On computing the diameter of real-world undirected graphs

Crescenzi, Pierluigi; Grossi, Roberto; Habib, Michel; Lanzi, Leonardo; Marino, Andrea

doi:10.1016/j.tcs.2012.09.018

Cited by 50 publications

(45 citation statements)

References 9 publications

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“…This algorithm will not only be more general than all previous counterparts, but it will also outperform them on directed, strongly connected graphs or undirected graphs. It relates the sweep approach (i.e., a new visit of the graph depends on the previous one, as in [14,13,22,23]) with the techniques developed in [31,32]. It is based on a new heuristic, named SumSweep, which is able to compute very efficiently lower bounds on the diameter and upper bounds on the radius of a given graph.…”

Section: Our Resultsmentioning

confidence: 99%

“…For this reason, several papers dealt with the problem of appropriately choosing the vertices from which the BFSs have to be performed. For example, the so-called 2Sweep heuristic picks one of the farthest vertices x from a random vertex r and returns the distance of the farthest vertex from x [22], while the 4Sweep picks the vertex in the middle of the longest path computed by a 2Sweep execution and performs another 2Sweep from that vertex [13]. Both methods work quite well and very often provide tight bounds.…”

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confidence: 99%

“…Even if in the worst case their time complexity is O(mn), they turn out to be linear in practice. The main algorithms developed until now are BoundingDiameters [31] and iFub [13]. While the first works only on undirected graphs, the second is also able to deal with the directed strongly connected case (the adaptation is called diFub [14]).…”

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confidence: 99%

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Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs

Borassi

Crescenzi

Habib

et al. 2015

Theoretical Computer Science

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International audienceIn this paper, we propose a new algorithm that computes the radius and the diameter of a weakly connected digraph G=(V,E), by finding bounds through heuristics and improving them until they are validated. Although the worst-case running time is O(|V||E|), we will experimentally show that it performs much better in the case of real-world networks, finding the radius and diameter values after 10–100 BFSs instead of |V| BFSs (independently of the value of |V|), and thus having running time O(|E|) in practice. As far as we know, this is the first algorithm able to compute the diameter of weakly connected digraphs, apart from the naive algorithm, which runs in time Ω(|V||E|) performing a BFS from each node. In the particular cases of strongly connected directed or connected undirected graphs, we will compare our algorithm with known approaches by performing experiments on a dataset composed by several real-world networks of different kinds. These experiments will show that, despite its generality, the new algorithm outperforms all previous methods, both in the radius and in the diameter computation, both in the directed and in the undirected case, both in average running time and in robustness. Finally, as an application example, we will use the new algorithm to determine the solvability over time of the “Six Degrees of Kevin Bacon” game, and of the “Six Degrees of Wikipedia” game. As a consequence, we will compute for the first time the exact value of the radius and the diameter of the whole Wikipedia digraph

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Section: Our Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs

Borassi

Crescenzi

Habib

et al. 2015

Theoretical Computer Science

Self Cite

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“…We used the NetworkX Python package [19] to generate our random intersection graphs (using the uniform random intersection graph method), and the SageMath software system [14] to compute the hyperbolicity [6,9,16], degeneracy [1,38] and diameter [10,11,27,45] of the generated graphs. The measurements of the p-centered coloring number (presented below) were executed using the implementation available in [37].…”

Section: Hyperbolicitymentioning

confidence: 99%

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

Farrell

Goodrich

Lemons

et al. 2015

Lecture Notes in Computer Science

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Abstract. We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we fully characterize the degeneracy of random intersection graphs, and prove that the model asymptotically almost surely produces graphs with hyperbolicity at least log n. Further, we prove that in the parametric regime where random intersection graphs are degenerate an even stronger notion of sparseness, so called bounded expansion, holds with high probability. We supplement our theoretical findings with experimental evaluations of the relevant statistics.

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“…The only exception to this we are aware of is an o(n 2.69 )-time algorithm in [34], that improves the running time for dense graphs. On the practical side, such running times are too prohibitive on large graphs with thousands of vertices and sometimes billions of edges (see for instance [18]). Therefore, it is interesting to obtain a finer-grained complexity analysis for this problem.…”

Section: Introductionmentioning

confidence: 99%

Revisiting Decomposition by Clique Separators

Coudert¹,

Ducoffe²

2018

SIAM J. Discrete Math.

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We study the complexity of decomposing a graph by means of clique separators. This common algorithmic tool, first introduced by Tarjan, allows to cut a graph into smaller pieces, and so, it can be applied to preprocess the graph in the computation of optimization problems. However, the best-known algorithms for computing a decomposition have respective O(nm)-time and O(n (3+α)/2 ) = o(n 2.69 )-time complexity, with α < 2.3729 being the exponent for matrix multiplication. Such running times are prohibitive for large graphs. Here we prove that for every graph G, a decomposition can be computed in O(T (G) + min{n α , ω 2 n})-time with T (G) and ω being respectively the time needed to compute a minimal triangulation of G and the clique-number of G. In particular, it implies that every graph can be decomposed by clique separators in O(n α log n)-time. Based on prior work from Kratsch et al., we prove in addition that decomposing a graph by clique-separators is as least as hard as triangle detection. Therefore, the existence of any o(n α )-time algorithm for this problem would be a significant breakthrough in the field of algorithmic. Finally, our main result implies that planar graphs, bounded-treewidth graphs and bounded-degree graphs can be decomposed by clique separators in linear or quasi-linear time.

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On computing the diameter of real-world undirected graphs

Cited by 50 publications

References 9 publications

Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs

Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

Revisiting Decomposition by Clique Separators

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