Optimal non-adaptive algorithm for edge estimation

2025-12-12

AI Summary

This paper studies the non-adaptive estimation of the number of edges in an undirected simple graph, where only degree queries and random edge queries are permitted—no adaptive queries are allowed. We propose the first optimal non-adaptive algorithm that achieves a $(1 pm varepsilon)$-multiplicative approximation using $ ilde{O}(sqrt{n})$ queries on an $n$-vertex graph, attaining sublinear query complexity. Our method employs a dual random sampling strategy—jointly sampling vertices according to their degrees and sampling edges uniformly at random—augmented by precise probabilistic analysis and careful variance control. Crucially, we establish the first tight query complexity bound for edge estimation in this model: $Theta(sqrt{n})$, matching upper and lower bounds. This resolves, in full, the fundamental problem of edge estimation under non-adaptivity, providing both a definitive theoretical benchmark and a general technical framework for future work on graph property estimation.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Estimates graph edges using degree and random queries
• Achieves sublinear query complexity with O(√n) queries
• Proves optimality via matching lower bound for non-adaptive queries

Innovation

• Uses degree and random edge queries for estimation
• Achieves sublinear query complexity with O(sqrt(n)) queries
• Proves optimality with matching lower bound