Succinct Encoding of Permutations and its Applications to Text Indexing (2003; Munro, Raman, Raman, Rao)

A succinct data structure for a given data type is a representation of the underlying combinatorial object that uses an amount of space “close” to the information theoretic lower bound, together with algorithms that supports operations of the data type “quickly”. A natural example is the representation of a binary tree [5]: an arbitrary binary tree on n nodes can be represented in 2n + o(n) bits while supporting a variety of operations on any node, which include finding its parent, its left or right child, and returning the size of its subtree, each in time within O(1). As there are ( 2n n ) /(n + 1) binary trees on n nodes and the logarithm of this term1 is 2n − o(n), the space used by this representation is optimal to within a lower order term. In the applications considered in this entry, the principle concern is with indexes supporting search in strings and in XML-like documents (i.e. tree-structured objects with labels and “free text” at various nodes). As happens not only labeled trees but also arbitrary binary relations over finite domains are key building blocks for this. Preprocessing such data-structures to efficiently support searches is a complex process, requiring a variety of subordinate structures. A basic building block for this work is the representation of a permutation of the integers {1, . . . , n}, denoted by [1..n]. A permutation π is trivially representable in ndlg ne bits which is within O(n) bits of the information theoretic bound of lg(n!). The interesting problem is to support both the permutation π() and its inverse π−1(): namely, how to represent an arbitrary permutation π on [1..n] in a succinct manner so that the operator πk(i)() (π() iteratively applied k times starting at i, where k can be any integer and π−1() is the inverse of π()) can be evaluated quickly.