Sparsification—a technique for speeding up dynamic graph algorithms

Eppstein, David; Galil, Zvi; Italiano, Giuseppe F.; Nissenzweig, Amnon

doi:10.1145/265910.265914

Cited by 247 publications

(219 citation statements)

References 25 publications

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“…Hence, we would like to possess a criterion to check if a rewiring will lead to disconnectivity. In the field of dynamic graph algorithms, Eppstein et al [23] have proposed a technique to check for connectivity in O( N ) time for each link update (four updates per rewiring), which naturally beats any standard way of checking for graph connectivity.…”

Section: Where L H ⊆ L and Each Node I Has Precisely D I Neighbors (mentioning

confidence: 99%

Influence of assortativity and degree-preserving rewiring on the spectra of networks

et al. 2010

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Abstract. Newman's measure for (dis)assortativity, the linear degree correlation coefficient ρD, is reformulated in terms of the total number N k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases ρD conform our desired objective. Spectral metrics (eigenvalues of graph-related matrices), especially, the largest eigenvalue λ1 of the adjacency matrix and the algebraic connectivity μN−1 (second-smallest eigenvalue of the Laplacian) are powerful characterizers of dynamic processes on networks such as virus spreading and synchronization processes. We present various lower bounds for the largest eigenvalue λ1 of the adjacency matrix and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that the lower bounds for λ1 increase with ρD. A new upper bound for the algebraic connectivity μN−1 decreases with ρD. Applying the degree-preserving rewiring algorithm to various real-world networks illustrates that (a) assortative degree-preserving rewiring increases λ1, but decreases μN−1, even leading to disconnectivity of the networks in many disjoint clusters and that (b) disassortative degree-preserving rewiring decreases λ1, but increases the algebraic connectivity, at least in the initial rewirings.

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Section: Where L H ⊆ L and Each Node I Has Precisely D I Neighbors (mentioning

confidence: 99%

Influence of assortativity and degree-preserving rewiring on the spectra of networks

et al. 2010

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“…In [14] Kapoor and Ramesh proposed an algorithm for ranking all the spanning trees which requires O(nm +τ log n) time and O(nm + τ ) space. Recently, Eppstein et al [8] developed the sparsification technique and proposed an algorithm for generating k smallest spanning trees in compact form which requires O(n + m log n + k √ n log(m/n)) time.…”

Section: Preliminarymentioning

confidence: 99%

A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

Matsui¹

1997

Algorithmica

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In this paper we propose an algorithm for generating all the spanning trees in undirected graphs. The algorithm requires O(n + m + τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, this time complexity is optimal.Our algorithm follows a special rooted tree structure on the skeleton graph of the spanning tree polytope. The rule by which the rooted tree structure is traversed is irrelevant to the time complexity. In this sense, our algorithm is flexible.If we employ the depth-first search rule, we can save the memory requirement to O(n + m). A breadth-first implementation requires as much as O(m + τ n) space, but when a parallel computer is available, this might have an advantage. When a given graph is weighted, the best-first search rule provides a ranking algorithm for the minimum spanning tree problem. The ranking algorithm requires O(n + m + τ n) time and O(m + τ n) space when we have a minimum spanning tree. Introduction.In this paper we propose an algorithm which generates all the spanning trees explicitly in O(n + m + τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, the time complexity is optimal.Our algorithm generates all the spanning trees by following one-dimensional faces of the spanning tree polytope (the convex hull of the characteristic vectors of spanning trees). More precisely, our algorithm traverses a tree structure (a rooted spanning tree) on the skeleton graph of the spanning tree polytope. From the above, our algorithm can be considered as a specialization of the reverse search method proposed by Avis and Fukuda [4], which is a general scheme for solving enumeration problems.Our algorithm has an advantage in that the tree search rule by which we traverse the tree structure on the skeleton graph is irrelevant to the time complexity. Thus, various algorithms with O(n + m + τ n) time complexity can be easily constructed by using different tree search rules. In this sense, our algorithm is flexible. For example, employing the depth-first search rule we obtain an algorithm whose space complexity is O(n + m). A breadth-first implementation requires as much as O(m +τ n) space, but when a parallel computer is available, this might have an advantage. The two tree search rules can even be combined to construct an algorithm which runs parallel but requires a fixed amount of memory storage. When a weighted graph is given, our algorithm generates all the spanning trees in nondecreasing order of weight by employing best-first search rule. The

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“…For a variety of graph properties, we establish the existence of n-vertex graphs for which every strong certificate must have (n 2 ) edges, thereby ruling out the application of sparsification to design efficient deterministic dynamic algorithms for these problems. Existence of properties requiring quadratic size strong certificates was independently pointed out by Eppstein et al [10] where they indicate that matching requires (n 2 ) size certificates. We also show that similar bounds hold even when a more relaxed notion of certificates is used.…”

Section: Introductionmentioning

confidence: 95%

“…While much of the prior research in the design of dynamic graph algorithms involved problem-specific approaches [11]- [13], [22], [27], Eppstein et al [9] devised a general paradigm called sparsification that is useful in both designing new dynamic graph algorithms, as well as speeding up the existing ones. The sparsification technique is based on the notion of a sparse strong certificate which is a sparse graph whose structure is representative of the input graph with respect to the property of interest.…”

Section: Introductionmentioning

confidence: 99%

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On Certificates and Lookahead in Dynamic Graph Problems

1998

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Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closure, as well as some undirected graph problems such as maximum matchings and cuts. We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate complexity for the problems which have efficient dynamic algorithms.In many applications of dynamic (di)graph problems, a certain form of lookahead is available. Specifically, we consider the problems of assembly planning in robotics and the maintenance of relations in databases. These give rise to dynamic strong connectivity and dynamic transitive closure problems, respectively. We explain why it is reasonable, and indeed natural and desirable, to assume that lookahead is available in these two applications. Exploiting lookahead to circumvent their inherent complexity, we obtain efficient dynamic algorithms for strong connectivity and transitive closure.

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Sparsification—a technique for speeding up dynamic graph algorithms

Cited by 247 publications

References 25 publications

Influence of assortativity and degree-preserving rewiring on the spectra of networks

Influence of assortativity and degree-preserving rewiring on the spectra of networks

A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

On Certificates and Lookahead in Dynamic Graph Problems

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