On the hardness of distance oracle for sparse graph

TL;DR

This paper demonstrates that the set-intersection problem is computationally harder than constructing distance oracles for sparse graphs, establishing a complexity relationship between these problems.

Contribution

It establishes a formal reduction showing that efficient distance oracles imply solutions for set-intersection, highlighting the complexity hierarchy.

Findings

Set-intersection is harder than distance oracle construction on sparse graphs.

Efficient distance oracles would enable faster solutions for set-intersection.

The paper provides a theoretical link between graph distance queries and set-intersection complexity.

Abstract

In this paper we show that set-intersection is harder than distance oracle on sparse graphs. Given a collection of total size n which consists of m sets drawn from universe U, the set-intersection problem is to build a data structure which can answer whether two sets have any intersection. A distance oracle is a data structure which can answer distance queries on a given graph. We show that if one can build distance oracle for sparse graph G=(V,E), which requires s(|V|,|E|) space and answers a (2-\epsilon,c)-approximate distance query in time t(|V|,|E|) where (2-\epsilon) is a multiplicative error and c is a constant additive error, then, set-intersection can be solved in t(m+|U|,n) time using s(m+|U|,n) space.