Time Lower Bounds for Distributed Distance Oracles

Abstract

Distributed distance oracles consist of a labeling scheme which assigns a label to each node and a local data structure deployed to each node. When a node v wants to know the distance to a node u, it queries its local data structure with the label of u. The data structure returns an estimated distance to u, which must be larger than the actual distance but can be overestimated. The accuracy of the distance oracle is measured by stretch, which is defined as the maximum ratio between actual distances and estimated distances over all pairs (u, v).

In this paper, we focus on the time complexity of constructing distributed distance oracles with a given stretch. We show a number of time lower bounds depending on the stretch:

• Under the assumption that the popular combinatorial girth conjecture is true, any distributed algorithm constructing oracles with stretch 2t requires $\tilde{\Omega}(n^{1/(t+1)})$ rounds in unweighted graphs. This bound holds even if we only consider constant diameter graphs.

• For oracles with stretch 2t in weighted graphs, we have a lower bound of $\Omega(n^{\frac{1}{2} + \frac{1}{5t}})$ rounds, assuming the girth conjecture. This bound holds even if we only consider O(logn) diameter graphs.

• If we restrict the label size of oracles to o(n ^ε) bits, where ε = 1/2t(t + 1) in unweighted graphs and ε = (1/5t ²) in weighted graphs, the same lower bounds are obtained without assuming the girth conjecture.

To the best of our knowledge, this paper is the first that exhibits a non-trivial trade-off between time and stretch for distributed distance oracles.

This work is supported in part by KAKENHI No.25106507 and No.25289114.