Journal ArticleDOI
Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons
Leonidas J. Guibas,John Hershberger,Daniel Leven,Micha Sharir,Micha Sharir,Robert E. Tarjan,Robert E. Tarjan +6 more
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Given a triangulation of a simple polygonP, linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP are presented.Abstract:
Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.read more
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Dynamic Algorithms for Visibility Polygons in Simple Polygons
TL;DR: The authors devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and deletions to the simple polygon.
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Computing the $L_1$ Geodesic Diameter and Center of a Polygonal Domain
Sang Won Bae,Matias Korman,Joseph S. B. Mitchell,Yoshio Okamoto,Valentin Polishchuk,Haitao Wang +5 more
TL;DR: For a polygonal domain with holes and a total number of vertices, the fastest known algorithm for the problem was given in this paper with a complexity of O((n^4+n^2 h^4)/α(n)) where α(n) denotes the inverse Ackermann function.
Combinatorial Rigidity and Generation of Discrete Structures
Abstract: The well-known Maxwell rule asserts that, if a 2-dimensional bar-and-joint framework (i.e., a structure consisting of rigid bars connected by universal joints) is statically rigid, then the number of bars is at least twice the number of joints minus three. It is notable that this rule does not concern with how a structure is realized in the space; it suggests a deep dependence between the rigidity and the topology of structures. Deciding whether a structure is rigid or flexible is indeed the most basic problem in the structural engineering, and combinatorial rigidity can answer it only by looking at the topologies of structures. The most famous result in this context is the theorem of Laman proposed in 1970 which asserts the converse direction of the Maxwell rule for almost all situations, implying that the general behavior of 2-dimensional bar-and-joint frameworks can be captured by a combinatorial condition without looking at the geometry of frameworks. In this dissertation, with the aid of rigidity theory since Laman’s result, we prove new combinatorial characterizations of the rigidity of several types of structures and develop efficient algorithms for generating discrete structures based on combinatorial characterizations. First, we consider a problem of partitioning a graph into rooted-forests, and as a generalization of Tutte-Nash-Williams tree-packing theorem, we present a necessary and sufficient condition for a graph to be decomposed into edge-disjoint rooted-forests. This result leads to a new combinatorial characterization of the generic rigidity of 2-dimensional bar-joint-slider frameworks. In particular, we prove that, even though the directions of sliders are predetermined and degenerate (i.e., some sliders have the same direction), it is combinatorially decidable whether the framework is rigid or not. We consequently extend Laman’s counting theorem, Crapo’s 3tree2-partition theorem, and the Henneberg construction of bar-and-joint frameworks to bar-joint-slider frameworks. Next, we deal with the so-called Molecular conjecture posed by Tay and Whiteley in 1984. We solve this long-standing open problem affirmatively. In particular, we obtain a combinatorial characterization of the generic rigidity of panel-and-hinge frameworks in terms of the number of edge-disjoint spanning trees that can be packed into the underlying graphs. As a corollary, we obtain a combinatorial characterization of the generic rigidity of 3-dimensional bar-and-joint frameworks of the square of graphs. We then deal with the problem of enumerating 2-dimensional minimally rigid bar-andjoint frameworks connecting a given set of n joints. Based on the well-known reverse search paradigm, we present an algorithm for enumerating non-crossing minimally rigid frameworks in O(n3) time per output. Subsequently, we generalize the idea to develop a general enumeration technique that can apply to arbitrary non-crossing geometric graph classes. We show that our new technique provides not only faster algorithms for some enumeration problems compared with existing ones but also first algorithms for various problems that had not been considered to the best of our knowledge. In particular, using a generalization of Laman’s theorem to bar-joint-slider frameworks, we propose an efficient algorithm for enumerating all non-crossing k-degree-of-freedom mechanisms consisting of given joints some of which are connected with external environment by a set of sliders.
Journal Article
Approximating a Shortest Watchman Route
TL;DR: A fast algorithm for computing awatchman route in a simple polygon that is at most a constant factor longer than the shortest watchman route is presented.
Posted Content
Query-points visibility constraint minimum link paths in simple polygons
TL;DR: This work studies the query version of constrained minimum link paths between two points inside a simple polygon P with n vertices such that there is at least one point on the path, visible from a query point, and proposes an algorithm with O(n) preprocessing time.
References
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Data Structures and Network Algorithms
TL;DR: This paper presents a meta-trees tree model that automates the very labor-intensive and therefore time-heavy and therefore expensive process of manually selecting trees to grow in a graph.
Journal ArticleDOI
Self-adjusting binary search trees
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Optimal Search in Planar Subdivisions
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Optimal point location in a monotone subdivision
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Journal ArticleDOI
Euclidean shortest paths in the presence of rectilinear barriers
Der-Tsai Lee,Franco P. Preparata +1 more
TL;DR: The goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph, which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.