Ranking and unranking permutations in linear time

Abstract

A ranking function for the permutations on n symbols assigns a unique integer in the range [0, n! − 1] to each of the n! permutations. The corresponding unranking function is the inverse: given an integer between 0 and n! − 1, the value of the function is the permutation having this rank. We present simple ranking and unranking algorithms for permutations that can be computed using O(n) arithmetic operations.  2001 Elsevier Science B.V. All rights reserved.Keywords: Permutation; Ranking; Unranking; Algorithms; Combinatorial problems A permutation of order n is an arrangement of n symbols. For convenience when applying modular arithmetic, this paper considers permutations of {0, 1, 2, . . ., n − 1}. The set of all permutations over {0, 1, 2, . . ., n − 1} is denoted by S n .There are many applications that call for an array indexed by the permutations in S n [2]. One example is the development of programs that search for Hamilton cycles in particular types of Cayley graphs [10,11]. To do such indexing, what is desired is a bijective ranking function r that takes as input a permutation π and produces r(π), a number in the range 0, 1, . . ., n! − 1. The inverse of r is also often useful, and is called the unranking function.The traditional approach to this problem is to first define an ordering of permutations and then find ranking and unranking functions relative to that ordering.

Untitled

“…We feel this difference is worth highlighting, and hence we introduced the new terms indexing and reverse-indexing for this purpose. Note that some important prior work on ranking/unranking distinguishes between these notions [21].…”

Rank Modulation for Flash Memories

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