66,427 research outputs found

    Metric Embedding via Shortest Path Decompositions

    Full text link
    We study the problem of embedding shortest-path metrics of weighted graphs into p\ell_p spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth 11. General graph has an SPD of depth kk if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most k1k-1. In this paper we give an O(kmin{1p,12})O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})-distortion embedding for graphs of SPD depth at most kk. This result is asymptotically tight for any fixed p>1p>1, while for p=1p=1 it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth kk embed into p\ell_p with distortion O(kmin{1p,12})O(k^{\min\{\frac{1}{p},\frac{1}{2}\}}). For p=1p=1, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in kk; moreover, for other values of pp it gives the first embeddings whose distortion is independent of the graph size nn. Furthermore, we use the fact that planar graphs have SPD depth O(logn)O(\log n) to give a new proof that any planar graph embeds into 1\ell_1 with distortion O(logn)O(\sqrt{\log n}). Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor

    The Effect of Planarization on Width

    Full text link
    We study the effects of planarization (the construction of a planar diagram DD from a non-planar graph GG by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of nn-vertex graphs with bounded parameter value, all of whose planarizations have parameter value Ω(n)\Omega(n). However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm

    Get PDF
    Given an undirected graph and two disjoint vertex pairs s1,t1s_1,t_1 and s2,t2s_2,t_2, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting s1s_1 with t1t_1, and s2s_2 with t2t_2, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions. Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs

    Optimal randomized incremental construction for guaranteed logarithmic planar point location

    Full text link
    Given a planar map of nn segments in which we wish to efficiently locate points, we present the first randomized incremental construction of the well-known trapezoidal-map search-structure that only requires expected O(nlogn)O(n \log n) preprocessing time while deterministically guaranteeing worst-case linear storage space and worst-case logarithmic query time. This settles a long standing open problem; the best previously known construction time of such a structure, which is based on a directed acyclic graph, so-called the history DAG, and with the above worst-case space and query-time guarantees, was expected O(nlog2n)O(n \log^2 n). The result is based on a deeper understanding of the structure of the history DAG, its depth in relation to the length of its longest search path, as well as its correspondence to the trapezoidal search tree. Our results immediately extend to planar maps induced by finite collections of pairwise interior disjoint well-behaved curves.Comment: The article significantly extends the theoretical aspects of the work presented in http://arxiv.org/abs/1205.543
    corecore