Succinct Representations of Permutations

We investigate the problem of succinctly representing an arbitrary permutation, π, on {0, ..., n - 1} so that πk(i) can be computed quickly for any i and any (positive or negative integer) power k. A representation taking (1 + Ɛ)n lg n + O(1) bits suffices to compute arbitrary powers in constant time. A representation taking the optimal [lg n!] + o(n) bits can be used to compute arbitrary powers in O(lg n/lg lg n) time, or indeed in a minimal O(lg n) bit probes.

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