In-Place Sparse Suffix Sorting

Abstract

Suffix arrays encode the lexicographical order of all suffixes of a text and are often combined with the Longest Common Prefix array (LCP) to simulate navigational queries on the suffix tree in reduced space. In space-critical applications such as sparse and compressed text indexing, only information regarding the lexicographical order of a size-b subset of all n text suffixes is often needed. Such information can be stored space-efficiently (in b words) in the sparse suffix array (SSA). The SSA and its relative sparse LCP array (SLCP) can be used as a space-efficient substitute of the sparse suffix tree. Very recently, Gawrychowski and Kociumaka [11] showed that the sparse suffix tree (and therefore SSA and SLCP) can be built in asymptotically optimal O(b) space with a Monte Carlo algorithm running in O(n) time. The main reason for using the SSA and SLCP arrays in place of the sparse suffix tree is, however, their reduced space of b words each. This leads naturally to the quest for in-place algorithms building these arrays. Franceschini and Muthukrishnan [8] showed that the full suffix array can be built in-place and in optimal running time. On the other hand, finding sub-quadratic in-place algorithms for building the SSA and SLCP for general subsets of suffixes has been an elusive task for decades. In this paper, we give the first solution to this problem. We provide the first in-place algorithm building the full LCP array in O(n log n) expected time and the first Monte Carlo in-place algorithms building the SSA and SLCP in O(n + b log 2 n) expected time. We moreover describe the first in-place solution for the suffix selection problem: to compute the i-th smallest text suffix. In order to achieve these results, we show that we can quickly overwrite the text with a reversible and implicit data structure supporting Longest Common Extension queries in polylogarithmic time and text extraction in optimal time: this structure is strictly more powerful than a plain text representation and is of independent interest.

Optimal Substring-Equality Queries with Applications to Sparse Text Indexing

“…The work described in this paper is an extension of the data structure presented in [28] for supporting substring-equality queries within n log σ + Θ(log n) bits of space. We generalize this result to any alphabet size σ (the strategy [28] converts S to a binary string before building the data structure). As an immediate result of this generalization, we can abstract from the input representation and present our result as a string transformation.…”

“…As an immediate result of this generalization, we can abstract from the input representation and present our result as a string transformation. A second benefit of generalizing to any alphabet size is that we are able to reduce the space of the structure described in [28] to n log σ + Θ(log n) bits, therefore matching the information-theoretic lower bound for representing the underlying string with a prefix-free encoding.…”

Deterministic Sparse Suffix Sorting on Rewritable Texts

“…aab aab ccc aab aab ccc bbb bbb bbb type M type M type 1 type M type M type 1 type 1 Figure 32: ET(X 3 ) as defined in Ex. 35. The subtree of each node with name x i is equal to ET(X i ).…”

“…Gawrychowski and Kociumaka [16] presented an algorithm running with O(m) words of additional space in either O(n √ lg m) expected time, or in O(n) time as a Monte Carlo algorithm (i.e., the output is correct only with high probability). Most recently, Prezza [35] presented a Monte Carlo algorithm in the restore model [8] that runs with O(m) words of space in O(n + m lg 2 n) expected time.…”

“…Differences between ET(Y ) (left) and ET(Y ) (right) on the heights i and i+ 1, where Y = a 3 k (a 2 b) 3 k−1 ((a 2 b) 2 c 3 ) 3 k−2 · · · and Y = aY (defined in Ex 35…”