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16
Optimal system of loops on an orientable surface
 DISCRETE COMPUT. GEOM
, 2005
"... Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a system of loops for M. The resulting disk may be viewed as a polygon in which the ..."
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Cited by 37 (4 self)
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Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a system of loops for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertexedge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.
Tightening NonSimple Paths and Cycles on Surfaces
 SUBMITTED TO SIAM JOURNAL ON COMPUTING
"... We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity ..."
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Cited by 28 (9 self)
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We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity n, genus g ≥ 2, and no boundary, we construct in O(gn log n) time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity k in O(gnk) time, or the shortest cycle homotopic to a given cycle of complexity k in O(gnk log(nk)) time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.
Drawing with Fat Edges
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with “fat ” edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or ..."
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Cited by 22 (8 self)
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Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with “fat ” edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding polynomial time algorithm that uses the model. We focus on a restricted class of graphs that occur in VLSI wire routing and show how to extend the algorithm to general planar graphs. We show how to convert an arbitrary wire routing into a homotopically equivalent routing that maximizes the distance between any two wires. Among such, we obtain the routing with minimum total wire length. A homotopically equivalent routing that maximizes the distance between any two wires yields a graph drawing which maximizes edge thickness. Finally, our algorithm also allows for different edge weights, that is, the requirement for unit wire thickness can be removed.
Sensor Beams, Obstacles, and Possible Paths
, 2008
"... Summary. This paper introduces a problem in which an agent (robot, human, or animal) travels among obstacles and binary detection beams. The task is to determine the possible agent path based only on the binary sensor data. This is a basic filtering problem encountered in many settings, which may ar ..."
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Cited by 10 (9 self)
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Summary. This paper introduces a problem in which an agent (robot, human, or animal) travels among obstacles and binary detection beams. The task is to determine the possible agent path based only on the binary sensor data. This is a basic filtering problem encountered in many settings, which may arise from physical sensor beams or virtual beams that are derived from other sensing modalities. Methods are given for three alternative representations: 1) the possible sequences of regions visited, 2) path descriptions up to homotopy class, and 3) numbers of times winding around obstacles. The solutions are adapted to the minimal sensing setting; therefore, precise estimation, distances, and coordinates are replaced by topological expressions. Applications include sensorbased forensics, assisted living, security, and environmental monitoring. 1
Thick NonCrossing Paths and MinimumCost Flows in Polygonal Domains
"... We study the problem of finding shortest noncrossing thick paths in a polygonal domain, where a thick path is the Minkowski sum of a usual (zerothickness, or thin) path and a disk. Given K pairs of terminals on the boundary of a simple ngon, we compute in O(n + K) time a representation of the set ..."
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Cited by 5 (3 self)
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We study the problem of finding shortest noncrossing thick paths in a polygonal domain, where a thick path is the Minkowski sum of a usual (zerothickness, or thin) path and a disk. Given K pairs of terminals on the boundary of a simple ngon, we compute in O(n + K) time a representation of the set of K shortest noncrossing thick paths joining the terminal pairs; using the representation, any particular path can be output in time proportional to its complexity. We compute K shortest thick noncrossing paths in a polygon with h holes in O ` (K + 1) h h! poly(n, K) ´ time, using an efficient method to compute any one of the K thick paths if the “threadings ” of all paths amidst the holes are specified. We show that if h is not constant, the problem is NPhard; we also show the hardness of approximation. We give a pseudopolynomialtime algorithm for some rectilinear versions of the problem. We apply our thick paths algorithms to obtain the first algorithmic results for the minimumcost continuous flow problem — an extension of the standard discrete minimumcost network flow problem to continuous domains. The results are based on showing a continuous analog of the Network
Computational topology
 Algorithms and Theory of Computation Handbook
, 2010
"... According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in t ..."
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Cited by 3 (2 self)
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According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in turn, undertakes the challenge of studying topology using a computer.
Testing contractibility in planar Rips complexes
 In Proc. Symp. on Comp. Geom. (SoCG) 2008
"... The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Ou ..."
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Cited by 3 (1 self)
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The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest noncontractible cycle in a planar Rips complex in O(n 2 log n + mn) time.
HOMOTOPIC FRÉCHET DISTANCE BETWEEN CURVES OR, WALKING YOUR DOG IN THE WOODS IN POLYNOMIAL TIME
, 2008
"... The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requi ..."
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Cited by 3 (0 self)
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The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomialtime algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.
Pointed binary encompassing trees, in
 Proc. 9th SWAT, in: Lecture Notes in Comput. Sci
, 2004
"... Abstract. We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T, that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex v ∈ T is pointed, that is, v has an incident ang ..."
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Cited by 1 (0 self)
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Abstract. We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T, that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex v ∈ T is pointed, that is, v has an incident angle greater than π. Such a tree can be completed to a minimum pseudotriangulation. In particular, it follows that every set of disjoint line segments has a minimum pseudotriangulation of bounded vertex degree. 1
Shortest NonCrossing Walks in the Plane
, 2010
"... Let G be a plane graph with nonnegative edge weights, and let k terminal pairs be specified on h face boundaries. We present an algorithm to find k noncrossing walks in G of minimum total length that connect all terminal pairs, if any such walks exist, in 2 O(h2) n log k time. The computed walks m ..."
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Cited by 1 (0 self)
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Let G be a plane graph with nonnegative edge weights, and let k terminal pairs be specified on h face boundaries. We present an algorithm to find k noncrossing walks in G of minimum total length that connect all terminal pairs, if any such walks exist, in 2 O(h2) n log k time. The computed walks may overlap but may not cross each other or themselves. Our algorithm generalizes a result of Takahashi, Suzuki, and Nikizeki [Algorithmica 1996] for the special case h ≤ 2. We also describe an algorithm for the corresponding geometric problem, where the terminal points lie on the boundary of h polygonal obstacles of total complexity n, again in 2 O(h2) n time, generalizing an algorithm of Papadopoulou [Int. J. Comput. Geom. Appl. 1999] for the special case h ≤ 2. In both settings, shortest noncrossing walks can have complexity exponential in h. We also describe algorithms to determine in O(n) time whether the terminal pairs can be connected by any noncrossing walks.