190 research outputs found

    Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford

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    A \emph{metric tree embedding} of expected \emph{stretch~α1\alpha \geq 1} maps a weighted nn-node graph G=(V,E,ω)G = (V, E, \omega) to a weighted tree T=(VT,ET,ωT)T = (V_T, E_T, \omega_T) with VVTV \subseteq V_T such that, for all v,wVv,w \in V, dist(v,w,G)dist(v,w,T)\operatorname{dist}(v, w, G) \leq \operatorname{dist}(v, w, T) and operatornameE[dist(v,w,T)]αdist(v,w,G)operatorname{E}[\operatorname{dist}(v, w, T)] \leq \alpha \operatorname{dist}(v, w, G). Such embeddings are highly useful for designing fast approximation algorithms, as many hard problems are easy to solve on tree instances. However, to date the best parallel (polylogn)(\operatorname{polylog} n)-depth algorithm that achieves an asymptotically optimal expected stretch of αO(logn)\alpha \in \operatorname{O}(\log n) requires Ω(n2)\operatorname{\Omega}(n^2) work and a metric as input. In this paper, we show how to achieve the same guarantees using polylogn\operatorname{polylog} n depth and O~(m1+ϵ)\operatorname{\tilde{O}}(m^{1+\epsilon}) work, where m=Em = |E| and ϵ>0\epsilon > 0 is an arbitrarily small constant. Moreover, one may further reduce the work to O~(m+n1+ϵ)\operatorname{\tilde{O}}(m + n^{1+\epsilon}) at the expense of increasing the expected stretch to O(ϵ1logn)\operatorname{O}(\epsilon^{-1} \log n). Our main tool in deriving these parallel algorithms is an algebraic characterization of a generalization of the classic Moore-Bellman-Ford algorithm. We consider this framework, which subsumes a variety of previous "Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss it in depth. In our tree embedding algorithm, we leverage it for providing efficient query access to an approximate metric that allows sampling the tree using polylogn\operatorname{polylog} n depth and O~(m)\operatorname{\tilde{O}}(m) work. We illustrate the generality and versatility of our techniques by various examples and a number of additional results

    Three problems on well-partitioned chordal graphs

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    In this work, we solve three problems on well-partitioned chordal graphs. First, we show that every connected (resp., 2-connected) well-partitioned chordal graph has a vertex that intersects all longest paths (resp., longest cycles). It is an open problem [Balister et al., Comb. Probab. Comput. 2004] whether the same holds for chordal graphs. Similarly, we show that every connected well-partitioned chordal graph admits a (polynomial-time constructible) tree 3-spanner, while the complexity status of the Tree 3-Spanner problem remains open on chordal graphs [Brandstädt et al., Theor. Comput. Sci. 2004]. Finally, we show that the problem of finding a minimum-size geodetic set is polynomial-time solvable on well-partitioned chordal graphs. This is the first example of a problem that is NP -hard on chordal graphs and polynomial-time solvable on well-partitioned chordal graphs. Altogether, these results reinforce the significance of this recently defined graph class as a tool to tackle problems that are hard or unsolved on chordal graphs.acceptedVersio

    Graph Sketches: Sparsification, Spanners, and Subgraphs

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    When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., those based on linear projections of the data. These are applicable in many models including various parallel, stream, and compressed sensing settings. A rich body of analytic and empirical work exists for sketching numerical data such as the frequencies of a set of entities. Our work investigates graph sketching where the graphs of interest encode the relationships between these entities. The main challenge is to capture this richer structure and build the necessary synopses with only linear measurements. In this paper we consider properties of graphs including the size of the cuts, the distances between nodes, and the prevalence of dense sub-graphs. Our main result is a sketch-based sparsifier construction: we show that O̅(nε-2) random linear projections of a graph on n nodes suffice to (1 + ε) approximate all cut values. Similarly, we show that O(ε-2) linear projections suffice for (additively) approximating the fraction of induced sub-graphs that match a given pattern such as a small clique. Finally, for distance estimation we present sketch-based spanner constructions. In this last result the sketches are adaptive, i.e., the linear projections are performed in a small number of batches where each projection may be chosen dependent on the outcome of earlier sketches. All of the above results immediately give rise to data stream algorithms that also apply to dynamic graph streams where edges are both inserted and deleted. The non-adaptive sketches, such as those for sparsification and subgraphs, give us single-pass algorithms for distributed data streams with insertion and deletions. The adaptive sketches can be used to analyze MapReduce algorithms that use a small number of rounds

    08081 Abstracts Collection -- Data Structures

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    From February 17th to 22nd 2008, the Dagstuhl Seminar 08081 ``Data Structures\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. It brought together 49 researchers from four continents to discuss recent developments concerning data structures in terms of research but also in terms of new technologies that impact how data can be stored, updated, and retrieved. During the seminar a fair number of participants presented their current research. There was discussion of ongoing work, and in addition an open problem session was held. This paper first describes the seminar topics and goals in general, then gives the minutes of the open problem session, and concludes with abstracts of the presentations given during the seminar. Where appropriate and available, links to extended abstracts or full papers are provided

    Parameterized complexity of the spanning tree congestion problem

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    We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.publishedVersio
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