Fast Dynamic Weight Matchings in Convex Bipartite Graphs

We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V,E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense. Despite this difficulty, we develop a data structure which maintains the set of vertices that participate in a maximum matching in O(log2 |V|) amortized time per update and reports the status of a vertex (matched or unmatched) in constant worst-case time. Our structure can report the mate of a matched vertex in the maximum matching in worst-case O(min{k log2 |V |+log |V|, |V| log |V|}) time, where k is the number of update operations since the last query for the same pair of vertices was made. In addition, we give an O(√|V| log2 |V|)-time amortized bound for this pair query.

Confluently Persistent Deques via Data-Structural Bootstrapping

We introduce data-structural bootstrapping, a technique to design data structures recursively, and use it to design confluently persistent deques. Our data structure requires O(log* k) worst-case time and space per deletion, where k is the total number of deque operations, and constant worst-case time and space for other operations. Further, the data structure allows a purely functional implementation, with no side effects. This improves a previous result of Driscoll, Sleator, and Tajan (in Proceedings 2nd Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 89-99).

Obtaining a matching in a graph satisfying a certain objective is an important class of graph problems. Matching algorithms have received attention for several decades. However, while there are efficient algorithms to obtain a maximum weight matching, not much is known about the maximum weight maximum cardinality, and maximum cardinality maximum weight matching problems for general graphs. Our contribution in this work is to show that for bounded weight input graphs one can obtain an algorithm for both maximum weight maximum cardinality (for real weights), and maximum cardinality maximum weight matching (for integer weights) by modifying the input and running the existing maximum weight matching algorithm. Also, given the current state of the art in maximum weight matching algorithms, we show that, for bounded weight input graphs, both maximum weight maximum cardinality, and maximum cardinality maximum weight matching have algorithms of similar complexities to that of maximum weight matching. Subsequently, we also obtain approximation algorithms for maximum weight maximum cardinality, and maximum cardinality maximum weight matching.

On suffix extensions in suffix trees

We introduce a new framework for computing approximate maximum weight matchings. Our primary focus is on the fully dynamic setting, where there is a large gap between the guarantees of the best known algorithms for computing weighted and unweighted matchings. Indeed, almost all current weighted matching algorithms that reduce to the unweighted problem lose a factor of two in the approximation ratio. In contrast, in other sublinear models such as the distributed and streaming models, recent work has largely closed this weighted/unweighted gap. For bipartite graphs, we almost completely settle the gap with a general reduction that converts any algorithm for α-approximate unweighted matching to an algorithm for (1−)α-approximate weighted matching, while only increasing the update time by an O(logn) factor for constant . We also show that our framework leads to significant improvements for non-bipartite graphs, though not in the form of a universal reduction. In particular, we give two algorithms for weighted non-bipartite matching: 1. A randomized (Las Vegas) fully dynamic algorithm that maintains a (1/2−)-approximate maximum weight matching in worst-case update time O(polylog n) with high probability against an adaptive adversary. Our bounds are essentially the same as those of the unweighted algorithm of Wajc [STOC 2020]. 2. A deterministic fully dynamic algorithm that maintains a (2/3−)-approximate maximum weight matching in amortized update time O(m1/4). Our bounds are essentially the same as those of the unweighted algorithm of Bernstein and Stein [SODA 2016]. A key feature of our framework is that it uses existing algorithms for unweighted matching as black-boxes. As a result, our framework is simple and versatile. Moreover, our framework easily translates to other models, and we use it to derive new results for the weighted matching problem in streaming and communication complexity models.

The delay and throughput characteristics of a packet switch depend mainly on the queueing scheme and the scheduling algorithm deployed at the switch. Early research on scheduling algorithms has mainly focused on maximum weight matching scheduling algorithms. It is known that maximum weight matching algorithms guarantee the stability of input-queued switches, but are impractical due to their high computational complexity. Later research showed that the less complex maximal matching algorithms can stabilize input-queued switches when they are deployed with a speed-up of two. For practical purposes, neither a high computational complexity nor a speed-up of two is desirable.In this paper, we investigate the application of matching algorithms that approximate maximum weight matching algorithms to scheduling problems. We show that while having a low computational complexity, they guarantee the stability of input queued switches when they are deployed with a moderate speed-up.In particular, we show that the improve_matching algorithm stabilizes input-queued switches when it is deployed with a speed-up of 3<over>2+ε.In a second step, we further improve on these results by proposing a class of maximal weight matching algorithms that stabilize an input-queued switch without any speed-up.Whereas initial research has only focused on scheduling algorithms that guarantee the stability of a single switch, recent work has shown how scheduling algorithms for single switches can be modified in order to design distributed scheduling algorithms that stabilize networks of input-queued switches. Using those results, we show that the switching algorithms proposed in this paper do not only stabilize a single switch, but also networks of input-queued switches.

Fully Dynamic Algorithms for Maintaining Shortest Paths Trees

Reflected min-max heaps

Open AccessOpen Access licenseAboutSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail Go to SectionOpen AccessOpen Access license HomeStochastic SystemsVol. 2, No. 2 Diffusion Approximation for an Input-queued Switch Operating under a Maximum Weight Matching PolicyW. N. Kang, R. J. WilliamsW. N. Kang, R. J. WilliamsPublished Online:19 Dec 2012https://doi.org/10.1287/12-SSY061AbstractFor N ≥ 2, we consider an N × N input-queued switch operating under a maximum weight matching policy. We establish a diffusion approximation for a (2N − 1)-dimensional workload process associated with this switch when all input ports and output ports are heavily loaded. The diffusion process is a semimartingale reflecting Brownian motion living in a polyhedral cone with N2 boundary faces, each of which has an associated constant direction of reflection. Our proof builds on our own prior work [13] on an invariance principle for semimartingale reflecting Brownian motions in piecewise smooth domains and on a multiplicative state space collapse result for switched networks established by Shah and Wischik in [19]. Previous Back to Top Next FiguresReferencesRelatedInformationCited ByHeavy-Traffic Analysis of Queueing Systems with No Complete Resource PoolingDaniela Andrea Hurtado Lange, Siva Theja Maguluri2 March 2022 | Mathematics of Operations Research, Vol. 0, No. 0Fluctuation Bounds for the Max-Weight Policy with Applications to State Space CollapseArsalan Sharifnassab, John N. Tsitsiklis, S. Jamaloddin Golestani9 April 2020 | Stochastic Systems, Vol. 10, No. 3Collaboration and Multitasking in Networks: Prioritization and Achievable CapacityItai Gurvich, Jan A. Van Mieghem19 May 2017 | Management Science, Vol. 64, No. 5 Volume 2, Issue 2December 2012Pages 232-446 Article Information Metrics Downloaded 48 times in the past 12 months Information Received:January 01, 2012Published Online:December 19, 2012 Copyright © 2012, The author(s)Cite asW. N. Kang, R. J. Williams (2012) Diffusion Approximation for an Input-queued Switch Operating under a Maximum Weight Matching Policy. Stochastic Systems 2(2):277-321. https://doi.org/10.1287/12-SSY061 KeywordsInput-queued switchmaximum weight matching policydiffusion approximationheavy trafficsemimartingale reflecting Brownian motion (SRBM)PDF download

The problem of computing a matching of maximum weight in a given edge-weighted graph is not known to be P-hard or in RNC. This paper presents four parallel approximation algorithms for this problem. The first is an RNC-approximation scheme, i.e., an RNC algorithm that computes a matching of weight at least 1 - Ɛ times the maximum for any given constant Ɛ > 0. The second one is an NC approximation algorithm achieving an approximation ratio of 1/2+Ɛ for any fixed Ɛ > 0. The third and fourth algorithms only need to know the total order of weights, so they are useful when the edge weights require a large amount of memories to represent. The third one is an NC approximation algorithm that finds a matching of weight at least 2/3Δ+2 times the maximum, where Δ is the maximum degree of the graph. The fourth one is an RNC algorithm that finds a matching of weight at least 1/2Δ+4 times the maximum on average, and runs in O(logΔ) time, not depending on the size of the graph.

Many dynamic graph algorithms have an amortized update time, rather than a stronger worst-case guarantee. But amortized data structures are not suitable for real-time systems, where each individual operation has to be executed quickly. For this reason, there exist many recent randomized results that aim to provide a guarantee stronger than amortized expected. The strongest possible guarantee for a randomized algorithm is that it is always correct (Las Vegas) and has high-probability worst-case update time, which gives a bound on the time for each individual operation that holds with high probability. In this article, we present the first polylogarithmic high-probability worst-case time bounds for the dynamic spanner and the dynamic maximal matching problem. (1) For dynamic spanner, the only known o ( n ) worst-case bounds were O ( n 3/4 ) high-probability worst-case update time for maintaining a 3-spanner and O ( n 5/9 ) for maintaining a 5-spanner. We give a O (1) k log 3 ( n ) high-probability worst-case time bound for maintaining a ( 2k-1 )-spanner, which yields the first worst-case polylog update time for all constant k . (All the results above maintain the optimal tradeoff of stretch 2k-1 and Õ( n 1+1/k ) edges.) (2) For dynamic maximal matching, or dynamic 2-approximate maximum matching, no algorithm with o(n) worst-case time bound was known and we present an algorithm with O (log 5 ( n )) high-probability worst-case time; similar worst-case bounds existed only for maintaining a matching that was (2+ϵ)-approximate, and hence not maximal. Our results are achieved using a new approach for converting amortized guarantees to worst-case ones for randomized data structures by going through a third type of guarantee, which is a middle ground between the two above: An algorithm is said to have worst-case expected update time ɑ if for every update σ, the expected time to process σ is at most ɑ. Although stronger than amortized expected, the worst-case expected guarantee does not resolve the fundamental problem of amortization: A worst-case expected update time of O(1) still allows for the possibility that every 1/ f(n) updates requires ϴ ( f(n) ) time to process, for arbitrarily high f(n) . In this article, we present a black-box reduction that converts any data structure with worst-case expected update time into one with a high-probability worst-case update time: The query time remains the same, while the update time increases by a factor of O (log 2(n) ). Thus, we achieve our results in two steps: (1) First, we show how to convert existing dynamic graph algorithms with amortized expected polylogarithmic running times into algorithms with worst-case expected polylogarithmic running times. (2) Then, we use our black-box reduction to achieve the polylogarithmic high-probability worst-case time bound. All our algorithms are Las-Vegas-type algorithms.

Given a node x at depth itd in a rooted tree LevelAncestor (x, i) returns the ancestor to x in depth d — i. We show how to maintain a tree under addition of new leaves so that updates and level ancestor queries are being performed in worst case constant time. Given a forest of trees with n nodes where edges can be added, m queries and updates take O(mα(m, n)) time. This solves two open problems (P.F. Dietz, Finding level-ancestors in dynamic trees, LNCS, 519:32–40, 1991). In a tree with node weights, min(x, y) report the node with minimum weight on the path between the nodes x and y. We can substitute the LevelAncestor query with min, without increasing the complexity for updates and queries. Previously such results have been known only for special cases (e.g. R.E. Tarjan. Applications of path compression on balanced trees. J.ACM, 26(4):690–715, 1979).

A parallel matching algorithm for convex bipartite graphs and applications to scheduling

An efficient parallel algorithm for scheduling interval ordered tasks