On the hardness of distance oracle for sparse graph

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− ǫ, c)-approximate distance query in time t(|V |, |E|) where 2 − ǫ 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)