Computing the Maximum Detour of a Plane Geometric Graph in Subquadratic Time

by Christian Wulff-Nilsen

Abstract:

Let G be a plane geometric graph where each edge is a line segment. We consider the problem of computing the maximum detour of G, defined as the maximum over all pairs of distinct points (vertices as well as interior points of edges) p and q of G of the ratio between the distance between p and q in G and the Euclidean distance ||pq||2. The fastest known algorithm for this problem has Θ(n2) running time where n is the number of vertices. We show how to obtain O(n3/2(log n)3) expected running time. We also show that if G has bounded treewidth, its maximum detour can be computed in O(n(log n)3) expected time.

## Tuesday, November 30, 2010

## Sunday, November 21, 2010

### JoCG paper

Posted by
Joachim
at
7:43 PM

New paper published in JoCG:

COMPUTING MULTIDIMENSIONAL PERSISTENCE

Gunnar Carlsson, Gurjeet Singh, Afra J. Zomorodian

ABSTRACT

The theory of multidimensional persistence captures the topology of a multifiltration - a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence. We recast this computation as a problem within computational commutative algebra and utilize algorithms from this area to solve it. While the resulting problem is EXPSPACE-complete and the standard algorithms take doubly-exponential time, we exploit the structure inherent withing multifiltrations to yield practical algorithms. We implement all algorithms in the paper and provide statistical experiments to demonstrate their feasibility.

COMPUTING MULTIDIMENSIONAL PERSISTENCE

Gunnar Carlsson, Gurjeet Singh, Afra J. Zomorodian

ABSTRACT

The theory of multidimensional persistence captures the topology of a multifiltration - a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence. We recast this computation as a problem within computational commutative algebra and utilize algorithms from this area to solve it. While the resulting problem is EXPSPACE-complete and the standard algorithms take doubly-exponential time, we exploit the structure inherent withing multifiltrations to yield practical algorithms. We implement all algorithms in the paper and provide statistical experiments to demonstrate their feasibility.

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