Approximate Query Processing over Static Sets and Sliding Windows

TL;DR

This paper introduces efficient data structures for approximate set and multiset indexing supporting rank and select queries, including sliding window scenarios, with theoretical bounds and practical algorithms.

Contribution

It presents new succinct data structures and lower bounds for approximate rank/select over static and dynamic sets, extending to sliding windows with near-optimal space and constant-time operations.

Findings

Supports approximate rank and select with bounded error.

Provides algorithms for sliding window sum queries with constant time.

Achieves space efficiency close to the theoretical lower bounds.

Abstract

Indexing of static and dynamic sets is fundamental to a large set of applications such as information retrieval and caching. Denoting the characteristic vector of the set by B, we consider the problem of encoding sets and multisets to support approximate versions of the operations rank(i) (i.e., computing sum_{j <= i}B[j]) and select(i) (i.e., finding min{p | rank(p) >= i}) queries. We study multiple types of approximations (allowing an error in the query or the result) and present lower bounds and succinct data structures for several variants of the problem. We also extend our model to sliding windows, in which we process a stream of elements and compute suffix sums. This is a generalization of the window summation problem that allows the user to specify the window size at query time. Here, we provide an algorithm that supports updates and queries in constant time while requiring just (1+o(1)) factor more space than the fixed-window summation algorithms.