16,341 research outputs found

    Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

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    A greedily routable region (GRR) is a closed subset of R2\mathbb R^2, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201

    On traces of tensor representations of diagrams

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    Let TT be a set, of {\em types}, and let \iota,o:T\to\oZ_+. A {\em TT-diagram} is a locally ordered directed graph GG equipped with a function τ:V(G)→T\tau:V(G)\to T such that each vertex vv of GG has indegree ι(τ(v))\iota(\tau(v)) and outdegree o(τ(v))o(\tau(v)). (A directed graph is {\em locally ordered} if at each vertex vv, linear orders of the edges entering vv and of the edges leaving vv are specified.) Let VV be a finite-dimensional \oF-linear space, where \oF is an algebraically closed field of characteristic 0. A function RR on TT assigning to each t∈Tt\in T a tensor R(t)∈V∗⊗ι(t)⊗V⊗o(t)R(t)\in V^{*\otimes \iota(t)}\otimes V^{\otimes o(t)} is called a {\em tensor representation} of TT. The {\em trace} (or {\em partition function}) of RR is the \oF-valued function pRp_R on the collection of TT-diagrams obtained by `decorating' each vertex vv of a TT-diagram GG with the tensor R(τ(v))R(\tau(v)), and contracting tensors along each edge of GG, while respecting the order of the edges entering vv and leaving vv. In this way we obtain a {\em tensor network}. We characterize which functions on TT-diagrams are traces, and show that each trace comes from a unique `strongly nondegenerate' tensor representation. The theorem applies to virtual knot diagrams, chord diagrams, and group representations

    On the horseshoe conjecture for maximal distance minimizers

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    We study the properties of sets Σ\Sigma having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ⊂R2\Sigma \subset \mathbb{R}^2 satisfying the inequality \mbox{max}_{y \in M} \mbox{dist}(y,\Sigma) \leq r for a given compact set M⊂R2M \subset \mathbb{R}^2 and some given r>0r > 0. Such sets can be considered shortest possible pipelines arriving at a distance at most rr to every point of MM which in this case is considered as the set of customers of the pipeline. We prove the conjecture of Miranda, Paolini and Stepanov about the set of minimizers for MM a circumference of radius R>0R>0 for the case when r<R/4.98r < R/4.98. Moreover we show that when MM is a boundary of a smooth convex set with minimal radius of curvature RR, then every minimizer Σ\Sigma has similar structure for r<R/5r < R/5. Additionaly we prove a similar statement for local minimizers.Comment: 25 pages, 21 figure

    Optimally Efficient Prefix Search and Multicast in Structured P2P Networks

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    Searching in P2P networks is fundamental to all overlay networks. P2P networks based on Distributed Hash Tables (DHT) are optimized for single key lookups, whereas unstructured networks offer more complex queries at the cost of increased traffic and uncertain success rates. Our Distributed Tree Construction (DTC) approach enables structured P2P networks to perform prefix search, range queries, and multicast in an optimal way. It achieves this by creating a spanning tree over the peers in the search area, using only information available locally on each peer. Because DTC creates a spanning tree, it can query all the peers in the search area with a minimal number of messages. Furthermore, we show that the tree depth has the same upper bound as a regular DHT lookup which in turn guarantees fast and responsive runtime behavior. By placing objects with a region quadtree, we can perform a prefix search or a range query in a freely selectable area of the DHT. Our DTC algorithm is DHT-agnostic and works with most existing DHTs. We evaluate the performance of DTC over several DHTs by comparing the performance to existing application-level multicast solutions, we show that DTC sends 30-250% fewer messages than common solutions

    Characterizations of Decomposable Dependency Models

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    Decomposable dependency models possess a number of interesting and useful properties. This paper presents new characterizations of decomposable models in terms of independence relationships, which are obtained by adding a single axiom to the well-known set characterizing dependency models that are isomorphic to undirected graphs. We also briefly discuss a potential application of our results to the problem of learning graphical models from data.Comment: See http://www.jair.org/ for any accompanying file

    Circular Networks from Distorted Metrics

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    Trees have long been used as a graphical representation of species relationships. However complex evolutionary events, such as genetic reassortments or hybrid speciations which occur commonly in viruses, bacteria and plants, do not fit into this elementary framework. Alternatively, various network representations have been developed. Circular networks are a natural generalization of leaf-labeled trees interpreted as split systems, that is, collections of bipartitions over leaf labels corresponding to current species. Although such networks do not explicitly model specific evolutionary events of interest, their straightforward visualization and fast reconstruction have made them a popular exploratory tool to detect network-like evolution in genetic datasets. Standard reconstruction methods for circular networks, such as Neighbor-Net, rely on an associated metric on the species set. Such a metric is first estimated from DNA sequences, which leads to a key difficulty: distantly related sequences produce statistically unreliable estimates. This is problematic for Neighbor-Net as it is based on the popular tree reconstruction method Neighbor-Joining, whose sensitivity to distance estimation errors is well established theoretically. In the tree case, more robust reconstruction methods have been developed using the notion of a distorted metric, which captures the dependence of the error in the distance through a radius of accuracy. Here we design the first circular network reconstruction method based on distorted metrics. Our method is computationally efficient. Moreover, the analysis of its radius of accuracy highlights the important role played by the maximum incompatibility, a measure of the extent to which the network differs from a tree.Comment: Submitte

    Non-Uniform Robust Network Design in Planar Graphs

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    Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on \textbf{bulk-robustness}, a model recently introduced~\cite{bulk} for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used in~\cite{bulk} to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.Comment: 17 pages, 2 figure
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