The Simultaneous Conjugacy Problem In The Symmetric Group

The transitive simultaneous conjugacy problem asks whether there exists a permutation tau is an element of S-n such that b(j) = tau(-1)a(j)tau holds for all j = 1, 2,..., d, where a(1), a(2),..., a(d) and b(1), b(2),..., b(d) are given sequences of d permutations in S-n, each of which generates a transitive subgroup of S-n. As from mid 70' it has been known that the problem can be solved in O(dn(2)) time. An algorithm with running time O(dn log(dn)), proposed in late 80', does not work correctly on all input data. In this paper we solve the transitive simultaneous conjugacy problem in O(n(2) log d/log n+ dn log n) time and O(n(3/2) + dn) space. Experimental evaluation on random instances shows that the expected running time of our algorithm is considerably better, perhaps even nearly linear in n at given d.