Optimal Indexes for Sparse Bit Vectors

TL;DR

This paper develops optimal data structures for efficiently supporting Rank and Select queries on sparse bit vectors, balancing index size and query time in a read-only memory model.

Contribution

It introduces asymptotically optimal density-sensitive trade-offs for Rank and Select operations on sparse bit vectors, considering both vector size and density.

Findings

Achieves asymptotically optimal trade-offs between index size and query time.

Provides solutions particularly effective when the number of 1 bits is much smaller than vector length.

Extends understanding of succinct data structures for sparse data.

Abstract

We consider the problem of supporting Rank() and Select() operations on a bit vector of length m with n 1 bits. The problem is considered in the succinct index model, where the bit vector is stored in "read-only" memory and an additional data structure, called the index, is created during pre-processing to help answer the above queries. We give asymptotically optimal density-sensitive trade-offs, involving both m and n, that relate the size of the index to the number of accesses to the bit vector (and processing time) needed to answer the above queries. The results are particularly interesting for the case where n = o(m).