A linear-time algorithm for finding a central vertex of a chordal graph

Chepoi, Victor; Dragan, Feodor F.

doi:10.1007/bfb0049406

Cited by 33 publications

(49 citation statements)

References 12 publications

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“…It was also shown in [11,12] that for rectilinear link distance after two FP scans, the returned distance d(v, w) is a good approximation of the diameter of P and that a cut c passing via a middle edge of any shortest (v, w)-path is close to the center of P. Then using methods of computational geometry it is possible to compute in linear time a central point of P (another linear time algorithm for this problem has been proposed in [39]). As shown in [13], this approach can be appropriately modified to compute the centers of chordal graphs in linear time O(|E|). It was also shown that diam(G) ≥ 2rad(G) − 2 holds for such graphs.…”

Section: Related Work On Diameters and Centersmentioning

confidence: 99%

“…In computational geometry, the diameter and center problems have been investigated for point sets in two-or high-dimensional vector spaces endowed with usual metrics [17,22,38,45] and for polygonal or polyhedral domains with the geodesic [31,41] or link metrics [19,35,44]. For graphs and networks, efficient algorithms for these problems are known for several classes of graphs [6,13,15,18,21,30]. Most of these algorithms are based on geometric properties of classes of graphs in question.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

Chepoi

Dragan

Estellon

et al. 2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points u, v, w, x, the two larger of the sums d (u, v) + d(w, x), d(u, w) + d(v, x), d(u, x) + d(v, w) differ by at most 2δ. Given a finite set S of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of S with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for S with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs G = (V, E) with uniformly bounded degrees of vertices, the exact center of S can be computed in linear time O(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δ log 2 n). This construction has an additive error comparable with that given by Gromov for npoint δ-hyperbolic spaces, but can be implemented in O(|E|) time (instead of O(n 2 )). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.

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Section: Related Work On Diameters and Centersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

Chepoi

Dragan

Estellon

et al. 2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

Self Cite

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“…In directed graphs, depending on the application, there may be multiple definitions for "closeness": a node can be close in the sense that it has short paths to other nodes ("source"), from other nodes ("target"), or even to and then back from other nodes ("roundtrip"). That is, there are several natural definitions of both radius and diameter for directed graphs; all of them are well-studied [28,40,27,36,6,30,26,35,11,12,57,58,23,38,54,47,25,2,17] (and many others). Even estimating the diameter and radius of a network efficiently is useful in practical applications (e.g.…”

mentioning

confidence: 99%

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

Abboud¹,

Williams²,

Wang³

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

136

409

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The radius and diameter are fundamental graph parameters, with several natural definitions for directed graphs. Each definition is well-motivated in a variety of applications. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. However, solving APSP on n-node graphs requires Ω(n 2 ) time even in sparse graphs. We study the question: when can diameter and radius in sparse graphs be solved in truly subquadratic time, and when is such an algorithm unlikely? Motivated by our conditional lower bounds on computing these measures exactly in truly subquadratic time, we search for approximation and fixed parameter subquadratic algorithms, and alternatively, for reasons why they do not exist.We find that:• Most versions of Diameter and Radius can be solved in truly subquadratic time with optimal approximation guarantees, under plausible assumptions. For example, there is a 2-approximation algorithm for directed Radius with one-way distances that runs inÕ(m √ n) time, while a (2 − δ)-approximation algorithm in O(n 2−ε ) time is considered unlikely.• On graphs with treewidth k, we can solve all versions in 2 O(k log k) n 1+o(1) time. We show that these algorithms are near optimal since even a (3/2 − δ)-approximation algorithm that runs in time 2 o(k) n 2−ε would refute plausible assumptions.Two conceptual contributions of this work that we hope will incite future work are: the introduction of a Fixed Parameter Tractability in P framework, and the statement of a differently-quantified variant of the Orthogonal Vectors Conjecture, which we call the Hitting Set Conjecture. * The full version of the paper can be found at: http://arxiv.org/abs/1506. 01799. A.A and V

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“…Note that a ratio of 1/2 can easily be achieved by choosing an arbitrary vertex (the eccentricity of any vertex is at least one-half the diameter of the graph) and performing a BFS starting at this vertex. It follows also from the results in [1,7] (see also paper [16] which surveyed recent results related to the computation of exact and approximate distances in graphs) that the diameter problem in unweighted, undirected graphs can be solved in Õ (min{n 3 , n 7/3 }) time with an additive error of at most 2 without matrix multiplication. Here, Õ ( f ) means O( f polylog(n)).…”

Section: Introductionmentioning

confidence: 94%

On the power of BFS to determine a graph's diameter

2003

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Recently, considerable effort has been spent on showing that Lexicographic Breadth First Search (LBFS) can be used to determine a tight bound on the diameter of graphs from various restricted classes. In this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximum-sized cycle that may appear as an induced subgraph. We show that, on graphs that have no induced cycle of size greater than k, BFS finds an estimate of the diameter that is no worse than diam(G) ؊ k/2.

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A linear-time algorithm for finding a central vertex of a chordal graph

Cited by 33 publications

References 12 publications

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

On the power of BFS to determine a graph's diameter

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