Online Bipartite Matching with Amortized O(log 2 n) Replacements

A. Bernstein, J. Holm, E. Rotenberg
{"title":"Online Bipartite Matching with Amortized O(log 2 n) Replacements","authors":"A. Bernstein, J. Holm, E. Rotenberg","doi":"10.1145/3344999","DOIUrl":null,"url":null,"abstract":"In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log 2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n) lower bound. The previous best strategy known achieved amortized O(√ n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O(n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O(log 2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O(min { L log2 n , √ nlog n}) reassignments. We also show that this is close to tight, because Ω (min { L, √ n}) reassignments can be necessary.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"2005 1","pages":"1 - 23"},"PeriodicalIF":0.0000,"publicationDate":"2017-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"34","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the ACM (JACM)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3344999","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 34

Abstract

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log 2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n) lower bound. The previous best strategy known achieved amortized O(√ n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O(n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O(log 2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O(min { L log2 n , √ nlog n}) reassignments. We also show that this is close to tight, because Ω (min { L, √ n}) reassignments can be necessary.
具有O(log 2 n)个平摊替换的在线二部匹配
在带替换的在线二部匹配问题中,给定了二部分割的一边的所有顶点,另一边的顶点带着所有的关联边一个接一个地到达。目标是在保持最大匹配的同时尽量减少对匹配的更改(替换)数量。我们证明贪心算法总是从新插入的顶点(表示SAP协议)获取最短的扩展路径,每次插入最多使用平摊O(log 2 n)次替换,其中n是插入顶点的总数。这是第一个实现任何替换策略的多对数替换数的分析,几乎匹配Ω (log n)的下界。先前已知的最佳策略实现了平摊O(√n)次替换[Bosek, Leniowski, Sankowski, Zych, FOCS 2014]。特别是对于SAP协议,除了在特殊情况下,没有什么比平凡的O(n)界更好的了。我们的分析立即表明,对于有容量分配问题,O(log 2 n)重新分配的上界是相同的,其中双分区静态侧的每个顶点都初始化为服务多个顶点的能力。我们还分析了最小化最大服务器负载的问题。我们证明,如果最终的图具有最大的服务器负载L,那么SAP协议将平摊O(min {L log2 n,√nlog n})个重分配。我们还证明了这是接近紧密的,因为Ω (min {L,√n})重赋值可能是必要的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信