graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent...
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graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in Ologn flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O1 and Ologn flips nearly matching out-degree bounds to their respective amortized solutions.
This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamicgraph. Specifically, our algorithms maintain the number of paths 1 of length three...
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graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent...
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ISBN:
(纸本)9783959770545
graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in Ologn flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O1 and Ologn flips nearly matching out-degree bounds to their respective amortized solutions.
Consider a distributed task where the communication network is fixed but the local inputs given to the nodes of the distributed system may change over time. In this work, we explore the following question: if some of ...
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Consider a distributed task where the communication network is fixed but the local inputs given to the nodes of the distributed system may change over time. In this work, we explore the following question: if some of the local inputs change, can an existing solution be updated efficiently, in a dynamic and distributed manner? To address this question, we define the batch dynamic CONGEST model in which we are given a bandwidth-limited communication network and a dynamic edge labelling defines the problem input. The task is to maintain a solution to a graph problem on the labeled graph under batch changes. We investigate, when a batch of alpha edge label changes arrive, - how much time as a function of.. we need to update an existing solution, and - how much information the nodes have to keep in local memory between batches in order to update the solution quickly. Our work lays the foundations for the theory of input-dynamic distributed network algorithms. We give a general picture of the complexity landscape in this model, design both universal algorithms and algorithms for concrete problems, and present a general framework for lower bounds. In particular, we derive non-trivial upper bounds for two selected, contrasting problems: maintaining a minimum spanning tree and detecting cliques.
The problem of (Delta+1)-vertex coloring a graph of maximum degree Delta has been extremely well studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known u...
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The problem of (Delta+1)-vertex coloring a graph of maximum degree Delta has been extremely well studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA'18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized algorithm for maintaining a (Delta + 1)-coloring with O(log Delta) expected amortized update time. In this article, we present an improved randomized algorithm for (Delta + 1)-coloring that achieves O(1) amortized update time and show that this bound holds not only in expectation but also with high probability. Our starting point is the state-of-the-art randomized algorithm for maintaining a maximal matching (Solomon, FOCS'16). We carefully build on the approach of Solomon, but, due to inherent differences between the maximal matching and (Delta + 1)-coloring problems, we need to deviate significantly from it in several crucial and highly nontrivial points.
Finding the centrality measures of nodes in a graph is a problem of fundamental importance due to various applications from social networks, biological networks, and transportation networks. Given the large size of su...
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Finding the centrality measures of nodes in a graph is a problem of fundamental importance due to various applications from social networks, biological networks, and transportation networks. Given the large size of such graphs, it is natural to use parallelism as a recourse. Several studies show how to compute the various centrality measures of nodes in a graph on parallel architectures, including multi-core systems and GPUs. However, as these graphs evolve and change, it is pertinent to study how to update the centrality measures on changes to the underlying graph. In this article, we show novel parallel algorithms for updating the betweenness- and closeness-centrality values of nodes in a dynamicgraph. Our algorithms process a batch of updates in parallel by extending the approach of handling a single update for betweenness- and closeness-centrality. For the latter, we also introduce techniques based on traversals of the block-cut tree of a graph. Besides, our algorithms incorporate mechanisms to exploit the structural properties of graphs for enhanced performance. We implement our algorithms on two parallel architectures: an Intel 24-core CPU and an Nvidia Tesla V100 GPU. To the best of our knowledge, we are the first to show GPU algorithms for the above two problems. In addition, we conduct detailed experiments to study the impact of various parameters associated with our algorithms and their implementation. Our results on a collection of real-world graphs indicate that our algorithms achieve a significant speedup over corresponding state-of-the-art algorithms.
We consider the dynamicgraph coloring problem restricted to the class of interval graphs in the incremental and fully dynamic setting. The input consists of a sequence of intervals that are to be either colored, or d...
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We consider the dynamicgraph coloring problem restricted to the class of interval graphs in the incremental and fully dynamic setting. The input consists of a sequence of intervals that are to be either colored, or deleted, if previously colored. For the incremental setting, we consider the well studied optimal online algorithm (KT-algorithm) for interval coloring due to Kierstead and Trotter [ I]. We present the following results on the dynamic interval coloring problem. Any direct implementation of the KT-algorithm requires Omega(Delta(2)) time per interval in the worst case. There exists an incremental algorithm which supports insertion of an interval in amortized O (logn + Delta) time per update and maintains a proper coloring using at most 3 omega - 2 colors. There exists a fully dynamic algorithm which supports insertion of an interval in O (logn + Delta log omega) update time and deletion of an interval in O (Delta(2) logn) update time in the worst case and maintains a proper coloring using at most 3 omega - 2 colors. The KT-algorithm crucially uses the maximum clique size in an induced subgraph in the neighborhood of a given vertex. We show that the problem of computing the induced subgraph among the neighbors of a given vertex has the same hardness as the online boolean matrix vector multiplication problem [2]. We show that Any algorithm that computes the induced subgraph among the neighbors of a given vertex requires at least quadratic time unless the OMy conjecture [2] is false. Finally, we obtain the following result on the OMy conjecture. If the matrix and the vectors in the online sequence have the consecutive ones property, then the OMy conjecture [2] is false. (C) 2020 Elsevier B.V. All rights reserved.
We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Delta+1)-vertex coloring of a graph with maximum degree at most Delta. This improves upon t...
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We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Delta+1)-vertex coloring of a graph with maximum degree at most Delta. This improves upon the previous O(log Delta)-time algorithm by Bhattacharya et al. (SODA 2018). Our algorithm uses an approach based on assigning random ranks to vertices and does not need to maintain a hierarchical graph decomposition. We show that our result does not only have optimal running time but is also optimal in the sense that already deciding whether a Delta-coloring exists in a dynamically changing graph with maximum degree at most Delta takes Omega(logn) time per operation.
In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The best sequential algorithm for dynam...
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ISBN:
(纸本)9781450361842
In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The best sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves O(1g(2) n) amortized time per edge insertion or deletion, and O(1g n) time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where A is the average batch size of all deletions, our algorithm achieves O(1g n lg(1 + n/A)) expected amortized work per edge insertion and deletion and O (1g(3) n) depth w.h.p. Our algorithm answers a batch of k connectivity queries in O(k lg(1 + n/ k)) expected work and O(1g n) depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.
We give two fully dynamicalgorithms that maintain a (1 + epsilon)-approximation of the weight M of a minimum spanning forest (MSF) of an n-node graph G with edges weights in [1, W], for any epsilon > 0. (1) Our de...
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We give two fully dynamicalgorithms that maintain a (1 + epsilon)-approximation of the weight M of a minimum spanning forest (MSF) of an n-node graph G with edges weights in [1, W], for any epsilon > 0. (1) Our deterministic algorithm takes O(W2 log W /epsilon(3)) worst-case update time, which is O (1) if both W and E are constants. (2) Our randomized (Monte -Carlo style) algorithm works with high probability and runs in worst-case O (log W /epsilon(4)) update time if W = O((m*)(1/6)/log(2/3) n), where m* is the minimum number of edges in the graph throughout all the updates. It works even against an adaptive adversary. We complement our algorithmic results with two cell-probe lower bounds for dynamically maintaining an approximation of the weight of an MSF of a graph. (C) 2021 Elsevier Inc. All rights reserved.
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