In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version o... more In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fr\'echet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fr\'echet distance between two polygonal curves in time $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ 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 $O(n^{2-\varepsilon})$, for some $\varepsilon > 0$. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fr\'echet distance on a word RAM.
We show that the vertices of any plane graph in which every face is incident to at least g vertic... more We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by (3g − 5)/4 colors so that every color
We give lower bounds for the combinatorial complexity of the Voronoi diagram of polygonal curves ... more We give lower bounds for the combinatorial complexity of the Voronoi diagram of polygonal curves under the discrete Frechet distance. We show that the Voronoi diagram of n curves in R^d with k vertices each, has complexity Omega(n^{dk}) for dimension d=1,2 and Omega(n^{d(k-1)+2}) for d>2.
International Journal of Computational Geometry Applications, 2008
Polygonal chains are fundamental objects in many applications like pattern recognition and protei... more Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension (d = 2, 3) under the discrete Fréchet distance. Given n polygonal chains C in d-dimension (d = 2, 3), each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VD F (C). Our main results are summarized as follows.
We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivi... more We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivial class of surfaces: simple polygons. For this, we show that it suffices to consider homeomorphisms that map an arbitrary triangulation of one polygon to the other polygon such that diagonals of the triangulation are mapped to shortest paths in the other polygon.
Curve matching is a fundamental problem that occurs in many applications. In this paper, we study... more 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.
We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivi... more We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivial class of surfaces: simple polygons. For this, we show that it suffices to consider homeomorphisms that map an arbitrary triangulation of one polygon to the other polygon such that diagonals of the triangulation are mapped to shortest paths in the other polygon.
Polygonal chains are fundamental objects in many applications like pattern recognition and protei... more Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fr\`{e}chet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in $d$-dimension ($d=2,3$) under the discrete Fr\`{e}chet distance. Given $n$ polygonal chains ${\cal C}$ in $d$-dimension ($d=2,3$), each with at most $k$ vertices, we prove fundamental properties of such a Voronoi diagram {\em VD}$_F({\cal C})$ by presenting the first known upper and lower bounds for {\em VD}$_F({\cal C})$.
We show that it is NP-hard to decide the Fréchet distance between (i) non-intersecting polygons w... more We show that it is NP-hard to decide the Fréchet distance between (i) non-intersecting polygons with holes embedded in the plane, (ii) 2d terrains, and (iii) self-intersecting simple polygons in 2d, which can be unfolded in 3d. The only previously known NP-hardness result for 2d surfaces was based on self-intersecting polygons with an unfolding in 4d. In contrast to this
Proceedings of the 16th International Symposium on Graph Drawing, 2009
A binary tanglegram is a pair S, T of binary trees whose leaf sets are in one-to-one corresponden... more A binary tanglegram is a pair S, T of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn with no edge crossing and that the inter-tree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We show that the problem is hard even if both trees are complete binary trees. For this case we give an O(n 3 )-time 2-approximation and a new and simple fixedparameter algorithm. We prove that under the Unique Games Conjecture there is no constant-factor approximation for general binary trees. We show that the maximization version of the problem for general binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.
We study the complexity of computing the Frechet distance (also called dog-leash distance) betwee... more We study the complexity of computing the Frechet distance (also called dog-leash 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 ...
Given a piecewise monotone function f : R → R and a real value T min , we develop an algorithm th... more Given a piecewise monotone function f : R → R and a real value T min , we develop an algorithm that finds an interval of length at least T min for which the average value of f is minimized. The run-time of the algorithm is linear in the number of monotone pieces of f if certain operations are available in constant time for f . We use this algorithm to solve a basic problem arising in the analysis of trajectories: Finding the most similar subtrajectories of two given trajectories, provided that the duration is at least T min . Since the precise solution requires complex operations, we also give a simple (1+ε)-approximation algorithm in which these operations are not needed.
The Frechet distance is a distance measure for pa- rameterized curves or surfaces. Using a discre... more The Frechet distance is a distance measure for pa- rameterized curves or surfaces. Using a discrete ap- proximation, we show that for triangulated surfaces it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a monotone decreas- ing sequence of rationals converging to the result. It follows that the decision problem, whether the Frechet distance of two given surfaces lies below some speci- fied value, is recursively enumerable.
ABSTRACT The movement of animals, people, and vehicles is embedded in a geographic context. This ... more ABSTRACT The movement of animals, people, and vehicles is embedded in a geographic context. This context both enables and limits movement. Most analysis algorithms for trajectories have so far ignored context: trajectories are analyzed in an otherwise empty space. This severely lim-its the applicability of algorithmic methods. In this paper we present a model for (geographic) context that allows us to integrate context into the analysis of movement data. Based on this model we develop simple but efficient context-aware similarity measures. We validate our approach by applying these measures to hurricane trajectories.
In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version o... more In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fr\'echet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fr\'echet distance between two polygonal curves in time $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ 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 $O(n^{2-\varepsilon})$, for some $\varepsilon > 0$. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fr\'echet distance on a word RAM.
We show that the vertices of any plane graph in which every face is incident to at least g vertic... more We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by (3g − 5)/4 colors so that every color
We give lower bounds for the combinatorial complexity of the Voronoi diagram of polygonal curves ... more We give lower bounds for the combinatorial complexity of the Voronoi diagram of polygonal curves under the discrete Frechet distance. We show that the Voronoi diagram of n curves in R^d with k vertices each, has complexity Omega(n^{dk}) for dimension d=1,2 and Omega(n^{d(k-1)+2}) for d>2.
International Journal of Computational Geometry Applications, 2008
Polygonal chains are fundamental objects in many applications like pattern recognition and protei... more Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension (d = 2, 3) under the discrete Fréchet distance. Given n polygonal chains C in d-dimension (d = 2, 3), each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VD F (C). Our main results are summarized as follows.
We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivi... more We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivial class of surfaces: simple polygons. For this, we show that it suffices to consider homeomorphisms that map an arbitrary triangulation of one polygon to the other polygon such that diagonals of the triangulation are mapped to shortest paths in the other polygon.
Curve matching is a fundamental problem that occurs in many applications. In this paper, we study... more 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.
We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivi... more We present the first polynomial-time algorithm for computing the Fréchet distance for a non-trivial class of surfaces: simple polygons. For this, we show that it suffices to consider homeomorphisms that map an arbitrary triangulation of one polygon to the other polygon such that diagonals of the triangulation are mapped to shortest paths in the other polygon.
Polygonal chains are fundamental objects in many applications like pattern recognition and protei... more Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fr\`{e}chet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in $d$-dimension ($d=2,3$) under the discrete Fr\`{e}chet distance. Given $n$ polygonal chains ${\cal C}$ in $d$-dimension ($d=2,3$), each with at most $k$ vertices, we prove fundamental properties of such a Voronoi diagram {\em VD}$_F({\cal C})$ by presenting the first known upper and lower bounds for {\em VD}$_F({\cal C})$.
We show that it is NP-hard to decide the Fréchet distance between (i) non-intersecting polygons w... more We show that it is NP-hard to decide the Fréchet distance between (i) non-intersecting polygons with holes embedded in the plane, (ii) 2d terrains, and (iii) self-intersecting simple polygons in 2d, which can be unfolded in 3d. The only previously known NP-hardness result for 2d surfaces was based on self-intersecting polygons with an unfolding in 4d. In contrast to this
Proceedings of the 16th International Symposium on Graph Drawing, 2009
A binary tanglegram is a pair S, T of binary trees whose leaf sets are in one-to-one corresponden... more A binary tanglegram is a pair S, T of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn with no edge crossing and that the inter-tree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We show that the problem is hard even if both trees are complete binary trees. For this case we give an O(n 3 )-time 2-approximation and a new and simple fixedparameter algorithm. We prove that under the Unique Games Conjecture there is no constant-factor approximation for general binary trees. We show that the maximization version of the problem for general binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.
We study the complexity of computing the Frechet distance (also called dog-leash distance) betwee... more We study the complexity of computing the Frechet distance (also called dog-leash 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 ...
Given a piecewise monotone function f : R → R and a real value T min , we develop an algorithm th... more Given a piecewise monotone function f : R → R and a real value T min , we develop an algorithm that finds an interval of length at least T min for which the average value of f is minimized. The run-time of the algorithm is linear in the number of monotone pieces of f if certain operations are available in constant time for f . We use this algorithm to solve a basic problem arising in the analysis of trajectories: Finding the most similar subtrajectories of two given trajectories, provided that the duration is at least T min . Since the precise solution requires complex operations, we also give a simple (1+ε)-approximation algorithm in which these operations are not needed.
The Frechet distance is a distance measure for pa- rameterized curves or surfaces. Using a discre... more The Frechet distance is a distance measure for pa- rameterized curves or surfaces. Using a discrete ap- proximation, we show that for triangulated surfaces it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a monotone decreas- ing sequence of rationals converging to the result. It follows that the decision problem, whether the Frechet distance of two given surfaces lies below some speci- fied value, is recursively enumerable.
ABSTRACT The movement of animals, people, and vehicles is embedded in a geographic context. This ... more ABSTRACT The movement of animals, people, and vehicles is embedded in a geographic context. This context both enables and limits movement. Most analysis algorithms for trajectories have so far ignored context: trajectories are analyzed in an otherwise empty space. This severely lim-its the applicability of algorithmic methods. In this paper we present a model for (geographic) context that allows us to integrate context into the analysis of movement data. Based on this model we develop simple but efficient context-aware similarity measures. We validate our approach by applying these measures to hurricane trajectories.
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Papers by Maike Buchin