We introduce a fast parallel approximation algorithm for the positive linear programming optimization problem, i.e. the special case of the linear programming optimization problem where the input constraint matrix and constraint vector consist entirely of positive entries. The algorithm is elementary, and has a simple parallel implementation that runs in polylog time using a linear number of processors.
{"title":"A parallel approximation algorithm for positive linear programming","authors":"M. Luby, N. Nisan","doi":"10.1145/167088.167211","DOIUrl":"https://doi.org/10.1145/167088.167211","url":null,"abstract":"We introduce a fast parallel approximation algorithm for the positive linear programming optimization problem, i.e. the special case of the linear programming optimization problem where the input constraint matrix and constraint vector consist entirely of positive entries. The algorithm is elementary, and has a simple parallel implementation that runs in polylog time using a linear number of processors.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"20 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114022860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The classic problem of searching an ordered list becomes more complicated when the answers to queries are not reliable. In the prefix-bounded error model, the number of incorrect responses at no point exceeds r times the number of questions, where r is a fixed positive fraction strictly less than ~. We present efficient algorithms for searching in this error model, and we also establish nontrivial lower bounds for sorting, maxima finding, and related problems.
{"title":"Comparison-based search in the presence of errors","authors":"Ryan S. Borgstrom, S. R. Kosaraju","doi":"10.1145/167088.167129","DOIUrl":"https://doi.org/10.1145/167088.167129","url":null,"abstract":"The classic problem of searching an ordered list becomes more complicated when the answers to queries are not reliable. In the prefix-bounded error model, the number of incorrect responses at no point exceeds r times the number of questions, where r is a fixed positive fraction strictly less than ~. We present efficient algorithms for searching in this error model, and we also establish nontrivial lower bounds for sorting, maxima finding, and related problems.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117027805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally checkable labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: There are nontrivial LCL problems that have local algorithms. There is a variant of the dining philosophers problem that can be solved locally. Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. It is undecidable, in general, whether a given LCL has a local algorithm. However, it is decidable whether a given LCL has an algorithm that operates in a given time $t$. Any LCL problem that has a local algorithm has one that is order-invariant (the algorithm depends only on the order of the processor IDs).
{"title":"What can be computed locally?","authors":"M. Naor, L. Stockmeyer","doi":"10.1145/167088.167149","DOIUrl":"https://doi.org/10.1145/167088.167149","url":null,"abstract":"The purpose of this paper is a study of computation that can be done locally in a distributed network, where \"locally\" means within time (or distance) independent of the size of the network. Locally checkable labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: \u0000There are nontrivial LCL problems that have local algorithms. \u0000There is a variant of the dining philosophers problem that can be solved locally. \u0000Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. \u0000It is undecidable, in general, whether a given LCL has a local algorithm. \u0000However, it is decidable whether a given LCL has an algorithm that operates in a given time $t$. \u0000Any LCL problem that has a local algorithm has one that is order-invariant (the algorithm depends only on the order of the processor IDs).","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126963552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We initiate a study of security in asynchronous networks. We consider a completely asynchronous network where every two parties are connected via a private channel, and some of the parties may be faulty. We start by defining secure computation in this model. Our definition adapts the underlying principles of defining security (i.e., comparing the computation to a computation in the presence of a trusted party) to the asynchronous model. In particular, our definition takes into account the fact that the computation must be completed even if we never hear from the faulty parties. Next, we show that whatever can be securely computed in an asynchronous network in the presence of a trusted party, can be securely computed in a network in which no such trusted party exists. We distinguish two types of faults. In case of Fail-Stop faults, our construction is valid as long as the faulty parties constitute less than a thzr-d of the parties in the network. In case of general (i.e., Byzantine) faults, our construction requires that the faulty parties are less than a fourth fraction. In both cases, the resilience of our construction is optimal. Our construction generalizes known synchronous constructions by Ben-Or, Goldwasser and Wigderson. In addition, we introduce and implement several new asynchronous primitives. Among these, we note an errorless asynchronous verifiable secret sharing scheme, an asynchronous agreement on a large set that is contained in the dynamical y growing inputs of all non-faulty parties, and an on-line error-correcting procedure. * email: benor@cs.huji.ac.il t ~~ail.. canetti@tx .technion.ac.il %email: odedrfiks.technion. ac.il. Supported by grant no. 8900312 from the United States — Israel Binational Science Foundation, Jerusalem, Israel Permission to copy without fee all or part of this material ia granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appaar, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwiee, or to republish, requiree a fee and/or specific permission. 25th ACM STOC ‘93-5 /93/CA, USA G 1993 ACM 0-89791 -591 -7/93 /0005 [0052 . ..$1 .50 Computer Science Dept. of Computer Science
我们开始研究异步网络的安全性。我们考虑一个完全异步的网络，其中每两方都通过专用通道连接，其中一些方可能出现故障。我们首先在这个模型中定义安全计算。我们的定义将定义安全性的基本原则(即，将计算与存在可信方的计算进行比较)应用于异步模型。特别是，我们的定义考虑了这样一个事实，即即使我们从未收到错误方的消息，计算也必须完成。接下来，我们将展示在存在可信方的异步网络中可以安全计算的任何内容，在不存在此类可信方的网络中也可以安全计算。我们区分出两种类型的断层。在Fail-Stop故障的情况下，我们的构造是有效的，只要故障方占网络各方的比例小于1 / 3。在一般(即拜占庭)故障的情况下，我们的构造要求故障方小于四分之一。在这两种情况下，我们的建筑的弹性都是最佳的。我们的构造推广了Ben-Or, Goldwasser和Wigderson已知的同步构造。此外，我们还引入并实现了几个新的异步原语。其中，我们注意到一个无错误异步可验证秘密共享方案，一个包含在所有无错误方的动态y增长输入中的大集合上的异步协议，以及一个在线纠错过程。*邮箱:benor@cs.huji.ac.il t ~~ail..canetti@tx .technion.ac。电子邮件:odedrfiks.technion。ac.il。资助项目:8900312(美国-以色列两国科学基金会，耶路撒冷，以色列)允许免费复制本材料的全部或部分，前提是这些副本不是为了直接的商业利益而制作或分发的，ACM版权声明和出版物的标题及其日期，并通知复制是由计算机械协会许可的。复制他人，或重新发布，需要支付费用和/或特定许可。25 ACM STOC ' 93-5 /93/CA, USA G 1993 ACM 0-89791 -591 -7/93 /0005[0052 .…]$ 1.50计算机科学计算机科学系
{"title":"Asynchronous secure computation","authors":"M. Ben-Or, R. Canetti, Oded Goldreich","doi":"10.1145/167088.167109","DOIUrl":"https://doi.org/10.1145/167088.167109","url":null,"abstract":"We initiate a study of security in asynchronous networks. We consider a completely asynchronous network where every two parties are connected via a private channel, and some of the parties may be faulty. We start by defining secure computation in this model. Our definition adapts the underlying principles of defining security (i.e., comparing the computation to a computation in the presence of a trusted party) to the asynchronous model. In particular, our definition takes into account the fact that the computation must be completed even if we never hear from the faulty parties. Next, we show that whatever can be securely computed in an asynchronous network in the presence of a trusted party, can be securely computed in a network in which no such trusted party exists. We distinguish two types of faults. In case of Fail-Stop faults, our construction is valid as long as the faulty parties constitute less than a thzr-d of the parties in the network. In case of general (i.e., Byzantine) faults, our construction requires that the faulty parties are less than a fourth fraction. In both cases, the resilience of our construction is optimal. Our construction generalizes known synchronous constructions by Ben-Or, Goldwasser and Wigderson. In addition, we introduce and implement several new asynchronous primitives. Among these, we note an errorless asynchronous verifiable secret sharing scheme, an asynchronous agreement on a large set that is contained in the dynamical y growing inputs of all non-faulty parties, and an on-line error-correcting procedure. * email: benor@cs.huji.ac.il t ~~ail.. canetti@tx .technion.ac.il %email: odedrfiks.technion. ac.il. Supported by grant no. 8900312 from the United States — Israel Binational Science Foundation, Jerusalem, Israel Permission to copy without fee all or part of this material ia granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appaar, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwiee, or to republish, requiree a fee and/or specific permission. 25th ACM STOC ‘93-5 /93/CA, USA G 1993 ACM 0-89791 -591 -7/93 /0005 [0052 . ..$1 .50 Computer Science Dept. of Computer Science","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126290528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of locality based graph coloring. This problem is motivated by the problem of assigning time slots for broadcast in mobile packet radio networks. This problem has also been studied in the context of distributed and parallel graph coloring [4, 6, 9, 8]. In this problem, one has to design a coloring algorithm that assigns a color to a vertex based on the label of the vertex and the labels on its neighbors. Linial proved an upper bound of O(A2 log n) and a lower bound of fl(log log n) on the number of colors needed to locally color an n-vertex graph with maximum vertex degree A [9, 8]. His main motivation was that repeated application of local coloring gives a fast algorithm for distributed coloring. He proved that one could get a A2 coloring in O(log* n) steps this way. In this paper we improve upon the bounds for the problem of local coloring. Using a new characterization in terms of a family of set systems we design a randomized algorithm for the problem and prove an upper bound of O(A. 2A log log n). An important question left open in Linial’s paper was the case of large A. The best lower bound was A + 1. Linial observed that a result of Erdos, Frankl and Furedi implied that his method cannot be applied to reduce the number of colors to below (A~2). We obtain lower bounds that match the upper bounds within a factor that is poly-logarithmic in terms of these bounds. Of particular interest we have very precise bounds for the case when A > 2+. These bounds are useful to obtain a heuristic estimate on the *Researchsupported in part by Ketan Mulmuley’s Packard Fellowship. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of tha Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-51931CA, LJSA
{"title":"Locality based graph coloring","authors":"M. Szegedy, S. Vishwanathan","doi":"10.1145/167088.167156","DOIUrl":"https://doi.org/10.1145/167088.167156","url":null,"abstract":"We study the problem of locality based graph coloring. This problem is motivated by the problem of assigning time slots for broadcast in mobile packet radio networks. This problem has also been studied in the context of distributed and parallel graph coloring [4, 6, 9, 8]. In this problem, one has to design a coloring algorithm that assigns a color to a vertex based on the label of the vertex and the labels on its neighbors. Linial proved an upper bound of O(A2 log n) and a lower bound of fl(log log n) on the number of colors needed to locally color an n-vertex graph with maximum vertex degree A [9, 8]. His main motivation was that repeated application of local coloring gives a fast algorithm for distributed coloring. He proved that one could get a A2 coloring in O(log* n) steps this way. In this paper we improve upon the bounds for the problem of local coloring. Using a new characterization in terms of a family of set systems we design a randomized algorithm for the problem and prove an upper bound of O(A. 2A log log n). An important question left open in Linial’s paper was the case of large A. The best lower bound was A + 1. Linial observed that a result of Erdos, Frankl and Furedi implied that his method cannot be applied to reduce the number of colors to below (A~2). We obtain lower bounds that match the upper bounds within a factor that is poly-logarithmic in terms of these bounds. Of particular interest we have very precise bounds for the case when A > 2+. These bounds are useful to obtain a heuristic estimate on the *Researchsupported in part by Ketan Mulmuley’s Packard Fellowship. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of tha Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-51931CA, LJSA","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134480944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A minimum cut is a set of edges of minimum weight whose removal disconnects a given graph. Minimum cut algorithms historically applied duality with maximum flows and thus had the same 0 (inn) running time as maximum flow algorithms. More recent algorithms which are not based on maximum flows also require fl (inn) time. In this paper, we present the first algorithm that breaks the tl(mn) “max-flow barrier” for finding minimum cuts in weighted undirected graphs. We give a strongly polynomial randomized algorithm which finds a minimum cut with high probability in 0(n2 log3 n) time. This suggests that the rein-cut problem might be fundamentally easier to solve than the maximum flow problem. Our algorithm can be implemented in 72JUC using only nz processors—this is the first efficient 7UfC algorithm for the rein-cut problem. Our algorithm is simple and uses no complicated data structures.
{"title":"An Õ(n2) algorithm for minimum cuts","authors":"David R Karger, C. Stein","doi":"10.1145/167088.167281","DOIUrl":"https://doi.org/10.1145/167088.167281","url":null,"abstract":"A minimum cut is a set of edges of minimum weight whose removal disconnects a given graph. Minimum cut algorithms historically applied duality with maximum flows and thus had the same 0 (inn) running time as maximum flow algorithms. More recent algorithms which are not based on maximum flows also require fl (inn) time. In this paper, we present the first algorithm that breaks the tl(mn) “max-flow barrier” for finding minimum cuts in weighted undirected graphs. We give a strongly polynomial randomized algorithm which finds a minimum cut with high probability in 0(n2 log3 n) time. This suggests that the rein-cut problem might be fundamentally easier to solve than the maximum flow problem. Our algorithm can be implemented in 72JUC using only nz processors—this is the first efficient 7UfC algorithm for the rein-cut problem. Our algorithm is simple and uses no complicated data structures.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129035064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
N. Cesa-Bianchi, Y. Freund, D. Helmbold, D. Haussler, R. Schapire, Manfred K. Warmuth
We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called `experts''. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show how this leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context. We also extend our analysis to the case in which log loss is used instead of the expected number of mistakes.
{"title":"How to use expert advice","authors":"N. Cesa-Bianchi, Y. Freund, D. Helmbold, D. Haussler, R. Schapire, Manfred K. Warmuth","doi":"10.1145/167088.167198","DOIUrl":"https://doi.org/10.1145/167088.167198","url":null,"abstract":"We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called `experts''. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show how this leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context. We also extend our analysis to the case in which log loss is used instead of the expected number of mistakes.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128906179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that any randomized space(S) algorithm which uses only poly(S) random bits can be simulated deterministically in space(S), for S(n) ~ log n. Of independent interest is our main technical tool: a procedure which extracts randomness from a defective random source using a small additional number of truly random bits.
{"title":"More deterministic simulation in logspace","authors":"N. Nisan, David Zuckerman","doi":"10.1145/167088.167162","DOIUrl":"https://doi.org/10.1145/167088.167162","url":null,"abstract":"We show that any randomized space(S) algorithm which uses only poly(S) random bits can be simulated deterministically in space(S), for S(n) ~ log n. Of independent interest is our main technical tool: a procedure which extracts randomness from a defective random source using a small additional number of truly random bits.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128021126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The ultimate goal of the Human Genome Project and many other efforts in molecular biology is to sequence the chromosomal DNA of humans and other species and elucidate the genetic information contained therein. A less ambitious intermediate goal is to construct physical maps of our 23 pairs of chromosomes. A physical map specifies the locations of markers identifiable fragments of DNA along the DNA molecule. These markers provide a kind of random access to the linear DNA molecule. To locate a feature of interest such as a gene, an experimenter can start at a marker that is known to be close to the gene and “walk” along the DNA until the gene is identified.
{"title":"Mapping the genome: some combinatorial problems arising in molecular biology","authors":"R. Karp","doi":"10.1145/167088.167170","DOIUrl":"https://doi.org/10.1145/167088.167170","url":null,"abstract":"The ultimate goal of the Human Genome Project and many other efforts in molecular biology is to sequence the chromosomal DNA of humans and other species and elucidate the genetic information contained therein. A less ambitious intermediate goal is to construct physical maps of our 23 pairs of chromosomes. A physical map specifies the locations of markers identifiable fragments of DNA along the DNA molecule. These markers provide a kind of random access to the linear DNA molecule. To locate a feature of interest such as a gene, an experimenter can start at a marker that is known to be close to the gene and “walk” along the DNA until the gene is identified.","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116619752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that, for any fixed k, there exists an optimal O(n)-space compact representation of a k-connected graph G with n vertices, such that one can determine in O(1) time whether two vertices areconnectedbyk+l vertex-dkjoint paths, or are separated by k vertices/edges. Previously, the existence of such compact representations was known only for k <3. 1 Summary of Results A fundamental issue for the fault-tolerance and reliabilityofnetworksis determining the existence of multiple disjoint paths connecting two nodes. In this paper we investigate the problem of constructing a compact representation of a graph so that one can test quickly for the existence of such paths. *Research supported in part by the National ScienceFoundation under grant CCR9007851, by the U.S. Army Research Office under grant DAAL03-91-G-o035, by the Office of Naval Research and the Defense Advanced Research Projects Agency under contract NOO014-91-J-4052, ARPA order 8225, by the NATO Scientific Affairs Division under collaborative research grant 911016, by the Pro get t o Finalimato Sisterni Informatici e Calcolo ParaUelo of the Italian National Research Council, and by the Esprit II BRA of the European Community (project ALCOM). t Department of Computer Science, University of Newcastle, Callaghan, New South Wales 2308, Australia. The work of this author was carried out in part while visiting the United States Coast Guard Academy and Brown University. t Dip=tirnento & Inforrnatica e %ternist.ica, Universit& & Roma ‘[La Sapienza”, Via Salaria 113, 00198 Rome, Italy, 5Department of Computer Science, Texas A&M University, College Station, TX 77843-3112. f Depmtment of Computer Science, Brown University, providence, RI 02912–1910. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-5193 /CA, USA 9 1993 ACM 0-89791 -591 -7/93 /0005 /0794 . ..$1.!50 Given a graph G, we say that vertices v’ and vII of G are k-connected if there are k vertex-diajoint paths between vi and v“. Graph G is said to be kconnected if every pair of vertices is k-connected. Let G be a k-connected graph. A k-cut of G ia a k-tuple U={tl,... t} , k , where each t~ is a vertex or an edge, such that removing U disconnects G. By Menger’a theorem [5], G is (k + I )-connected if and only if it has no k-cut. We say that a k-cut U separates subsets V’ and V“ of vertices if U n V’ = U n V“ = 0, and there ia no connected component of G — U containing vertices of both V’ and V“. A (k + I)-connectivity query for vertices v’ and v“ of a k-connected graph G, denoted (k + I)-Paths(v’, v“), returns -true if v’ and v“ are (k + I)-connected, and otherwise retu
我们证明，对于任意固定的k，存在一个具有n个顶点的k连通图G的最优O(n)空间紧表示，使得人们可以在O(1)时间内确定两个顶点是由k+ 1个顶点连接路径连接，还是由k个顶点/边分开。以前，只有在k <3时才知道这种紧表示的存在性。网络容错和可靠性的一个基本问题是确定连接两个节点的多条不相交路径的存在性。在本文中，我们研究了构造一个图的紧表示的问题，以便人们可以快速地测试这样的路径的存在性。*部分研究由国家科学基金会(CCR9007851)、美国陆军研究办公室(daal03 -91- g - 035)、海军研究办公室和国防高级研究计划局(no014 -91- j -4052)、ARPA订单8225、北约科学事务部(911016)、意大利国家研究委员会(nrc)最终信息系统和Calcolo ParaUelo项目资助。欧共体的Esprit II BRA (ALCOM项目)。t纽卡斯尔大学计算机科学系，澳大利亚新南威尔士州卡拉汉2308本作者的部分工作是在访问美国海岸警卫队学院和布朗大学期间进行的。it + it + it + it + it + it + it + it + it + it + it + it + it。5 .德克萨斯农工大学计算机科学系，大学城，德克萨斯州77843-3112。f布朗大学计算机科学系，加州普罗维登斯02912-1910允许免费复制本材料的全部或部分，前提是这些副本不是为了直接的商业利益而制作或分发的，必须出现ACM版权声明、出版物的标题和日期，并注明复制是由计算机协会许可的。以其他方式复制或重新发布需要付费和/或特定许可。25 ACM STOC ' 93-5193 /CA, USA 9 1993 ACM 0-89791 -591 -7/93 /0005 /0794…$1 !50给定一个图G，如果在vi和v”之间有k个顶点不相交的路径，我们说G的顶点v '和vII是k连通的。如果图G的每个顶点对都是k连通的，则称图G是连通的。设G是一个k连通图。k元组U={tl，…t}， k，其中每个t~是一个顶点或一条边，使得移除U断开G。根据门格尔定理[5]，G是(k + I)连通的当且仅当它没有k-cut。我们说，如果U n V ' = U n V ' = 0，则k-cut U将顶点的子集V '和V "分开，并且G - U中不存在同时包含V '和V "顶点的连通分量。对于k连通图G的顶点v '和v "，表示为(k + I)-Paths(v '， v ")的(k + I)-连通性查询，如果v '和v "是(k + I)连通，则返回-true，否则返回分隔v '和v "的k-cut。本文证明了对于任意固定的k，存在(k + 1)个连通顶点对的最优紧表示。也就是说，我们提出了一个O(rz)空间的数据结构，它支持在O(1)时间内对k个连接的n顶点图进行(k + l)个连接查询。在此之前，这种紧表示的存在性仅在k <3时才被建立[1,3,7,9]。虽然可以有f2(rz2)个k-cuts[2]，但我们的数据结构是基于O (n)个k-cuts的递归分解图，这样任何两个不(k+ 1)连接的顶点都被这个aet的k-cut分开。这种递归分解使用了名为“wheels”和“flowers”的结构，它们重新解释和推广了先前在[3]中为3连通图定义的相应结构。如果我们考虑边缘连接，LG。，边连接路径和仅由边组成的k-cut，那么问题就变成了回答边连通性查询的问题，对此，可以从边k-cut的表示中得到一个有效的数据结构[4](参见[6])。顶点连通性问题通常比相应的边连通性问题更难。我们的结果表明，对于任何固定k的(k + 1)-连通性查询，情况并非如此。
{"title":"Reinventing the wheel: an optimal data structure for connectivity queries","authors":"R. Cohen, G. Battista, A. Kanevsky, R. Tamassia","doi":"10.1145/167088.167152","DOIUrl":"https://doi.org/10.1145/167088.167152","url":null,"abstract":"We show that, for any fixed k, there exists an optimal O(n)-space compact representation of a k-connected graph G with n vertices, such that one can determine in O(1) time whether two vertices areconnectedbyk+l vertex-dkjoint paths, or are separated by k vertices/edges. Previously, the existence of such compact representations was known only for k <3. 1 Summary of Results A fundamental issue for the fault-tolerance and reliabilityofnetworksis determining the existence of multiple disjoint paths connecting two nodes. In this paper we investigate the problem of constructing a compact representation of a graph so that one can test quickly for the existence of such paths. *Research supported in part by the National ScienceFoundation under grant CCR9007851, by the U.S. Army Research Office under grant DAAL03-91-G-o035, by the Office of Naval Research and the Defense Advanced Research Projects Agency under contract NOO014-91-J-4052, ARPA order 8225, by the NATO Scientific Affairs Division under collaborative research grant 911016, by the Pro get t o Finalimato Sisterni Informatici e Calcolo ParaUelo of the Italian National Research Council, and by the Esprit II BRA of the European Community (project ALCOM). t Department of Computer Science, University of Newcastle, Callaghan, New South Wales 2308, Australia. The work of this author was carried out in part while visiting the United States Coast Guard Academy and Brown University. t Dip=tirnento & Inforrnatica e %ternist.ica, Universit& & Roma ‘[La Sapienza”, Via Salaria 113, 00198 Rome, Italy, 5Department of Computer Science, Texas A&M University, College Station, TX 77843-3112. f Depmtment of Computer Science, Brown University, providence, RI 02912–1910. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-5193 /CA, USA 9 1993 ACM 0-89791 -591 -7/93 /0005 /0794 . ..$1.!50 Given a graph G, we say that vertices v’ and vII of G are k-connected if there are k vertex-diajoint paths between vi and v“. Graph G is said to be kconnected if every pair of vertices is k-connected. Let G be a k-connected graph. A k-cut of G ia a k-tuple U={tl,... t} , k , where each t~ is a vertex or an edge, such that removing U disconnects G. By Menger’a theorem [5], G is (k + I )-connected if and only if it has no k-cut. We say that a k-cut U separates subsets V’ and V“ of vertices if U n V’ = U n V“ = 0, and there ia no connected component of G — U containing vertices of both V’ and V“. A (k + I)-connectivity query for vertices v’ and v“ of a k-connected graph G, denoted (k + I)-Paths(v’, v“), returns -true if v’ and v“ are (k + I)-connected, and otherwise retu","PeriodicalId":280602,"journal":{"name":"Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing","volume":"76 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123159906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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