Approximate Data Structures with Applications

TL;DR

This paper introduces approximate variants of the van Emde Boas data structure that provide faster query responses with controlled approximation guarantees, enabling more efficient algorithms for graph and geometric problems.

Contribution

It presents novel approximate data structures supporting standard operations with near-constant or very fast time, and demonstrates their application to classical algorithms like Prim's, Dijkstra's, and Graham's convex hull.

Findings

Prim's algorithm runs in linear time with approximation guarantees.

Dijkstra's algorithm runs in O(mloglogn) time with approximation.

Graham's convex hull algorithm achieves constant amortized time per operation.

Abstract

This paper explores the notion of approximate data structures, which return approximately correct answers to queries, but run faster than their exact counterparts. The paper describes approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure (min, max, successor, predecessor, and existence queries, as well as insertion and deletion), except that answers to queries are approximate. The variants support all operations in constant time provided the performance guarantee is 1+1/polylog(n), and in O(loglogn) time provided the performance guarantee is 1+1/polynomial(n), for n elements in the data structure. Applications described include Prim's minimum-spanning-tree algorithm, Dijkstra's single-source shortest paths algorithm, and an on-line variant of Graham's convex hull algorithm. To obtain output which approximates the desired output with the performance guarantee tending to 1, Prim's algorithm requires only linear time, Dijkstra's algorithm requires O(mloglogn) time, and the on-line variant of Graham's algorithm requires constant amortized time per operation.