Optimal Trade-Off for Succinct String Indexes

TL;DR

This paper establishes tight bounds for string indexing operations like select and rank, introduces a novel technique for proving lower bounds on succinct data structures, and improves existing theoretical results.

Contribution

It presents a general method for deriving lower bounds on succinct data structures based on access patterns, applicable beyond the specific problems studied.

Findings

Matching upper and lower bounds for select and rank operations.

A new technique for lower bound proofs based on access patterns.

Improved bounds over previous work by Golynski and Barbay et al.

Abstract

Let s be a string whose symbols are solely available through access(i), a read-only operation that probes s and returns the symbol at position i in s. Many compressed data structures for strings, trees, and graphs, require two kinds of queries on s: select(c, j), returning the position in s containing the jth occurrence of c, and rank(c, p), counting how many occurrences of c are found in the first p positions of s. We give matching upper and lower bounds for this problem, improving the lower bounds given by Golynski [Theor. Comput. Sci. 387 (2007)] [PhD thesis] and the upper bounds of Barbay et al. [SODA 2007]. We also present new results in another model, improving on Barbay et al. [SODA 2007] and matching a lower bound of Golynski [SODA 2009]. The main contribution of this paper is to introduce a general technique for proving lower bounds on succinct data structures, that is based on the access patterns of the supported operations, abstracting from the particular operations at hand. For this, it may find application to other interesting problems on succinct data structures.