On the Succinct Representation of Unlabeled Permutations

Abstract

We investigate the problem of succinctly representing an arbitrary unlabeled permutation π, so that πk(i) can be computed quickly for any i and any integer power k. We consider the problem in several scenarios: - Labeling schemes where we assign labels to elements and the query is to be answered by just examining the labels of the queried elements: We show that a label space of (formula presented) i is necessary and sufficient. In other words, 2 lg n bits of space are necessary and sufficient for representing each of the labels. - Succinct data structures for the problem where we assign labels to the n elements from the label set {1,…, cn} where c ≥ 1: We show that Θ(√ n) bits are necessary and sufficient to represent the permutation. Moreover, we support queries in such a structure in O(1) time in the standard word-RAM model. - Succinct data structures for the problem where we assign labels to the n elements from the label set {1,…, cn1+ε} where c is a constant and 0 < ε < 1: We show that Θ(n(1−ε)/2) bits are necessary and sufficient to represent the permutation. We can also support queries in such a structure in O(1) time in the standard word-RAM model.