Embedding graphs on the torus is a problem with both theoretical and practical importance. It is required to embed a graph on the torus for solving many application problems such as VLSI design, graph drawing and so o...
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ISBN:
(纸本)9781467397971
Embedding graphs on the torus is a problem with both theoretical and practical importance. It is required to embed a graph on the torus for solving many application problems such as VLSI design, graph drawing and so on. Polynomial time and exponential time algorithms for embedding graphs on the torus are known. However, the polynomial time algorithms are very complex and their implementation has been a challenge for a long time. On the other hand, the implementations of some exponential time algorithms are known but they are not efficient for large graphs in practice. To develop an efficient practical tool for embedding graphs on the torus, we propose a new exponential time algorithm for embedding graphs on the torus. Compared with a well used previous exponential time algorithm, our algorithm has a better practical running time.
Given an undirected weighted connected graph G = (V, E) with vertex set V and edge set E and a designated vertex r is an element of V, we consider the problem of constructing a spanning tree in G that balances both th...
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ISBN:
(纸本)9788360810668
Given an undirected weighted connected graph G = (V, E) with vertex set V and edge set E and a designated vertex r is an element of V, we consider the problem of constructing a spanning tree in G that balances both the minimum spanning tree and the shortest paths tree rooted at r. Formally, for any two constants a, alpha, beta >= 1, we consider the problem of computing an (alpha, beta)-balanced spanning tree T in G, in the sense that, (i) for every vertex v is an element of V, the distance between r and v in T is at most alpha times the shortest distance between the two vertices in G, and (ii) the total weight of T is at most beta times that of the minimum tree weight in G. It is well known that, for any alpha, beta >= 1, the problem of deciding whether G contains an (alpha, beta)-balanced spanning tree is NP-complete [ 15]. Consequently, given any alpha >= 1 (resp., beta >= 1), the problem of finding an (alpha, beta)-balanced spanning tree that minimizes beta (resp., alpha) is NP-complete. In this paper, we present efficient genetic algorithms for these problems. Our experimental results show that the proposed algorithm returns high quality balanced spanning trees.
We present deterministic distributed algorithms for computing approximate maximum cardinality matchings and approximate maximum weight matchings. Our algorithm for the unweighted case computes a matching whose size is...
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ISBN:
(纸本)9781450329286
We present deterministic distributed algorithms for computing approximate maximum cardinality matchings and approximate maximum weight matchings. Our algorithm for the unweighted case computes a matching whose size is at least (1- epsilon) times the optimal in Delta(O(1/epsilon)) + O (1/epsilon(2))center dot log*(n) rounds where n is the number of vertices in the graph and Delta is the maximum degree. Our algorithm for the edge weighted case computes a matching whose weight is at least (1- epsilon) times the optimal in log(min{1/w(min),n/epsilon})(O(1/epsilon)) (Delta(O(1/epsilon))+log*(n)) rounds for edge-weights in [w(min), 1]. The best previous algorithms for both the unweighted case and the weighted case are by Lotker, Patt-Shamir, and Pettie (SPAA 2008). For the unweighted case they give a randomized (1- epsilon)-approximation algorithm that runs in O((log(n))/epsilon(3)) rounds. For the weighted case they give a randomized (1/2- epsilon)-approximation algorithm that runs in O(log(epsilon(-1)) center dot log(n)) rounds. Hence, our results improve on the previous ones when the parameters Delta, epsilon and w(min), are constants (where we reduce the number of runs from O(log(n)) to O(log*(n))), and more generally when Delta, 1/epsilon and 1/w(min) are sufficiently slowly increasing functions of n. Moreover, our algorithms are deterministic rather than randomized.
The spanning centrality of an edge e in an undirected graph G is the fraction of the spanning trees of G that contain e. Despite its appealing definition and apparent value in certain applications in computational bio...
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ISBN:
(纸本)9781450334693
The spanning centrality of an edge e in an undirected graph G is the fraction of the spanning trees of G that contain e. Despite its appealing definition and apparent value in certain applications in computational biology, spanning centrality hasn't so far received a wider attention as a measure of edge centrality. We may partially attribute this to the perceived complexity of computing it, which appears to be prohibitive for very large networks. Contrary to this intuition, spanning centrality can in fact be approximated arbitrary well by very efficient near-linear time algorithms due to Spielman and Srivastava, combined with progress in linear system solvers. In this article we bring theory into practice, with careful and optimized implementations that allow the fast computation of spanning centrality in very large graphs with millions of nodes. With this computational tool in our disposition, we demonstrate experimentally that spanning centrality is in fact a useful tool for the analysis of large networks. Specifically, we show that, relative to common centrality measures, spanning centrality is more effective in identifying edges whose removal causes a higher disruption in an information propagation procedure, while being very resilient to noise, in terms of both the edges scores and the resulting edge ranking.
Finding dense subgraphs in large graphs is a key primitive in a variety of real-world application domains, encompassing social network analytics, event detection, biology, and finance. In most such applications, one t...
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ISBN:
(纸本)9781450333177
Finding dense subgraphs in large graphs is a key primitive in a variety of real-world application domains, encompassing social network analytics, event detection, biology, and finance. In most such applications, one typically aims at finding several (possibly overlapping) dense subgraphs which might correspond to communities in social networks or interesting events. While a large amount of work is devoted to finding a single densest subgraph, perhaps surprisingly, the problem of finding several dense subgraphs with limited overlap has not been studied in a principled way, to the best of our knowledge. In this work we define and study a natural generalization of the densest subgraph problem, where the main goal is to find at most k subgraphs with maximum total aggregate density, while satisfying an upper bound on the pairwise Jaccard coefficient between the sets of nodes of the subgraphs. After showing that such a problem is NP-Hard, we devise an efficient algorithm that comes with provable guarantees in some cases of interest, as well as, an efficient practical heuristic. Our extensive evaluation on large real-world graphs confirms the efficiency and effectiveness of our algorithms.
One of the most obvious features of ad hoc communication is the analyze of the relationships between the ad hoc network users and their need for communication. On that point, the determination of topologies for effici...
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ISBN:
(纸本)9783319188027;9783319188010
One of the most obvious features of ad hoc communication is the analyze of the relationships between the ad hoc network users and their need for communication. On that point, the determination of topologies for efficient broadcast based on property of users of ad hoc networks has attracted a growing interest. In the current work, we propose a method to build a virtual topology that exploits the property of community structure in ad hoc network. The first phase of the proposed method constructs a clustering tree based on structural weight of nodes, while maintaining capacity-efficient links. In the second phase, the algorithm determines a community backbone in order to ensure efficient transmission coverage. Results confirm the generation of a good topology.
We develop an efficient parallel algorithm for answering shortest-path queries in planar graphs and implement it on a multi-node CPU-GPU clusters. The algorithm uses a divide-and-conquer approach for decomposing the i...
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ISBN:
(纸本)9783319265209;9783319265193
We develop an efficient parallel algorithm for answering shortest-path queries in planar graphs and implement it on a multi-node CPU-GPU clusters. The algorithm uses a divide-and-conquer approach for decomposing the input graph into small and roughly equal subgraphs and constructs a distributed data structure containing shortest distances within each of those subgraphs and between their boundary vertices. For a planar graph with n vertices, that data structure needs O(n) storage per processor and allows queries to be answered in O(n(1/4)) time.
Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximable due to conne...
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ISBN:
(纸本)9781450334693
Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem, the poly-time solvable densest subgraph problem which maximizes the average degree over all possible subgraphs "lies at the core of large scale data mining"[10]. However, frequently the densest subgraph problem fails in detecting large near-cliques in networks. In this work, we introduce the k-clique densest subgraph problem, k >= 2. This generalizes the well studied densest subgraph problem which is obtained as a special case for k = 2. For k = 3 we obtain a novel formulation which we refer to as the triangle densest subgraph problem : given a graph G (V, E), find a subset of vertices S* such that tau (S*) = (max)(S subset of V) t (S)/vertical bar S vertical bar, where t (S) is the number of triangles induced by the set S. On the theory side, we prove that for any k constant, there exist an exact polynomial time algorithm for the k-clique densest subgraph problem. Furthermore, we propose an efficient 1/k-approximation algorithm which generalizes the greedy peeling algorithm of Asahiro and Charikar [8, 18] for k = 2. Finally, we show how to implement efficiently this peeling framework on MapReduce for any k >= 3, generalizing the work of Bahmani, Kumar and Vassilvitskii for the case k = 2 [10]. On the empirical side, our two main findings are that (i) the triangle densest subgraph is consistently closer to being a large near-clique compared to the densest subgraph and (ii) the peeling approximation algorithms for both k = 2 and k = 3 achieve on real-world networks approximation ratios closer to 1 rather than the pessimistic 1 k guarantee. An interesting consequence of our work is that triangle counting, a well-studied computational problem in the context of social network analysis can be used to detect large near-
Many graph algorithms are based on depth-first search (DFS). The formalizations of such algorithms typically share many common ideas. In this paper, we summarize these ideas into a framework in Isabelle/HOL. Building ...
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ISBN:
(纸本)9781450332965
Many graph algorithms are based on depth-first search (DFS). The formalizations of such algorithms typically share many common ideas. In this paper, we summarize these ideas into a framework in Isabelle/HOL. Building on the Isabelle Refinement Framework, we provide support for a refinement based development of DFS based algorithms, from phrasing and proving correct the abstract algorithm, over choosing an adequate implementation style (e. g., recursive, tail-recursive), to creating an executable algorithm that uses efficient data structures. As a case study, we verify DFS based algorithms of different complexity, from a simple cyclicity checker, over a safety property model checker, to complex algorithms like nested DFS and Tarjan's SCC algorithm.
We present a new distributed model for graph computations motivated by limited information sharing. Two or more independent entities have collected large social graphs. They wish to compute the result of running graph...
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ISBN:
(纸本)9781479986484
We present a new distributed model for graph computations motivated by limited information sharing. Two or more independent entities have collected large social graphs. They wish to compute the result of running graph algorithms on the entire set of relationships. Because the information is sensitive or economically valuable, they do not wish to simply combine the information in a single location. We consider two models for computing the solution to graph algorithms in this setting: 1) limited-sharing: the two entities can share only a polylogarithmic size subgraph;2) low-trust: the entities must not reveal any information beyond the query answer, assuming they are all honest but curious. We believe this model captures realistic constraints on cooperating autonomous data centers. We have results for both models for s-t connectivity, one of the simplest graph problems that requires global information in the worst case. In the limited-sharing model, our results exploit social network structure. Standard communication complexity gives polynomial lower bounds on s-t connectivity for general graphs. However, if the graph for each data center has a giant component and these giant components intersect, then we can overcome this lower bound, computing s-t connectivity while exchanging O(log(2) n) bits for a constant number of data centers. We can also test the assumption that the giant components overlap using O(log(2) n) bits provided the (unknown) overlap is sufficiently large. The second result is in the low trust model. We give a secure multi-party computation (MPC) algorithm that 1) does not make cryptographic assumptions when there are 3 or more entities;and 2) is efficient, especially when compared to the usual garbled circuit approach. The entities learn only the yes/no answer. No party learns anything about the others' graph, not even node names. This algorithm does not require any special graph structure. This secure MPC result for s-t connectivity is one of the f
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