11 research outputs found

    Approximating the Diameter of Planar Graphs in Near Linear Time

    Get PDF
    We present a (1+ϵ)(1+\epsilon)-approximation algorithm running in O(f(ϵ)nlog4n)O(f(\epsilon)\cdot n \log^4 n) time for finding the diameter of an undirected planar graph with non-negative edge lengths

    On The Center Sets and Center Numbers of Some Graph Classes

    Full text link
    For a set SS of vertices and the vertex vv in a connected graph GG, maxxSd(x,v)\displaystyle\max_{x \in S}d(x,v) is called the SS-eccentricity of vv in GG. The set of vertices with minimum SS-eccentricity is called the SS-center of GG. Any set AA of vertices of GG such that AA is an SS-center for some set SS of vertices of GG is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, Km,nK_{m,n}, KneK_n-e, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center number which is defined as the number of distinct center sets of a graph and determine the center number of some graph classes

    Fast approximation of centrality and distances in hyperbolic graphs

    Full text link
    We show that the eccentricities (and thus the centrality indices) of all vertices of a δ\delta-hyperbolic graph G=(V,E)G=(V,E) can be computed in linear time with an additive one-sided error of at most cδc\delta, i.e., after a linear time preprocessing, for every vertex vv of GG one can compute in O(1)O(1) time an estimate e^(v)\hat{e}(v) of its eccentricity eccG(v)ecc_G(v) such that eccG(v)e^(v)eccG(v)+cδecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta for a small constant cc. We prove that every δ\delta-hyperbolic graph GG has a shortest path tree, constructible in linear time, such that for every vertex vv of GG, eccG(v)eccT(v)eccG(v)+cδecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of GG, the smaller its eccentricity is. We also show that the distance matrix of GG with an additive one-sided error of at most cδc'\delta can be computed in O(V2log2V)O(|V|^2\log^2|V|) time, where c<cc'< c is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

    Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates

    Get PDF
    We introduce notions of certificates allowing to bound eccentricities in a graph. In particular , we revisit radius (minimum eccentricity) and diameter (maximum eccentricity) computation and explain the efficiency of practical radius and diameter algorithms by the existence of small certificates for radius and diameter plus few additional properties. We show how such computation is related to covering a graph with certain balls or complementary of balls. We introduce several new algorithmic techniques related to eccentricity computation and propose algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various real-world graphs showing that these parameters appear to be low in practice. We also obtain refined results in the case where the input graph has low doubling dimension, has low hyperbolicity, or is chordal

    Beyond Helly graphs: the diameter problem on absolute retracts

    Full text link
    Characterizing the graph classes such that, on nn-vertex mm-edge graphs in the class, we can compute the diameter faster than in O(nm){\cal O}(nm) time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph HH of a graph GG is called a retract of GG if it is the image of some idempotent endomorphism of GG. Two necessary conditions for HH being a retract of GG is to have HH is an isometric and isochromatic subgraph of GG. We say that HH is an absolute retract of some graph class C{\cal C} if it is a retract of any GCG \in {\cal C} of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized O~(mn)\tilde{\cal O}(m\sqrt{n}) time. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of kk-chromatic graphs, for every fixed k3k \geq 3. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively

    Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates

    Get PDF
    We introduce notions of certificates allowing to bound eccentricities in a graph. In particular , we revisit radius (minimum eccentricity) and diameter (maximum eccentricity) computation and explain the efficiency of practical radius and diameter algorithms by the existence of small certificates for radius and diameter plus few additional properties. We show how such computation is related to covering a graph with certain balls or complementary of balls. We introduce several new algorithmic techniques related to eccentricity computation and propose algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various real-world graphs showing that these parameters appear to be low in practice. We also obtain refined results in the case where the input graph has low doubling dimension, has low hyperbolicity, or is chordal
    corecore