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141
Computing the Fréchet Distance with a Retractable Leash Kevin Buchin ∗ Maike Buchin
"... All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; then they use this oracle to find the optimum among a finite set of critical values. We present a novel approach that avoids the detour through the ..."
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All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; then they use this oracle to find the optimum among a finite set of critical values. We present a novel approach that avoids the detour through the decision version. We demonstrate its strength by presenting a quadratic time algorithm for the Fréchet distance between polygonal curves in Rd under polyhedral distance functions, including L1 and L∞. We also get a (1 + )approximation of the Fréchet distance under the Euclidean metric. For the exact Euclidean case, our framework currently gives an algorithm with running time O(n2 log2 n). However, we conjecture that it may eventually lead to a faster exact algorithm. 1
Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations Are Equivalent
, 2011
"... We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree ..."
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Cited by 4 (2 self)
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quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous [40] and Buchin and Mulzer [10]. Our main tool for the second algorithm is the wellseparated pair decomposition (WSPD) [13], a structure that has
Detecting Commuting Patterns by Clustering Subtrajectories
, 2008
"... In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in spee ..."
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Cited by 31 (12 self)
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In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the ‘longest’ subtrajectory cluster is as hard as MaxClique to compute and approximate.
Computing the Fréchet Distance Between Simple Polygons
, 2007
"... We present the first polynomialtime algorithm for computing the Fréchet distance for a nontrivial class of surfaces: simple polygons, i.e., the area enclosed by closed simple polygonal curves, which may lie in different planes. For this, we show that we can restrict the set of maps realizing the F ..."
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Cited by 28 (12 self)
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We present the first polynomialtime algorithm for computing the Fréchet distance for a nontrivial class of surfaces: simple polygons, i.e., the area enclosed by closed simple polygonal curves, which may lie in different planes. For this, we show that we can restrict the set of maps realizing the Fréchet distance, and develop an algorithm for computing the Fréchet distance using the algorithm for curves, techniques for computing shortest paths in a simple polygon, and dynamic programming.
RealTime Expressive Rendering of City Models
 Seventh International Conference on Information Visualization, Proceedings IEEE 2003 Information Visualization
, 2003
"... City models have become central elements for visually communicating spatial information related to urban areas and have manifold applications. Our realtime nonphotorealistic rendering technique aims at abstract, comprehensible, and vivid drawings of assemblies of polygonal 3D urban objects. It tak ..."
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Cited by 22 (4 self)
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City models have become central elements for visually communicating spatial information related to urban areas and have manifold applications. Our realtime nonphotorealistic rendering technique aims at abstract, comprehensible, and vivid drawings of assemblies of polygonal 3D urban objects. It takes into account related principles in cartography, cognition, and nonphotorealism. Technically, the geometry of a building is rendered using expressive line drawings to enhance the edges, twotone or threetone shading to draw the faces, and simulated shadows. The edge enhancement offers several degrees of freedom, such as interactively changing the style, width, tilt, color, transparency, and length of the strokes. Traditional drawings of cities and panoramas inspired the tone shading that achieves a pleasing visual color effect. The rendering technique can be applied not only to city models but to polygonal shapes in general. 1.
How Difficult is it to Walk the Dog?
 IN PROC. 23RD EUROPEAN WORKSHOP ON COMPUTATIONAL GEOMETRY
, 2007
"... We study the complexity of computing the Fréchet distance (also called dogleash distance) between two polygonal curves with a total number of n vertices. For two polygonal curves in the plane we prove an Ω(n log n) lower bound for the decision problem in the algebraic computation tree model allowin ..."
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Cited by 18 (5 self)
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We study the complexity of computing the Fréchet distance (also called dogleash distance) between two polygonal curves with a total number of n vertices. For two polygonal curves in the plane we prove an Ω(n log n) lower bound for the decision problem in the algebraic computation tree model allowing arithmetic operations and tests. Up to now only a O(n 2) upper bound for the decision problem was known. The Ω(n log n) lower bound extends to variants of the Fréchet distance such as the weak as well as the discrete Fréchet distance. For the onedimensional case we give a lineartime algorithm to solve the decision problem for the weak Fréchet distance between onedimensional polygonal curves.
Exact Algorithm for Partial Curve Matching via the Fréchet Distance
 Proc. 20th ACMSIAM Symposium on Discrete Algorithms
, 2009
"... Curve matching is a fundamental problem that occurs in many applications. In this paper, we study the problem of measuring partial similarity between curves. Specifically, given two curves, we wish to maximize the total length of subcurves that are close to each other, where closeness is measured by ..."
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Cited by 24 (4 self)
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Curve matching is a fundamental problem that occurs in many applications. In this paper, we study the problem of measuring partial similarity between curves. Specifically, given two curves, we wish to maximize the total length of subcurves that are close to each other, where closeness is measured by the Fréchet distance, a common distance measure for curves. The resulting maximal length is called the partial Fréchet similarity between the two input curves. Given two polygonal curves P and Q in IR d of size m and n, respectively, we present the first exact algorithm that runs in polynomial time to compute Fδ(P, Q), the partial Fréchet similarity between P and Q, under the L1 and L ∞ norms. Specifically, we formulate the problem of computing Fδ(P, Q) as a longest path problem, and solve it in O(mn(m + n) log(mn)) time, under the L1 or L∞ norm, using a “shortestpath map ” type decomposition. To the best of our knowledge, this is the first paper to study this natural definition of partial curve similarity in the continuous setting (with all points in the curve considered), and present a polynomialtime exact algorithm for it. 1
On the number of cycles in planar graphs
 in Proc. 13th COCOON
, 2007
"... Abstract. We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262 n simple cycles and at least 2.0845 n Hamilton cycles. Based on c ..."
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Cited by 18 (5 self)
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Abstract. We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262 n simple cycles and at least 2.0845 n Hamilton cycles. Based on counting arguments for perfect matchings we prove that 2.3404 n is an upper bound for the number of Hamiltonian cycles. Moreover, we obtain upper bounds for the number of simple cycles of a given length with a face coloring technique. Combining both, we show that there is no planar graph with more than 2.8927 n simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46. 1
Four soviets walk the dog  with an application to Alt’s conjecture
 CORR
"... Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than ..."
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Cited by 20 (4 self)
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Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n 2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here n denotes the number of vertices on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUMhard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n 2 √ log n(log log n) 3/2) on a pointer machine and in time O(n 2 (log log n) 2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n 2−ε), for some ε> 0. This provides evidence that the decision problem may not be 3SUMhard after all and reveals an intriguing new aspect of this wellstudied problem.
Flow Map Layout via Spiral Trees
"... Abstract—Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visua ..."
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Cited by 13 (1 self)
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Abstract—Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding selfintersections. Most flow maps are still drawn by hand and only few automated methods exist. Some of the known algorithms do not support edgebundling and those that do, cannot guarantee crossingfree flows. We present a new algorithmic method that uses edgebundling and computes crossingfree flows of high visual quality. Our method is based on socalled spiral trees, a novel type of Steiner tree which uses logarithmic spirals. Spiral trees naturally induce a clustering on the targets and smoothly bundle lines. Our flows can also avoid obstacles, such as map features, region outlines, or even the targets. We demonstrate our approach with extensive experiments. Index Terms—Flow maps, Automated Cartography, Spiral Trees. 1
Results 1  10
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141