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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 18 (5 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.
Finding Long and Similar Parts of Trajectories
, 2011
"... A natural timedependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly selfintersecting lines with tim ..."
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Cited by 11 (4 self)
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A natural timedependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly selfintersecting lines with time stamps. For the case when a minimum duration of the subtrajectories is specified and the subtrajectories must start at corresponding times, we give a lineartime algorithm. The algorithm is based on a result of independent interest: We present a lineartime algorithm to find, for a piecewise monotone function, an interval of at least a given length that has minimum average value. In the case that the subtrajectories may start at noncorresponding times, it appears difficult to give exact algorithms, even if the duration of the subtrajectories is fixed. For this case, we give (1 + ε)approximation algorithms, for both fixed duration and when only a minimum duration is specified. 1
Locally correct Fréchet matchings
 In Proc. 20th Annu. European Sympos. Algorithms, volume 7501 of Lecture Notes Comput. Sci
, 2012
"... Abstract. The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity ..."
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Cited by 4 (2 self)
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Abstract. The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to “natural ” matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N3 logN) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N2) algorithm to compute a locally correct discrete Fréchet matching. 1
Fréchet queries in geometric trees
 In Proceedings of the 21st European Symposium on Algorithms
, 2013
"... Let T be a tree that is embedded in the plane and let ∆> 0 be a real number. The aim is to preprocess T into a data structure, such that, for any query polygonal path Q, we can decide if T contains a path P whose Fréchet distance δF (P,Q) to Q is less than ∆. For any real number ε> 0, we pres ..."
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Cited by 4 (1 self)
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Let T be a tree that is embedded in the plane and let ∆> 0 be a real number. The aim is to preprocess T into a data structure, such that, for any query polygonal path Q, we can decide if T contains a path P whose Fréchet distance δF (P,Q) to Q is less than ∆. For any real number ε> 0, we present an efficient data structure that solves an approximate version of this problem, for the case when T is cpacked and each of the edges of T and Q has length Ω(∆): If the data structure returns NO, then there is no such path P. If it returns YES, then δF (P,Q) ≤ 2(1 + ε) ∆ if Q is a line segment, and δF (P,Q) ≤ 3 ∆ otherwise. 1
Improved algorithms for partial curve matching
 In Proc. 19th Annu. European Sympos. Algorithms (ESA
, 2011
"... Abstract. Back in 1995, Alt and Godau gave an efficient algorithm for deciding whether a given curve resembles some part of a larger curve under a fixed Fréchet distance, achieving a running time of O(nm log(nm)), for n and m being the number of segments in the two curves, respectively. We improve ..."
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Abstract. Back in 1995, Alt and Godau gave an efficient algorithm for deciding whether a given curve resembles some part of a larger curve under a fixed Fréchet distance, achieving a running time of O(nm log(nm)), for n and m being the number of segments in the two curves, respectively. We improve this longstanding result by presenting an algorithm that solves this decision problem in O(nm) time. Our solution is based on constructing a simple data structure which we call freespace map. Using this data structure, we obtain improved algorithms for several variants of the Fréchet distance problem, including the Fréchet distance between two closed curves, and the socalled minimum/maximum walk problems. We also improve the map matching algorithm of Alt et al. for the case when the map is a directed acyclic graph. 1
Computing the Fréchet distance with a retractable leash
 In Proc. 21st Annu. European Sympos. Algorithms (ESA
, 2013
"... Abstract 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 thr ..."
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Abstract 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 R d under polyhedral distance functions, including L 1 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(n 2 log 2 n). However, we conjecture that it may eventually lead to a faster exact algorithm.
U N I V E R S I
"... With the popularity of GPS equipped smartphone, large volumes of location data associated with mobile objects can be recorded anytime, anywhere. Extracting a user’s behavior patterns embedded helps us to tailor information feeds to better meet a user’s real needs with respect not only to spatiotemp ..."
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With the popularity of GPS equipped smartphone, large volumes of location data associated with mobile objects can be recorded anytime, anywhere. Extracting a user’s behavior patterns embedded helps us to tailor information feeds to better meet a user’s real needs with respect not only to spatiotemporal information but also to previous movement history. This research utilized the raw GPS readings in an attempt to generate semantically meaningful trajectory summaries by modeling individual movement pattern. One supervised movement pattern model for periodic daily routines and the other unsupervised one for all other activities were trained by knowledgelevel features including Points of Interest, transportation mode and mapmatched route. In addition, a familiarity index of routes was generated representing a user’s personal knowledge about the environment he/she lived in. i Acknowledgements I would like to thank my supervisor, William Mackaness, for his support and encouragement throughout the most different time of my project. I would also like to thank
AngryAnts: A Citizen Science Approach to Computing Accurate Average Trajectories∗
"... In this paper we describe a citizen science system for solving timeconsuming and laborintensive problems, using crowdsourcing and efficient geometric algorithms. Specifically, the system can be used to trace static objects in images (such as trees in an urban environment), or to generate trajector ..."
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In this paper we describe a citizen science system for solving timeconsuming and laborintensive problems, using crowdsourcing and efficient geometric algorithms. Specifically, the system can be used to trace static objects in images (such as trees in an urban environment), or to generate trajectories of moving objects in videos (such as ants in an ant colony). The traces of the static objects can provide quantitative measurements such as size, shape and appearance, for example in monitoring the health of the trees in New York City’s Million Trees Initiative. It is relatively easy to plant a million trees, but ensuring they are healthy and taken care of is a challenge on a different scale, and a challenge where citizen scientists can make a big difference. The ant trajectories extracted from videos of ant colonies are needed by biologists studying longitudinal behavioral patterns in insect colonies. Existing automated solutions are not good enough, and there is only so much data that even motivated students can annotate in the research lab. AngryAnts is our online application which displays short video segments, specifies which ant needs to be traced and allows the citizen scientist to enter the trajectory in a firstperson shooter style via mouse clicks. Submitted trajectories are verified using a ReCaptcha method, where part of the trajectory
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
JOURNAL OF SPATIAL INFORMATION SCIENCE Number 9 (2014), pp. 101–124 doi:10.5311/JOSIS.2014.9.179 RESEARCH ARTICLE Similarity of trajectories taking into account geographic context∗
"... Abstract: The movements of animals, people, and vehicles are embedded in a geographic context. This context influences the movement and may cause the formation of certain behavioral responses. Thus, it is essential to include context parameters in the study of movement and the development of movemen ..."
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Abstract: The movements of animals, people, and vehicles are embedded in a geographic context. This context influences the movement and may cause the formation of certain behavioral responses. Thus, it is essential to include context parameters in the study of movement and the development of movement pattern analytics. Advances in sensor technologies and positioning devices provide valuable data not only of moving agents but also of the circumstances embedding the movement in space and time. Developing knowledge discovery methods to investigate the relation between movement and its surrounding context is a major challenge in movement analysis today. In this paper we show how to integrate geographic context into the similarity analysis of movement data. For this, we discuss models for geographic context of movement data. Based on this we develop simple but efficient contextaware similarity measures for movement trajectories, which combine a spatial and a contextual distance. These are based on wellknown similarity measures for trajectories, such as the Hausdorff, Fréchet, or equal time distance. We validate our approach by applying these measures to movement data of hurricanes and albatross.