Showing papers by "Kurt Mehlhorn published in 2001"
TL;DR: It is shown that furthest site abstract Voronoi diagrams are trees, have linear size, and that, given a set of $n$ sites, the furthestsite abstract Vor onoi diagram can be computed by a randomized algorithm in expected time.
Abstract: Voronoi diagrams were introduced by R. Klein as a unifying approach to Voronoi diagrams. In this paper we study furthest site abstract Voronoi diagrams and give a unified mathematical and algorithmic treatment for them. In particular, we show that furthest site abstract Voronoi diagrams are trees, have linear size, and that, given a set of $n$ sites, the furthest site abstract Voronoi diagram can be computed by a randomized algorithm in expected time $O(n\log n)$.
67 citations
28 Aug 2001
TL;DR: A new separation bound for real algebraic expressions is proved and it is used in the sign test of the number type leda_real, a task vital for the implementation of geometric algorithms.
Abstract: Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k- th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions.We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda_real.
49 citations
01 Jan 2001
TL;DR: The number type leda_real provides exact computation for a subset of real algebraic numbers that is successfully used to solve precision and robustness problems in geometric computing and is particularly advantageous when used in combination with the computational geometry algorithms library CGAL.
Abstract: The number type leda_real provides exact computation for a subset of real algebraic numbers: Every integer is a leda_real, and leda_reals are closed under the basic arithmetic operations +, -, *, / and k-th root operations. leda_reals guarantee correct results in all comparison operations. The number type is available as part of the LEDA C++ software library of efficient data types and algorithms (LEDA, Mehlhorn and Nuher 2000). leda_reals provide user-friendly exact computation. All the internals are hidden to the user. A user can use leda_reals just like any buHt-in number type. The number type is successfully used to solve precision and robustness problems in geometric computing (Burnikel et al. 2000, Seel). It is particularly advantageous when used in combination with the computational geometry algorithms library CGAL.
31 citations
TL;DR: It is shown that the well-known random incremental construction of Clarkson and Shor18 can be adapted to provide efficient external-memory algorithms for some geometric problems.
Abstract: We show that the well-known random incremental construction of Clarkson and Shor18 can be adapted to provide efficient external-memory algorithms for some geometric problems. In particular, as the ...
23 citations
28 Aug 2001
TL;DR: A heuristic is described that leads to a significant improvement in running time for the weighted matching problem; in experiments a speed-up by up to a factor of 10 was achieved.
Abstract: We consider the single-source many-targets shortest-path (SSMTSP) problem in directed graphs with non-negative edge weights. A source node s and a target set T is specified and the goal is to compute a shortest path from s to a node in T. Our interest in the shortest path problem with many targets stems from its use in weighted bipartite matching algorithms. A weighted bipartite matching in a graph with n nodes on each side reduces to n SSMTSP problems, where the number of targets varies between n and 1.
The SSMTSP problem can be solved by Dijkstra's algorithm. We describe a heuristic that leads to a significant improvement in running time for the weighted matching problem; in our experiments a speed-up by up to a factor of 10 was achieved. We also present a partial analysis that gives some theoretical support for our experimental findings.
19 citations
05 Jan 2001
TL;DR: The flexibility and the use of this generic package for resource constrained network optimization problems is illustrated by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem.
Abstract: We present a generic package for resource constrained network optimization problems. We illustrate the flexibility and the use of our package by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem.
13 citations
09 Jan 2001
TL;DR: This paper gives a polynomial time algorithm testing configurability of dominance graphs (and thus satisfiability of normal dominance constraints) and shows that this problem can be reduced in linear time to the configuration problem of dominanceGraphs.
Abstract: Dominance constraints are logical tree descriptions originating from automata theory that have multiple applications in computational linguistics. The satisfiability problem of dominance constraints is NP-complete. In most applications, however, only normal dominance constraints are used. The satisfiability problem of normal dominance constraints can be reduced in linear time to the configuration problem of dominance graphs, as shown recently. In this paper, we give a polynomial time algorithm testing configurability of dominance graphs (and thus satisfiability of normal dominance constraints). Previous to our work no polynomial time algorithms were known.
10 citations
01 Jan 2001
TL;DR: In this article, the authors present a generic package for resource constrained network optimization problems, such as route planning, curve approximation, minimum cost reliability constrained spanning trees and table layout problem.
Abstract: We present a generic package for resource constrained network optimization problems. We illustrate the flexibility and the use of our package by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem.
7 citations
01 Jan 2001
4 citations
TL;DR: The area of algorithmics has made significant progress over the past ten years, and many improved algorithms were found, for example, the O(nm) barrier for maximum network flow, which resisted for more than three decades, was finally broken.
Abstract: The area of algorithmics has made significant progress over the past ten years. - New concepts were invented and flourished: probabilistically checkable proof, non-approximability, approximation algorithms, on-line algorithms, to name some of them. The books [Hoc96, ACG+99, BEY98] give accounts. - Many improved algorithms were found, for example, the O(nm) barrier for maximum network flow, which resisted for more than three decades, was finally broken [GR98]. - Programs became first class citizens. Ten years ago, algorithmicists considered programs only as an afterthought. Our work was done, when algorithms were formulated and analyzed. Today, algorithmicists also think about implementations, develop software libraries, and experiment with algorithms. Many algorithms are now available in software libraries. Think of CPLEX, MAPLE, MATHEMATICA, CGAL, LEDA, ILOG-solver. I was involved in the design of LEDA [LED] and CGAL [CGA] and want to share with you some of the experiences made. A much more detailed account can be found in [MN99].