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We consider the maximum matching problem in the semi-streaming model formalized by Feigenbaum, Kannan, McGregor, Suri, and Zhang that is inspired by giant graphs of today. As our main result, we give a two-pass (1/2 + 1/16)-approximation algorithm for triangle-free graphs and a two-pass (1/2 + 1/32)-approximation algorithm for general graphs; these improve the approximation ratios of 1/2 + 1/52 for bipartite graphs and 1/2 + 1/140 for general graphs by Konrad, Magniez, and Mathieu. In three passes, we are able to achieve approximation ratios of 1/2 + 1/10 for triangle-free graphs and 1/2 + 1/19.753 for general graphs. We also give a multi-pass algorithm where we bound the number of passes precisely—we give a (2/3 − ε)- approximation algorithm that uses 2/(3ε) passes for triangle-free graphs and 4/(3ε) passes for general graphs. Our algorithms are simple and combinatorial, use O(n log n) space, and (can be implemented to) have O(1) update time per edge. For general graphs, our multi-pass algorithm improves the best known deterministic algorithms in terms of the number of passes: * Ahn and Guha give a (2/3−ε)-approximation algorithm that uses O(log(1/ε)/ε2) passes, whereas our (2/3 − ε)-approximation algorithm uses 4/(3ε) passes; * they also give a (1 − ε)-approximation algorithm that uses O(log n · poly(1/ε)) passes, where n is the number of vertices of the input graph; although our algorithm is (2/3−ε)-approximation, our number of passes do not depend on n. Earlier multi-pass algorithms either have a large constant inside big-O notation for the number of passes or the constant cannot be determined due to the involved analysis, so our multi-pass algorithm should use much fewer passes for approximation ratios bounded slightly below 2/3.
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Streaming Algorithms with Few State Changes
In this paper, we study streaming algorithms that minimize the number of changes made to their internal state (i.e., memory contents). While the design of streaming algorithms typically focuses on minimizing space and update time, these metrics fail to capture the asymmetric costs, inherent in modern hardware and database systems, of reading versus writing to memory. In fact, most streaming algorithms write to their memory on every update, which is undesirable when writing is significantly more expensive than reading. This raises the question of whether streaming algorithms with small space and number of memory writes are possible. We first demonstrate that, for the fundamental Fpmoment estimation problem with p ≥ 1, any streaming algorithm that achieves a constant factor approximation must make Ω(n1-1/p) internal state changes, regardless of how much space it uses. Perhaps surprisingly, we show that this lower bound can be matched by an algorithm which also has near-optimal space complexity. Specifically, we give a (1+ε)-approximation algorithm for Fpmoment estimation that use a near-optimal ~Oε(n1-1/p) number of state changes, while simultaneously achieving near-optimal space, i.e., for p∈[1,2), our algorithm uses poly(log n,1/ε) bits of space for, while for p>2, the algorithm uses ~Oε(n1-1/p) space. We similarly design streaming algorithms that are simultaneously near-optimal in both space complexity and the number of state changes for the heavy-hitters problem, sparse support recovery, and entropy estimation. Our results demonstrate that an optimal number of state changes can be achieved without sacrificing space complexity.
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- Award ID(s):
- 2335411
- PAR ID:
- 10560522
- Publisher / Repository:
- Proc. ACM Manag. Data
- Date Published:
- Journal Name:
- Proceedings of the ACM on Management of Data
- Volume:
- 2
- Issue:
- 2
- ISSN:
- 2836-6573
- Page Range / eLocation ID:
- 1 to 28
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3651145
