Succinct representations of permutations and functions

  • Authors:

  • J. Ian Munro;Rajeev Raman;Venkatesh Raman;Srinivasa Rao S.

  • Affiliations:

  • School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada;Department of Computer Science, University of Leicester, Leicester, LE1 7RH, UK;Institute of Mathematical Sciences, Chennai, 600 113, India;School of Computer Science and Engineering, Seoul National University, Seoul, South Korea

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2012

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Abstract

We investigate the problem of succinctly representing an arbitrary permutation, @p, on {0,...,n-1} so that @p^k(i) can be computed quickly for any i and any (positive or negative) integer power k. A representation taking (1+@e)nlgn+O(1) bits suffices to compute arbitrary powers in constant time, for any positive constant @e@?1. A representation taking the optimal @?lgn!@?+o(n) bits can be used to compute arbitrary powers in O(lgn/lglgn) time. We then consider the more general problem of succinctly representing an arbitrary function, f:[n]-[n] so that f^k(i) can be computed quickly for any i and any integer power k. We give a representation that takes (1+@e)nlgn+O(1) bits, for any positive constant @e@?1, and computes arbitrary positive powers in constant time. It can also be used to compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time. We place emphasis on the redundancy, or the space beyond the information-theoretic lower bound that the data structure uses in order to support operations efficiently. A number of lower bounds have recently been shown on the redundancy of data structures. These lower bounds confirm the space-time optimality of some of our solutions. Furthermore, the redundancy of one of our structures ''surpasses'' a recent lower bound by Golynski [Golynski, SODA 2009], thus demonstrating the limitations of this lower bound.