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Minor Excluded Network Families Admit Fast Distributed Algorithms∗ Session 3D: Graphs and Population PODC’18, July 23-27, 2018, Egham, United Kingdom Minor Excluded Network Families Admit Fast Distributed Algorithms∗ Bernhard Haeupler Jason Li Goran Zuzic Carnegie Mellon University Carnegie Mellon University Carnegie Mellon University Pittsburgh, PA Pittsburgh, PA Pittsburgh, PA [email protected] [email protected] [email protected] ABSTRACT KEYWORDS Distributed network optimization problems, such as minimum span- Distributed Computing; CONGEST; Graph Structure Theorem; Mi- ning tree, minimum cut, and shortest path, are an active research nor Excluded Graphs; Shortcuts; Low-congestion Shortcuts area in distributed computing. This paper presents a fast distributed ACM Reference Format: algorithm for such problems in the CONGEST model, on networks Bernhard Haeupler, Jason Li, and Goran Zuzic. 2018. Minor Excluded Net- that exclude a fixed minor. work Families Admit Fast Distributed Algorithms. In PODC ’18: ACM Sym- On general graphs, many optimization problems, including the posium on Principles of Distributed Computing, July 23ś27, 2018, Egham, ones mentioned above, require Ω(˜ √n) rounds of communication United Kingdom. ACM, New York, NY, USA, 10 pages. https://doi.org/10. in the CONGEST model, even if the network graph has a much 1145/3212734.3212776 smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on 1 INTRODUCTION a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari Network optimization problems in the CONGEST model, such as and Haeupler [SODA’16]. Building off of their work, this paper Minimum Spanning Tree, Min-Cut or Shortest Path are an active proves that excluded minor graphs admit high-quality shortcuts, research area in theoretical distributed computing [8, 11, 22, 27]. leading to an O˜ (D2) round algorithm for the aforementioned prob- This paper provides a fast distributed algorithm for such problems lems, where D is the diameter of the network graph. To work with in excluded minor graphs in the CONGEST model. excluded minor graph families, we utilize the Graph Structure The- Recently, lower bounds have been established on many dis- orem of Robertson and Seymour. To the best of our knowledge, this tributed network optimization problems, including all of the men- is the first time the Graph Structure Theorem has been used foran tioned ones [5]. More specifically, each of these problems in the ˜ 1 algorithmic result in the distributed setting. CONGEST model require Ω(√n) rounds of communication to Even though the proof is involved, merely showing the existence solve. This holds even on graphs with small diameter, for example, of good shortcuts is sufficient to obtain simple, efficient distributed when the diameter is logarithmic in the number of nodes n. The algorithms. In particular, the shortcut framework can efficiently result is surprising since it is not immediately clear why a network construct near-optimal shortcuts and then use them to solve the of small diameter requires such a large number of communication optimization problems. This, combined with the very general family rounds when solving an optimization problem. of excluded minor graphs, which includes most other important On the positive side, the next major question in algorithmic graph classes, makes this result of significant interest. design is to determine whether one can bypass this barrier by re- stricting the class of network graphs. One immediate question that CCS CONCEPTS arises is, what family of graphs should one consider? Ideally, such a class of graphs should be inclusive enough to admit most łrealisticž · Mathematics of computing → Graphs and surfaces; · The- networks, yet be restrictive enough to disallow the pathological ory of computation → Distributed computing models; Distributed lower bound instances. algorithms; In our search for a restricted graph family to study, we focus on three criteria. First, we desire a family with a rich and rigorous ∗Supported in part by NSF grants CCF-1527110, CCF-1618280 and NSF CA- mathematical theory, so that our result is technically meaningful. REER award CCF-1750808. The full version of the paper is available at Second, the family should capture many networks in practice. And https://arxiv.org/abs/1801.06237 finally, we want robustness: a graph with a few added or perturbed edges and vertices should still remain in the family. Robustness is an important goal, since we want our graph family to be resistant to Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed noise. For example, planar graphs satisfy the first two criteria, but for profit or commercial advantage and that copies bear this notice and the full citation fail to be robust since often adding a single random edge will make on the first page. Copyrights for components of this work owned by others than the the graph non-planar. Indeed, most algorithms on planar graphs fail author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission completely when run on a planar graph with a few perturbed edges and/or a fee. Request permissions from [email protected]. and vertices. Next, one might try genus-bounded graphs, but they PODC ’18, July 23ś27, 2018, Egham, United Kingdom © 2018 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-1-4503-5795-1/18/07...$15.00 1Throughout this paper, O˜ ( ) and Ω(˜ ) hide polylogarithmic factors in n, the number https://doi.org/10.1145/3212734.3212776 of nodes in the network. · · 465 Session 3D: Graphs and Population PODC’18, July 23-27, 2018, Egham, United Kingdom also suffer from similar problems since adding a single randomly of [15] computes a shortcut competitive to the optimal one, a con- connected vertex can arbitrarily increase the genus. sequence is that the running time of this algorithm rarely depends A candidate graph family that fulfills all three conditions is the on the (large) constants appearing in the Graph Structure Theorem. family of excluded minor graphs, namely the graphs which do not In other words, while we can only prove that the constants in the have a fixed graph H as a minor. This family encompasses sev- running time are bounded by (some functions of) the constants eral classes of naturally occuring networks. For example, trees in the Graph Structure Theorem, the actual running time of the which exclude K3, planar graphs that capture the structure of two- algorithm is likely to be much smaller. In fact, for most excluded 2 dimensional maps exclude K5 and K3,3, and, series-parallel graphs minor networks, we expect the running time to be O˜ (D ) with a that capture many network backbones exclude K4 [1, 9]. Excluded small constant, or even O˜ (D). In contrast, algorithms that explic- minor graphs also have a history of deep results, including the itly compute a Graph Structure Theorem decomposition have an series of Graph Minor papers by Robertson and Seymour. inherent bottleneck in the form of the potentially huge constants In this paper, we provide efficient distributed algorithms for the of the Graph Structure Theorem. class of excluded minor graphs which break the Ω(˜ √n + D) lower This paper is structured as follows. After the introduction, we bound for general graphs, giving evidence that most practical net- begin with introducing the two main tools necessary for our main re- works admit efficient distributed algorithms. We show an O˜ (D2) sult, namely the Graph Structure Theorem and the low-congestion algorithm for MST and (1 + ϵ) approximate min-cut, among other shortcuts framework. Then, we prove the existence of good short- results. For networks having low diameter, such as D = poly(logn) cuts one step at a time, following the step-by-step construction in or D = no(1), our algorithms are optimal up to poly(logn) or no(1) the Graph Structure Theorem. factors, respectively. This is a significant improvement over previ- ous MST and min-cut algorithms, which run in Ω(√n) time even 1.1 Outline of the Proof on an excluded minor graph with D = poly(logn), such as a planar The goal of this section is to outline the proof of our main result, graph with an added vertex attached to every other node. without delving into the technical details. Some concepts will be left Our results use the framework of low-congestion shortcuts, undefined (e.g., shortcut quality, tree-restricted shortcuts), since the introduced by [12] and built on by [15], which is a combinatorial definitions are technical and require a lot of motivation beforehand. abstraction to designing distributed algorithms. It introduces a However, one can think of shortcuts as a combinatorial construction simple, combinatorial problem involving shortcuts on a graph, on a graph, and think of quality as a metric with which to measure and guarantees that a good quality solution to this combinatorial a solution. problem automatically translates to a simple, efficient distributed ( + ) Theorem 1. [Haeupler et al. [15, 16]] Suppose that a graph with algorithm for MST and 1 ϵ approximate min-cut, among other N N problems; the concepts of shortcuts and quality will be defined later. diameter D admits tree-restricted shortcuts of quality q : . ˜ → Actually, the algorithm is the same regardless of the network or the Then, there is an O(q(D))-round distributed algorithm for MST and + graph family; the purpose of the combinatorial shortcuts problem is (1 ϵ)-approximate min-cut for that graph. to prove that the algorithm runs efficiently on the graph or family.
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