We present several new algorithms for detecting short fixed length cycles in digraphs. The new algorithms utilize fast rectangular matrix multiplication algorithms together with a dynamic programming approach similar to the one used in the solution of the classical chain matrix product problem. The new algorithms are instantiations of a generic algorithm that we present for finding a directed Ck, i.e., a directed cycle of length k, in a digraph, for any fixed k ≥ 3. This algorithm partitions the prospective Ck's in the input digraph G = (V, E) into O(logk V) classes, according to the degrees of their vertices. For each cycle class we determine, in O(E ck log V) time, whether G contains a Ck from that class, where ck, = ck,(ω) is a constant that depends only on ω, the exponent of square matrix multiplication. The search for cycles from a given class is guided by the solution of a small dynamic programming problem. The total running time of the obtained deterministic algorithm is therefore O(Ecklogk+1 V). For C3, we get c 3 = 2ω/(ω + 1) < 1.41 where ω < 2.376 is the exponent of square matrix multiplication. This coincides with an existing algorithm of [AYZ97]. For C4 we get c4 = (4ω - 1)/(2ω + 1) < 1.48. We can dispense, in this case, of the polylogarithmic factor and get an O(E(4ω-1)/(2ω,+1)) = o(E1.48) time algorithm. This improves upon an O(E3/2) time algorithm of [AYZ97]. For C5 we get c5 = 3ω/(ω + 2) < 1.63. The obtained running time of O(E 3ω/(ω+2) log6 V) = o(E1.63) improves upon an O(E5/3) time algorithm of [AYZ97]. Determining ck for k ≥ 6 is a difficult task. We conjecture that ck = (k + 1)ω/(2ω + k - 1), for every odd k. The values of c k, for even k ≥ 6 seem to exhibit a much more complicated dependence on ω.
|Number of pages||7|
|State||Published - 2004|
|Event||Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms - New Orleans, LA., United States|
Duration: 11 Jan 2004 → 13 Jan 2004
|Conference||Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms|
|City||New Orleans, LA.|
|Period||11/01/04 → 13/01/04|