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arXiv · 2512.11994

Optimal non-adaptive algorithm for edge estimation

Arijit Bishnu·Debarshi Chanda·Buddha Dev Das·Arijit Ghosh·Gopinath Mishra
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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.

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Arijit Bishnu, Debarshi Chanda, Buddha Dev Das, Arijit Ghosh, Gopinath Mishra. 2025-12-12. Optimal non-adaptive algorithm for edge estimation. https://arxiv.org/abs/2512.11994

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