Minimal Indices for Successor Search

TL;DR

This paper introduces a new successor search data structure that significantly reduces index size to near optimal levels while maintaining optimal probe complexity, improving upon previous structures and enabling efficient weak prefix search.

Contribution

The paper presents a novel successor search data structure with minimal index size of O(n log w) bits and efficient single word indices, advancing space efficiency in predecessor search.

Findings

Reduces index size from O(n w^{4/5}) to O(n log w) bits.

Achieves optimal probe complexity with smaller index size.

Introduces efficient bit selectors for high out-degree single word indices.

Abstract

We give a new successor data structure which improves upon the index size of the P\v{a}tra\c{s}cu-Thorup data structures, reducing the index size from $O(n w^{4/5})$ bits to $O(n \log w)$ bits, with optimal probe complexity. Alternatively, our new data structure can be viewed as matching the space complexity of the (probe-suboptimal) $z$-fast trie of Belazzougui et al. Thus, we get the best of both approaches with respect to both probe count and index size. The penalty we pay is an extra $O(\log w)$ inter-register operations. Our data structure can also be used to solve the weak prefix search problem, the index size of $O(n \log w)$ bits is known to be optimal for any such data structure. The technical contributions include highly efficient single word indices, with out-degree $w/\log w$ (compared to the $w^{1/5}$ out-degree of fusion tree based indices). To construct such high efficiency single word indices we device highly efficient bit selectors which, we believe, are of independent interest.