06481 Abstracts Collection -- Geometric Networks and Metric Space Embeddings

The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space Embeddings\u27\u27 was held from November~26 to December~1, 2006 in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. In this paper we describe the seminar topics, we have compiled a list of open questions that were posed during the seminar, there is a list of all talks and there are abstracts of the presentations given during the seminar. Links to extended abstracts or full papers are provided where available

Stream implementation of serial morphological filters with approximated polygons

ISBN : 978-142448157-6International audienceThis paper describes an original stream implementation of serially composed morphological filters using approximated flat polygons. It strictly respects a sequential data access. Results are obtained with minimal latency while operating within minimal memory space; even for very large neighborhoods. This is interesting for serially composed advanced filters, such as Alternating Sequential Filters or granulometries. We show how the dedicated implementation on an FPGA allows obtaining a previously unequaled performance, opening an opportunity to use these operators in time-critical, high-end applications

Improving the dilation of a metric graph by adding edges

Most of the literature on spanners focuses on building the graph from scratch. This paper instead focuses on adding edges to improve an existing graph. A major open problem in this field is: given a graph embedded in a metric space, and a budget of k edges, which k edges do we add to produce a minimum-dilation graph? The special case where k=1 has been studied in the past, but no major breakthroughs have been made for k > 1. We provide the first positive result, an O(k)-approximation algorithm that runs in O(n^3 \log n) time

Feed-links for network extensions

Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis

A Near-Optimal Algorithm for Finding an Optimal Shortcut of a Tree

We consider the problem of finding a shortcut connecting two vertices of a graph that minimizes the diameter of the resulting graph. We present an O(n^2 log^3 n)-time algorithm using linear space for the case that the input graph is a tree consisting of n vertices. Additionally, we present an O(n^2 log^3 n)-time algorithm using linear space for a continuous version of this problem

Expansive Motions and the Polytope of Pointed Pseudo-Triangulations

We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an n-gon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1-dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a by-product a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially to the "Further remarks" in Section 5) + changed to final book format. This version is to appear in "Discrete and Computational Geometry -- The Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds), series "Algorithms and Combinatorics", Springer Verlag, Berli