Optimal non-adaptive algorithm for edge estimation

TL;DR

This paper introduces an optimal non-adaptive randomized algorithm that estimates the total number of edges in a graph with sublinear query complexity, using degree and edge queries, and proves its optimality.

Contribution

It presents the first optimal non-adaptive algorithm for edge estimation in graphs, matching the lower bound for query complexity.

Findings

Requires only rac{1}{2}rac{ ilde{O}(\

Establishes a matching lower bound for non-adaptive edge estimation.

Achieves sublinear query complexity of rac{1}{2}rac{ ilde{O}(\

Abstract

We present a simple nonadaptive randomized algorithm that estimates the number of edges in a simple, unweighted, undirected graph, possibly containing isolated vertices, using only degree and random edge queries. For an $n$-vertex graph, our method requires only $\widetilde{O}(\sqrt{n})$ queries, achieving sublinear query complexity. The algorithm independently samples a set of vertices and queries their degrees, and also independently samples a set of edges, using the answers to these queries to estimate the total number of edges in the graph. We further prove a matching lower bound, establishing the optimality of our algorithm and resolving the non-adaptive query complexity of this problem with respect to degree and random-edge queries.