In classic distributed graph problems, each instance on a graph specifies a space of feasible solutions (e.g. all proper (Δ + 1)-listcolorings of the graph), and the task of distributed algorithm is to construct a feasible solution using local information. We study distributed sampling and counting problems, in which each instance specifies a joint distribution of feasible solutions. The task of distributed algorithm is to sample from this joint distribution, or to locally measure the volume of the probability space via the marginal probabilities. The latter task is also known as inference, which is a local counterpart of counting. For self-reducible classes of instances, the following equivalences are established in the LOCAL model up to polylogarithmic factors: For all joint distributions, approximate inference and approximate sampling are computationally equivalent. For all joint distributions defined by local constraints, exact sampling is reducible to either one of the above tasks. If further, sequentially constructing a feasible solution is trivial locally, then all above tasks are easy if and only if the joint distribution exhibits strong spatial mixing. Combining with the state of the arts of strong spatial mixing, we obtain efficient sampling algorithms in the LOCAL model for various important sampling problems, including: an O( √ Δ log3 n)-round algorithm for exact sampling matchings in graphs with maximum degree Δ, and anO(log3 n)-round algorithm for sampling according to the hardcore model (weighted independent sets) in the uniqueness regime, which along with the Ω(diam) lower bound in [3] for sampling according to the hardcore model in the non-uniqueness regime, gives the first computational phase transition for distributed sampling.
{"title":"On Local Distributed Sampling and Counting","authors":"Weiming Feng, Yitong Yin","doi":"10.1145/3212734.3212757","DOIUrl":"https://doi.org/10.1145/3212734.3212757","url":null,"abstract":"In classic distributed graph problems, each instance on a graph specifies a space of feasible solutions (e.g. all proper (Δ + 1)-listcolorings of the graph), and the task of distributed algorithm is to construct a feasible solution using local information. We study distributed sampling and counting problems, in which each instance specifies a joint distribution of feasible solutions. The task of distributed algorithm is to sample from this joint distribution, or to locally measure the volume of the probability space via the marginal probabilities. The latter task is also known as inference, which is a local counterpart of counting. For self-reducible classes of instances, the following equivalences are established in the LOCAL model up to polylogarithmic factors: For all joint distributions, approximate inference and approximate sampling are computationally equivalent. For all joint distributions defined by local constraints, exact sampling is reducible to either one of the above tasks. If further, sequentially constructing a feasible solution is trivial locally, then all above tasks are easy if and only if the joint distribution exhibits strong spatial mixing. Combining with the state of the arts of strong spatial mixing, we obtain efficient sampling algorithms in the LOCAL model for various important sampling problems, including: an O( √ Δ log3 n)-round algorithm for exact sampling matchings in graphs with maximum degree Δ, and anO(log3 n)-round algorithm for sampling according to the hardcore model (weighted independent sets) in the uniqueness regime, which along with the Ω(diam) lower bound in [3] for sampling according to the hardcore model in the non-uniqueness regime, gives the first computational phase transition for distributed sampling.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123727161","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}
Pierre Aboulker, Marthe Bonamy, N. Bousquet, Louis Esperet
This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring n-vertex planar graphs with 7 colors in O(log n) rounds. Here, we show how to color planar graphs with 6 colors in polylog(n) rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every n-vertex planar graph with 4 colors in o(n) rounds.
{"title":"Distributed Coloring in Sparse Graphs with Fewer Colors","authors":"Pierre Aboulker, Marthe Bonamy, N. Bousquet, Louis Esperet","doi":"10.1145/3212734.3212740","DOIUrl":"https://doi.org/10.1145/3212734.3212740","url":null,"abstract":"This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring n-vertex planar graphs with 7 colors in O(log n) rounds. Here, we show how to color planar graphs with 6 colors in polylog(n) rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every n-vertex planar graph with 4 colors in o(n) rounds.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131107901","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 address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results. Our main hardness result is that an ∝-approximation for the minimum directed k-spanner problem for k ≥ 5 requires Ω(n/ √ ∝ logn) rounds using deterministic algorithms or Ω( √ n/ √ ∝ logn) rounds using randomized ones, in the Congest model of distributed computing. Combined with the constant-round O(nε )-approximation algorithm in the Local model of [Barenboim, Elkin and Gavoille, 2016], as well as a polylog-round (1 + ε )-approximation algorithm in the Local model that we show here, our lower bounds for the Congest model imply a strict separation between the Local and Congest models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem. Similarly, a separation between the directed and undirected cases is implied. We also prove a nearly-linear lower bound for the minimum weighted k-spanner problem for k ≥ 4, and we show lower bounds for the weighted 2-spanner problem. On the algorithmic side, apart from the aforementioned (1 + ε )- approximation algorithm for minimum k-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of O(log(m/n)) for a graph with n vertices andm edges, which matches the best known ratio for polynomial time sequential algorithms [Kortsarz and Peleg, 1994], and is tight if we restrict ourselves to polynomial local computations. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem. It also provides a Congest algorithm for the minimum dominating set problem, with a guaranteed O(log Δ) approximation ratio.
{"title":"Distributed Spanner Approximation","authors":"K. Censor-Hillel, Michal Dory","doi":"10.1145/3212734.3212758","DOIUrl":"https://doi.org/10.1145/3212734.3212758","url":null,"abstract":"We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results. Our main hardness result is that an ∝-approximation for the minimum directed k-spanner problem for k ≥ 5 requires Ω(n/ √ ∝ logn) rounds using deterministic algorithms or Ω( √ n/ √ ∝ logn) rounds using randomized ones, in the Congest model of distributed computing. Combined with the constant-round O(nε )-approximation algorithm in the Local model of [Barenboim, Elkin and Gavoille, 2016], as well as a polylog-round (1 + ε )-approximation algorithm in the Local model that we show here, our lower bounds for the Congest model imply a strict separation between the Local and Congest models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem. Similarly, a separation between the directed and undirected cases is implied. We also prove a nearly-linear lower bound for the minimum weighted k-spanner problem for k ≥ 4, and we show lower bounds for the weighted 2-spanner problem. On the algorithmic side, apart from the aforementioned (1 + ε )- approximation algorithm for minimum k-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of O(log(m/n)) for a graph with n vertices andm edges, which matches the best known ratio for polynomial time sequential algorithms [Kortsarz and Peleg, 1994], and is tight if we restrict ourselves to polynomial local computations. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem. It also provides a Congest algorithm for the minimum dominating set problem, with a guaranteed O(log Δ) approximation ratio.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130754039","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}
An atomic cross-chain swap is a distributed coordination task where multiple parties exchange assets across multiple blockchains, for example, trading bitcoin for ether. An atomic swap protocol guarantees (1) if all parties conform to the protocol, then all swaps take place, (2) if some coalition deviates from the protocol, then no conforming party ends up worse off, and (3) no coalition has an incentive to deviate from the protocol. A cross-chain swap is modeled as a directed graph D, whose vertexes are parties and whose arcs are proposed asset transfers. For any pair (D, L), where D = (V,A) is a strongly-connected directed graph and L ⊂ V a feedback vertex set for D, we give an atomic cross-chain swap protocol for D, using a form of hashed timelock contracts, where the vertexes in L generate the hashlocked secrets. We show that no such protocol is possible if D is not strongly connected, or if D is strongly connected but L is not a feedback vertex set. The protocol has time complexityO(diam(D)) and space complexity (bits stored on all blockchains) O(|A|2).
{"title":"Atomic Cross-Chain Swaps","authors":"M. Herlihy","doi":"10.1145/3212734.3212736","DOIUrl":"https://doi.org/10.1145/3212734.3212736","url":null,"abstract":"An atomic cross-chain swap is a distributed coordination task where multiple parties exchange assets across multiple blockchains, for example, trading bitcoin for ether. An atomic swap protocol guarantees (1) if all parties conform to the protocol, then all swaps take place, (2) if some coalition deviates from the protocol, then no conforming party ends up worse off, and (3) no coalition has an incentive to deviate from the protocol. A cross-chain swap is modeled as a directed graph D, whose vertexes are parties and whose arcs are proposed asset transfers. For any pair (D, L), where D = (V,A) is a strongly-connected directed graph and L ⊂ V a feedback vertex set for D, we give an atomic cross-chain swap protocol for D, using a form of hashed timelock contracts, where the vertexes in L generate the hashlocked secrets. We show that no such protocol is possible if D is not strongly connected, or if D is strongly connected but L is not a feedback vertex set. The protocol has time complexityO(diam(D)) and space complexity (bits stored on all blockchains) O(|A|2).","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129832928","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}
Distributed network optimization problems, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require Ω(√ n) rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA'16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an Õ(D^2) round algorithm for the aforementioned problems, where D is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can efficiently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.
{"title":"Minor Excluded Network Families Admit Fast Distributed Algorithms","authors":"Bernhard Haeupler, Jason Li, Goran Zuzic","doi":"10.1145/3212734.3212776","DOIUrl":"https://doi.org/10.1145/3212734.3212776","url":null,"abstract":"Distributed network optimization problems, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require Ω(√ n) rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA'16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an Õ(D^2) round algorithm for the aforementioned problems, where D is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can efficiently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121308035","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}
Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed optimization problems. Our main result is such a distributed algorithm for the fundamental primitive of computing simple functions over each part of a graph partition. From this algorithm we derive round- and message-optimal algorithms for multiple problems, including MST, Approximate Min-Cut and Approximate Single Source Shortest Paths, among others. On general graphs all of our algorithms achieve worst-case optimal Õ (D+√ n) round complexity and Õ (m) message complexity. Furthermore, our algorithms require an optimal Õ (D) rounds and Õ (n) messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.
{"title":"Round- and Message-Optimal Distributed Graph Algorithms","authors":"Bernhard Haeupler, D. E. Hershkowitz, David Wajc","doi":"10.1145/3212734.3212737","DOIUrl":"https://doi.org/10.1145/3212734.3212737","url":null,"abstract":"Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed optimization problems. Our main result is such a distributed algorithm for the fundamental primitive of computing simple functions over each part of a graph partition. From this algorithm we derive round- and message-optimal algorithms for multiple problems, including MST, Approximate Min-Cut and Approximate Single Source Shortest Paths, among others. On general graphs all of our algorithms achieve worst-case optimal Õ (D+√ n) round complexity and Õ (m) message complexity. Furthermore, our algorithms require an optimal Õ (D) rounds and Õ (n) messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125820213","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 consider graph coloring and related problems in the distributed message-passing model. em Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1 - hop-neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative (Δ + 1)-coloring algorithm requires Ω(Δ log Δ + log^* n) rounds, unless there exists "a very special type of coloring that can be very efficiently reduced" citeSV93. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms, and to explore other approaches to the coloring problem citeBE09,K09,B15,FHK16. The latter gave rise to faster algorithms, but their heavy machinery which is of non-locally-iterative nature made them far less suitable to various settings. In this paper we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative (Δ + 1)-coloring algorithm with running time O(Δ + log^* n), i.e., em below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing and bandwidth-restricted settings. This includes the following results. beginitemize ıtem We obtain self-stabilizing distributed algorithms for (Δ + 1)-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set and maximal matching with O(Δ + log^* n) time. This significantly improves previously-known results that have O(n) or larger running times citeGK10. ıtem We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O(Δ + log^* n) time and O(Δ)-edge-coloring in the Bit-Round model with O(Δ + log n) time. The factors of log^* n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously-known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. ıtem We obtain an arbdefective coloring algorithm with running time O(sqrt Δ + log^* n). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it in order to compute a proper (1 + ε)Δ-coloring within O(√ Δ + log^* n) time, and √(Δ + 1)√-coloring within √O(√ Δ log Δ log^* Δ + log^* n)√ time. This improves the recent state-of-the-art bounds of Barenboim from PODC'15 citeB15 and Fraigniaud et al. from FOCS'16 citeFHK16 by polylogarithmic factors. ıtem Our algorithms are applicable to the SET-LOCAL model citeHKMS15 (also known as the weak LOCAL model). In this model a relatively strong lower bound of √Ω(Δ^1/3 )√ is known for √(Δ + 1)√-coloring. However, most of the coloring algorithms do not work in this model. (In citeHKMS15 only Linial's √O(Δ^2)√-time algorithm and Kuhn-Wattenhofer √O(Δ log Δ)√-time algorithms are shown to work in it.) We obtain the first li
{"title":"Locally-Iterative Distributed (Δ+ 1): -Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models","authors":"Leonid Barenboim, Michael Elkin, Uri Goldenberg","doi":"10.1145/3212734.3212769","DOIUrl":"https://doi.org/10.1145/3212734.3212769","url":null,"abstract":"We consider graph coloring and related problems in the distributed message-passing model. em Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1 - hop-neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative (Δ + 1)-coloring algorithm requires Ω(Δ log Δ + log^* n) rounds, unless there exists \"a very special type of coloring that can be very efficiently reduced\" citeSV93. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms, and to explore other approaches to the coloring problem citeBE09,K09,B15,FHK16. The latter gave rise to faster algorithms, but their heavy machinery which is of non-locally-iterative nature made them far less suitable to various settings. In this paper we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative (Δ + 1)-coloring algorithm with running time O(Δ + log^* n), i.e., em below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing and bandwidth-restricted settings. This includes the following results. beginitemize ıtem We obtain self-stabilizing distributed algorithms for (Δ + 1)-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set and maximal matching with O(Δ + log^* n) time. This significantly improves previously-known results that have O(n) or larger running times citeGK10. ıtem We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O(Δ + log^* n) time and O(Δ)-edge-coloring in the Bit-Round model with O(Δ + log n) time. The factors of log^* n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously-known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. ıtem We obtain an arbdefective coloring algorithm with running time O(sqrt Δ + log^* n). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it in order to compute a proper (1 + ε)Δ-coloring within O(√ Δ + log^* n) time, and √(Δ + 1)√-coloring within √O(√ Δ log Δ log^* Δ + log^* n)√ time. This improves the recent state-of-the-art bounds of Barenboim from PODC'15 citeB15 and Fraigniaud et al. from FOCS'16 citeFHK16 by polylogarithmic factors. ıtem Our algorithms are applicable to the SET-LOCAL model citeHKMS15 (also known as the weak LOCAL model). In this model a relatively strong lower bound of √Ω(Δ^1/3 )√ is known for √(Δ + 1)√-coloring. However, most of the coloring algorithms do not work in this model. (In citeHKMS15 only Linial's √O(Δ^2)√-time algorithm and Kuhn-Wattenhofer √O(Δ log Δ)√-time algorithms are shown to work in it.) We obtain the first li","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133759586","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}
This paper gives drastically faster gossip algorithms to compute exact and approximate quantiles. Gossip algorithms, which allow each node to contact a uniformly random other node in each round, have been intensely studied and been adopted in many applications due to their fast convergence and their robustness to failures. Kempe et al. [24] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful O(logn + log 1 ε ) round algorithm to ε-approximate the sum of all values and an O(log2 n) round algorithm to compute the exact Φ-quantile, i.e., the ?Φn? smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact Φ-quantile problem which runs in O(logn) rounds. We furthermore show that one can achieve an exponential speedup if one allows for an ε-approximation. In particular, we give an O(log logn + log 1 ε ) round gossip algorithm which computes a value of rank between Φn and (Φ + ε)n at every node. Our algorithms are extremely simple and very robust - they can be operated with the same running times even if every transmission fails with a, potentially different, constant probability. We also give a matching Ω(log logn + log 1 ε ) lower bound which shows that our algorithm is optimal for all values of ε.
{"title":"Optimal Gossip Algorithms for Exact and Approximate Quantile Computations","authors":"Bernhard Haeupler, Jeet Mohapatra, Hsin-Hao Su","doi":"10.1145/3212734.3212770","DOIUrl":"https://doi.org/10.1145/3212734.3212770","url":null,"abstract":"This paper gives drastically faster gossip algorithms to compute exact and approximate quantiles. Gossip algorithms, which allow each node to contact a uniformly random other node in each round, have been intensely studied and been adopted in many applications due to their fast convergence and their robustness to failures. Kempe et al. [24] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful O(logn + log 1 ε ) round algorithm to ε-approximate the sum of all values and an O(log2 n) round algorithm to compute the exact Φ-quantile, i.e., the ?Φn? smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact Φ-quantile problem which runs in O(logn) rounds. We furthermore show that one can achieve an exponential speedup if one allows for an ε-approximation. In particular, we give an O(log logn + log 1 ε ) round gossip algorithm which computes a value of rank between Φn and (Φ + ε)n at every node. Our algorithms are extremely simple and very robust - they can be operated with the same running times even if every transmission fails with a, potentially different, constant probability. We also give a matching Ω(log logn + log 1 ε ) lower bound which shows that our algorithm is optimal for all values of ε.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121516241","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}
In wireless networks, consisting of battery-powered devices, energy is a costly resource and most of it is spent on transmitting messages. Broadcast is a problem where a message needs to be transmitted from one node to all other nodes of the network. We study algorithms that can work under limited energy measured as the maximum number of transmissions among all the stations. The goal of the paper is to study tradeoffs between time and energy complexity of broadcast problem in unknown multi-hop radio networks with no collision detection. We propose and analyse two new randomized energy-efficient algorithms. Our first algorithm works in time O((D+φ)n^1/φ . φ) with high probability and uses O(φ) energy per station for any φ ≤ log n/(2loglog n) for any graph with n nodes and diameter D. Our second algorithm works in time O((D+log n)log n) with high probability and uses O(log n/loglog n) energy. We prove that our algorithms are almost time-optimal for given energy limits for graphs with constant diameters by constructing lower bound on time of Ω(n^1/φ . φ). The lower bound shows also that any algorithm working in polylogaritmic time in n for all graphs needs energy Ω(log n/loglog n).
{"title":"Brief Announcement: Broadcast in Radio Networks, Time vs. Energy Tradeoffs","authors":"M. Klonowski, Dominik Pajak","doi":"10.1145/3212734.3212786","DOIUrl":"https://doi.org/10.1145/3212734.3212786","url":null,"abstract":"In wireless networks, consisting of battery-powered devices, energy is a costly resource and most of it is spent on transmitting messages. Broadcast is a problem where a message needs to be transmitted from one node to all other nodes of the network. We study algorithms that can work under limited energy measured as the maximum number of transmissions among all the stations. The goal of the paper is to study tradeoffs between time and energy complexity of broadcast problem in unknown multi-hop radio networks with no collision detection. We propose and analyse two new randomized energy-efficient algorithms. Our first algorithm works in time O((D+φ)n^1/φ . φ) with high probability and uses O(φ) energy per station for any φ ≤ log n/(2loglog n) for any graph with n nodes and diameter D. Our second algorithm works in time O((D+log n)log n) with high probability and uses O(log n/loglog n) energy. We prove that our algorithms are almost time-optimal for given energy limits for graphs with constant diameters by constructing lower bound on time of Ω(n^1/φ . φ). The lower bound shows also that any algorithm working in polylogaritmic time in n for all graphs needs energy Ω(log n/loglog n).","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133692838","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}
Determining the number of registers required for solving x-obstruction-free (or randomized wait-free) k-set agreement for x ≤ k is an open problem that highlights important gaps in our understanding of the space complexity of synchronization. In x-obstruction-free protocols, processes are required to return in executions where at most x processes take steps. The best known upper bound on the number of registers needed to solve this problem among n>k processes is n-k+x registers. No general lower bound better than 2 was known. We prove that any x-obstruction-free protocol solving k-set agreement among n > k processes must use n-x/k+1-x rfloor + 1 or more registers. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free k-set agreement. In particular, we show that, if a protocol uses fewer registers, then it is possible for k+1 processes to simulate the protocol and deterministically solve k-set agreement in a wait-free manner, which is impossible. An important aspect of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce an augmented snapshot object, which facilitates this. We also prove that any lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo termination. Hence, our lower bound of n-1/k + 1 for the obstruction-free case (i.e., x = 1) also holds for randomized wait-free protocols. In particular, we get a tight lower bound of exactly n registers for solving obstruction-free and randomized wait-free consensus.
{"title":"Revisionist Simulations: A New Approach to Proving Space Lower Bounds","authors":"Faith Ellen, Rati Gelashvili, Leqi Zhu","doi":"10.1145/3212734.3212749","DOIUrl":"https://doi.org/10.1145/3212734.3212749","url":null,"abstract":"Determining the number of registers required for solving x-obstruction-free (or randomized wait-free) k-set agreement for x ≤ k is an open problem that highlights important gaps in our understanding of the space complexity of synchronization. In x-obstruction-free protocols, processes are required to return in executions where at most x processes take steps. The best known upper bound on the number of registers needed to solve this problem among n>k processes is n-k+x registers. No general lower bound better than 2 was known. We prove that any x-obstruction-free protocol solving k-set agreement among n > k processes must use n-x/k+1-x rfloor + 1 or more registers. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free k-set agreement. In particular, we show that, if a protocol uses fewer registers, then it is possible for k+1 processes to simulate the protocol and deterministically solve k-set agreement in a wait-free manner, which is impossible. An important aspect of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce an augmented snapshot object, which facilitates this. We also prove that any lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo termination. Hence, our lower bound of n-1/k + 1 for the obstruction-free case (i.e., x = 1) also holds for randomized wait-free protocols. In particular, we get a tight lower bound of exactly n registers for solving obstruction-free and randomized wait-free consensus.","PeriodicalId":198284,"journal":{"name":"Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129383869","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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