9,205 research outputs found

    Planar Visibility: Testing and Counting

    Full text link
    In this paper we consider query versions of visibility testing and visibility counting. Let SS be a set of nn disjoint line segments in R2\R^2 and let ss be an element of SS. Visibility testing is to preprocess SS so that we can quickly determine if ss is visible from a query point qq. Visibility counting involves preprocessing SS so that one can quickly estimate the number of segments in SS visible from a query point qq. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, VS(s)V_S(s), of R2\R^2 that is weakly visible from a segment ss can be represented as the union of a set, CS(s)C_S(s), of O(n2)O(n^2) triangles, even though the complexity of VS(s)V_S(s) can be Ω(n4)\Omega(n^4). We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of SS visible from a point pp to the number of triangles in sSCS(s)\bigcup_{s\in S} C_S(s) that contain pp.Comment: 22 page

    Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

    Get PDF
    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

    Packing Plane Perfect Matchings into a Point Set

    Full text link
    Given a set PP of nn points in the plane, where nn is even, we consider the following question: How many plane perfect matchings can be packed into PP? We prove that at least log2n2\lceil\log_2{n}\rceil-2 plane perfect matchings can be packed into any point set PP. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem

    CiNCT: Compression and retrieval for massive vehicular trajectories via relative movement labeling

    Full text link
    In this paper, we present a compressed data structure for moving object trajectories in a road network, which are represented as sequences of road edges. Unlike existing compression methods for trajectories in a network, our method supports pattern matching and decompression from an arbitrary position while retaining a high compressibility with theoretical guarantees. Specifically, our method is based on FM-index, a fast and compact data structure for pattern matching. To enhance the compression, we incorporate the sparsity of road networks into the data structure. In particular, we present the novel concepts of relative movement labeling and PseudoRank, each contributing to significant reductions in data size and query processing time. Our theoretical analysis and experimental studies reveal the advantages of our proposed method as compared to existing trajectory compression methods and FM-index variants

    Algorithms for Subpath Convex Hull Queries and Ray-Shooting Among Segments

    Get PDF
    In this paper, we first consider the subpath convex hull query problem: Given a simple path ? of n vertices, preprocess it so that the convex hull of any query subpath of ? can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [SODA 90\u27] proposed a data structure of O(n) space and O(log n log log n) query time; reducing the query time to O(log n) increases the space to O(nlog log n). We present an improved result that uses O(n) space while achieving O(log n) query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in O(log n) time each. Our new result leads to improvements for several other problems. In particular, with the help of the above result, we present new algorithms for the ray-shooting problem among segments. Given a set of n (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of O(n log n) space that can answer each query in (?n log n) time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to O(n). All these are classical problems that have been studied extensively. Previously data structures of O?(?n) query time were known in early 1990s; nearly no progress has been made for over two decades. For all problems, our results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better
    corecore