42 research outputs found

    A Distributed Algorithm for Directed Minimum-Weight Spanning Tree

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    On The Multiparty Communication Complexity of Testing Triangle-Freeness

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    In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the coordinator\textit{coordinator} model, where we have kk players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or far\textit{far} from triangle-free? For general communication protocols, we show that O~(k(nd)1/4+k2)\tilde{O}(k(nd)^{1/4}+k^2) bits are sufficient to test triangle-freeness in graphs of size nn with average degree dd (the degree need not be known in advance). For simultaneous\textit{simultaneous} protocols, where there is only one communication round, we give a protocol that uses O~(kn)\tilde{O}(k \sqrt{n}) bits when d=O(n)d = O(\sqrt{n}) and O~(k(nd)1/3)\tilde{O}(k (nd)^{1/3}) when d=Ω(n)d = \Omega(\sqrt{n}); here, again, the average degree dd does not need to be known in advance. We show that for average degree d=O(1)d = O(1), our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free.Comment: To Appear in PODC 201

    Truthful Information Dissemination in General Asynchronous Networks

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    We give a protocol for information dissemination in asynchronous networks of rational players, where each player may have its own desires and preferences as to the outcome of the protocol, and players may deviate from the protocol if doing so achieves their goals. We show that under minimalistic assumptions, it is possible to solve the information dissemination problem in a truthful manner, such that no participant has an incentive to deviate from the protocol we design. Our protocol works in any asynchronous network, provided the network graph is at least 2-connected. We complement the protocol with two impossibility results, showing that 2-connectivity is necessary, and also that our protocol achieves optimal bit complexity. As an application, we show that truthful information dissemination can be used to implement a certain class of communication equilibria, which are equilibria that are typically reached by interacting with a trusted third party. Recent work has shown that communication equilibria can be implemented in synchronous networks, or in asynchronous, complete networks; we show that in some useful cases, our protocol yields a lightweight mechanism for implementing communication equilibria in any 2-connected asynchronous network

    Lower Bounds for Subgraph Detection in the CONGEST Model

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    In the subgraph-freeness problem, we are given a constant-sized graph H, and wish to de- termine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs. The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle Ck of length k ? 4, and constructing a graph H ? from Ck by replacing each edge {i, (i + 1) mod k} of the cycle with a connected graph Hi, subject to some constraints on the graphs H_{0}, . . .H_{k?1}. In this case we obtain a polynomial lower bound for the new graph H ?, depending on the size of the shortest cycle in H ? passing through the vertices of the original k-cycle

    The Communication Complexity of Set Intersection Under Product Distributions

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    The SkipTrie: low-depth concurrent search without rebalancing

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    To date, all concurrent search structures that can support predecessor queries have had depth logarithmic in m, the number of elements. This paper introduces the SkipTrie, a new concurrent search structure supporting predecessor queries in amortized expected O(log log u + c) steps, insertions and deletions in O(c log log u), and using O(m) space, where u is the size of the key space and c is the contention during the recent past. The SkipTrie is a probabilistically-balanced version of a y-fast trie consisting of a very shallow skiplist from which randomly chosen elements are inserted into a hash-table based x-fast trie. By inserting keys into the x-fast-trie probabilistically, we eliminate the need for rebalancing, and can provide a lock-free linearizable implementation. To the best of our knowledge, our proof of the amortized expected performance of the SkipTrie is the first such proof for a tree-based data structure.National Science Foundation (U.S.) (Grant CCF-1217921)United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Grant ER26116/DE-SC0008923)Oracle CorporationIntel Corporatio
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