Lena Schlipf

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

We consider the problem of computing the discrete Frechet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time \(2^{O(d^2)} m^2n^2\log^2(mn)\) the Frechet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O( mn log2(mn) + (m 2 + n 2)log(mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L ∞ distance, we give an O(dmn log(dmn))-time algorithm.

Finding largest rectangles in convex polygons

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O ( n 3 ) . We also give ( 1 - e ) -approximation algorithms that take time O ( e - 1 / 2 log ? n + e - 3 / 2 ) .

Shortest path to a segment and quickest visibility queries

We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.

Covering and piercing disks with two centers

We give exact and approximation algorithms for two-center problems when the input is a set D of disks in the plane. We first study the problem of finding two smallest congruent disks such that each disk in D intersects one of these two disks. Then we study the problem of covering the set D by two smallest congruent disks.

On Romeo and Juliet Problems: Minimizing Distance-to-Sight

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points

Largest inscribed rectangles in convex polygons

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Convex transversals

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SHORTEST PATH TO A SEGMENT AND QUICKEST VISIBILITY QUERIES

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Notes on Convex Transversals

in a simple polygon

Largest Inscribed Rectangles in Convex Polygons (Extended Abstract)

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