Finding All Bounded-Length Simple Cycles in a Directed Graph

A new efficient algorithm is presented for finding all simple cycles that satisfy a length constraint in a directed graph. When the number of vertices is nontrivial, most cycle-finding problems are of practical interest for sparse graphs only. We show that for a class of sparse graphs in which the vertex degrees are almost uniform, our algorithm can find all cycles of length less than or equal to k in O((c+ n)(k − 1)d) steps, where n is the number of vertices, c is the total number of cycles discovered, d is the average degree of the graph’s vertices, and k > 1. While our analysis for the running time addresses only a class of sparse graphs, we provide empirical and experimental evidence of the efficiency of the algorithm for general sparse graphs. This algorithm is a significant improvement over the only other deterministic algorithm for this problem known to us; it also lends itself to massive parallelism. Experimental results of a serial implementation on some large real-world graphs are presented.