The simultaneous conjugacy problem in the symmetric group
Brodnik, Andrej ; Malnič, Aleksander ; Požar, Rok, 1986-
The transitive simultaneous conjugacy problem asks whether there exists a permutation ▫$\tau \in S_n$▫ such that ▫$b_j = \tau^{-1} a_j \tau$▫ holds for all ▫$j = 1,2, \ldots , d$▫, where ▫$a_1, a_2, ...\ldots , a_d$▫ and ▫$b_1, b_2, \ldots , b_d$▫ 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 ▫$d$▫.
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