A (Very) Nearly Optimal Sketch for $k$-Edge Connectivity Certificates

2025-10-17

AI Summary

This paper studies the sublinear-space computation of $k$-edge-connectivity certificates in dynamic graph streams. For the problem of maintaining a certificate certifying $k$-edge connectivity under edge insertions and deletions, we propose a lightweight randomized sketch based on hashing and sparsification. Our sketch operates in the dynamic graph stream model and uses only $O(n log^2 n cdot max{k, log n log k})$ bits of space—improving upon the prior $O(kn log^3 n)$ bound and approaching the known theoretical lower bound. For $k = Omega(log n log log n)$, our algorithm achieves the optimal space complexity $Theta(kn log^2 n)$. For smaller $k$, the space bound deviates from optimality by at most a double-logarithmic factor, essentially completing the space complexity characterization of this problem.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

In this note, we present a simple algorithm for computing a emph{$k$-connectivity certificate} in dynamic graph streams. Our algorithm uses $O(n log^2 n cdot max{k, log n log k})$ bits of space which improves upon the $O(kn log^3 n)$-space algorithm of Ahn, Guha, and McGregor (SODA'12). For the values of $k$ that are truly sublinear, our space usage emph{very nearly} matches the known lower bound $Ω(n log^2 n cdot max{k, log n})$ established by Nelson and Yu (SODA'19; implicit) and Robinson (DISC'24). In particular, our algorithm fully settles the space complexity at $Θ(kn log^2{n})$ for $k = Ω(log n log log n)$, and bridges the gap down to only a doubly-logarithmic factor of $O(log log n)$ for a smaller range of $k = o(log n log log n)$.

Problem

• Develops space-efficient algorithm for k-connectivity certificates
• Improves upon previous dynamic graph streaming approaches
• Nearly matches known lower bounds for space complexity

Innovation

• Dynamic graph streams algorithm for k-connectivity certificates
• Space complexity improved to O(n log²n · max{k, logn logk})
• Nearly matches known lower bound for sublinear k values