Maximum Matching in Turnstile Streams

• Published in: Algorithms - ESA 2015 Vol. 9294; pp. 840 - 852
• Main Author: Konrad, Christian
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2015; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Bipartite Match; Computer programming / software development; Discrete mathematics; Edge Insertion; Input Graph; Maximum Match; Programming & scripting languages: general; Vertex Group
• ISBN: 3662483491; 9783662483497
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-662-48350-3_70

We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass 2-approximation streaming algorithm can be easily obtained with space O(n logn), where n denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space O(n3/2 − δ) is granted, for every δ > 0. Specifically, for every 0 ≤ ε ≤ 1, we show that in the one-pass turnstile streaming model, in order to compute a O(nε)-approximation, space Ω(n3/2 − 4ε) is required for constant error randomized algorithms, and, up to logarithmic factors, space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{\mathrm{O}}( n^{2-2\epsilon} )$\end{document} is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest.