| A faster deterministic algorithm for minimum cycle basis in directed graphs (2006) | |||||||||||||||
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| Abstract. We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have non-negative weights. A cycle in this graph is actually a cycle in the underlying undirected graph with edges traversable in both directions. A {−1,0,1} edge incidence vector is associated with each cycle: edges traversed by the cycle in the right direction get 1 and edges traversed in the opposite direction get-1. The vector space over Q generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for this vector space. We seek a cycle basis where the sum of weights of the cycles is minimum. The current fastest algorithm for computing a minimum cycle basis in a directed graph with m edges and n vertices runs in Õ(m ω+1 n) time (where ω < 2.376 is the exponent of matrix multiplication). Here we present an O(m 3 n + m 2 n 2 logn) algorithm. This algorithm is obtained by using fast matrix multiplication over rings and an efficient extension of Dijkstra’s algorithm to compute a shortest cycle in G whose dot product with a function on its edge set is non-zero. We also slightly improve the running time of the fastest randomized algorithm from O(m 2 nlogn) to O(m 2 n + mn 2 logn). 1 | |||||||||||||||
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