Kevin Buchin

Detecting Commuting Patterns by Clustering Subtrajectories

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 Frechet distance and the discrete Frechet 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.

Detecting single file movement

We study the problem of detecting a single file behavior in a set of trajectories. A group of entities is moving in single file if they are following each other, one behind the other. This movement pattern occurs often, among animals, humans, and vehicles. It is challenging to detect because it does not have a fixed layout.n In this paper we first model the notion of following behind, on which we base our definition of single file. We present efficient algorithms for detecting following behind and single file behaviors. We test and evaluate these algorithms on real and generated test data.

Polychromatic colorings of plane graphs

We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Àý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.

Drawing (Complete) Binary Tanglegrams

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 without edge crossings 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. n nWe prove that under the Unique Games Conjecture there is no constant-factor approximation for general binary trees. 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 fixed-parameter algorithm. We show that the maximization version of the dual 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.

Detecting Hotspots in Geographic Networks

geographical analysis tasks, such as crime hotspot detection. Given a network N (for example, a street, train, or highway network) together with a set of sites which are located on the network (for example, accident locations or crime scenes), we want to find a connected subnetwork F of N of small total length that contains many sites. That is, we are searching for a subnetwork F that spans a cluster of sites which are close with respect to the network distance.

Fine-grained complexity analysis of two classic TSP variants

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. n nOur first set of results is motivated by the Bitonic tsp problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamicprogramming exercise to solve this problem in O(n^2) time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic tsp in O(n*log^2(n)) time and its bottleneck version in O(n*log^3(n)) time. In the more general pyramidal tsp problem, the points to be visited are labeled 1, ..., n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) tsp problem also work for the pyramidal tsp problem in the plane. n nOur second set of results concerns the popular k-opt heuristic for tsp in the graph setting. More precisely, we study the k-opt decision problem, which asks whether a given tour can be improved by a k-opt move that replaces k edges in the tour by k new edges. A simple algorithm solves k-opt in O(n^k) time for fixed k. For 2-opt, this is easily seen to be optimal. For k = 3 we prove that an algorithm with a runtime of the form ~O(n^{3-epsilon}) exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. For general k-opt, it is known that a runtime of f(k)*n^{o(k/log(k))} would contradict the Exponential Time Hypothesis. The results for k = 2, 3 may suggest that the actual time complexity of k-opt is Theta(n^k). We show that this is not the case, by presenting an algorithm that finds the best k-move in O(n^{lfoor 2k/3 rfloor +1}) time for fixed k >= 3. This implies that 4-opt can be solved in O(n^3) time, matching the best-known algorithm for 3-opt. Finally, we show how to beat the quadratic barrier for k = 2 in two important settings, namely for points in the plane and when we want to solve 2-opt repeatedly

Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

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 (continuous/discrete) Frechet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension under the discrete Frechet distance. Given a set

Feed-links for network extensions

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Clusters in Aggregated Health Data

of npolygonal chains in d-dimension, each with at most kvertices, we prove fundamental properties of such a Voronoi diagram VD F (

VORONOI DIAGRAM OF POLYGONAL CHAINS UNDER THE DISCRETE FRÉCHET DISTANCE

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