Streaming Maximal Matching with Bounded Deletions
Sanjeev Khanna, Christian Konrad, Jacques Dark
TL;DR
This work initiates the study of Maximal Matching in bounded-deletion graph streams, where at most $K$ deletions occur. It provides a tight, polylog-factor-accurate characterization of the space needed: a randomized algorithm uses $\tilde{\Theta}(n \sqrt{K})$ space, while any deterministic algorithm needs $\tilde{\Theta}(nK)$ space, with a sharp contrast for $\alpha$-approximate Maximum Matching where the space becomes $\tilde{\Theta}(n+K)$. The key ideas combine a hierarchical maximal matching structure on insertions with a repair mechanism built from $\ell_0$-samplers, plus information-theoretic lower bounds via Embedded-Augmented-Index to prove optimal space bounds. The results show a smooth interpolation between insertion-only and fully dynamic regimes and introduce a near-optimal $O(n+K)$ deterministic approach for near-maximum matchings, with potential applicability to other graph problems under bounded deletions.
Abstract
We initiate the study of the Maximal Matching problem in bounded-deletion graph streams. In this setting, a graph $G$ is revealed as an arbitrary sequence of edge insertions and deletions, where the number of insertions is unrestricted but the number of deletions is guaranteed to be at most $K$, for some given parameter $K$. The single-pass streaming space complexity of this problem is known to be $Θ(n^2)$ when $K$ is unrestricted, where $n$ is the number of vertices of the input graph. In this work, we present new randomized and deterministic algorithms and matching lower bound results that together give a tight understanding (up to poly-log factors) of how the space complexity of Maximal Matching evolves as a function of the parameter $K$: The randomized space complexity of this problem is $\tildeΘ(n \cdot \sqrt{K})$, while the deterministic space complexity is $\tildeΘ(n \cdot K)$. We further show that if we relax the maximal matching requirement to an $α$-approximation to Maximum Matching, for any constant $α> 2$, then the space complexity for both, deterministic and randomized algorithms, strikingly changes to $\tildeΘ(n + K)$.