1/n) error probability, where n is a number of Vertexes, m is the number of edges. We use quantum dynamic programming approach (Khadiev, 2018) and Dürr and Høyer minimum search algorithm to speed up our search. Our algorithm is better than C. Dürr and coauthors’ quantum algorithm for an undirected graph. The time complexity of C. Dürr’s algorithm is \(O(\sqrt {nm} \,{(\log \;n)^2})\). The best known deterministic algorithm for the problem is based on a dynamic programming approach and its time complexity is O (n + m). At the same time, if we use algorithms for general graphs, then we do not get better results. The time complexity of best implementations of Dijkstra’s algorithm with Fibonacci heap is O (m + n log n).

中文翻译:

有向无环图中最短路径搜索的量子算法

在这项工作中,我们考虑了加权有向无环图(DAG)的著名“单源最短路径搜索”问题。我们建议使用时间复杂度\(O(\ sqrt {nm} \,\ log \; n)\)O(1 / n)错误概率的量子算法,其中n是顶点的数量,m是顶点的数量。边缘。我们使用量子动态规划方法(Khadiev,2018)以及Dürr和Høyer最小搜索算法来加快搜索速度。对于无向图,我们的算法优于C.Dürr和合著者的量子算法。C.Dürr算法的时间复杂度为\(O(\ sqrt {nm} \,{(\ log \; n)^ 2})\)。针对该问题的最著名的确定性算法基于动态规划方法,其时间复杂度为On + m)。同时,如果我们将算法用于一般图形,那么我们将无法获得更好的结果。具有斐波那契堆的Dijkstra算法的最佳实现的时间复杂度为Om + n log n)。
更新日期:2019-05-02
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