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Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was 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...

Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet...

Distribution-Sensitive Construction of the Greedy Spanner

log log n) expected time algorithm on uniformly distributed Quirijn W. Bouts and Kevin Buchin are supported by the Netherlands Organisation for Scientific Research (NWO) under Project Nos. 639.023.208

Computing the Greedy Spanner in Linear Space

routes in the graph are at most t times longer than the direct geometric distance. The spanners B Quirijn W. Bouts Kevin Buchin considered in literature have only O(n) edges as opposed to the O(n2

Vectors in a box

For an integer d ≥ 1, let τ(d) be the smallest integer with the following property: if v 1, v 2, . . . , v t is a sequence of t ≥ 2 vectors in [−1, 1] d with \({{\bf v}_1+{\bf v}_2+\cdots+{\bf v}_t \in [-1,1]^d}\) , then there is a set \({S\subseteq \{1,2,\ldots,t\}}\) of indices, 2 ≤ |S| ≤ τ(d), such that \({\sum_{i \in S}{\bf v}_i \in [-1,1]^d}\) . The quantity τ(d) was...

Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and Extended

Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection ℛ of input regions known in advance. Building on recent work by Löffler and Snoeyink, we show how to leverage our knowledge of ℛ for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, e.g., overlapping disks...

Median Trajectories

We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed...

Processing aggregated data: the location of clusters in health data

author(s) and source are credited. Kevin Buchin received his Ph.D in computer science from the Free University Berlin in Germany. He was a postdoctoral researcher at Utrecht University, Netherlands, and

Drawing (Complete) Binary Tanglegrams

A binary tanglegram is a drawing of a pair of rooted 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...

Polychromatic Colorings of Plane Graphs

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 appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ⌊(3g+1)/4⌋ colors. We further show that...