2,214 research outputs found
Algorithms for the power-p Steiner tree problem in the Euclidean plane
We study the problem of constructing minimum power- Euclidean -Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of (where ), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio of the beaded-MST heuristic satisfies . We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the case
The Unreasonable Success of Local Search: Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems
in the plane? We prove that local search with neighborhoods of magnitude
is an approximation scheme for the following problems in the
Euclidian plane: TSP with random inputs, Steiner tree with random inputs,
facility location (with worst case inputs), and bicriteria -median (also
with worst case inputs). The randomness assumption is necessary for TSP
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Colored Non-Crossing Euclidean Steiner Forest
Given a set of -colored points in the plane, we consider the problem of
finding trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
, this is the well-known Euclidean Steiner tree problem. For general ,
a -approximation algorithm is known, where is the
Steiner ratio.
We present a PTAS for , a -approximation algorithm
for , and two approximation algorithms for general~, with ratios
and
Concurrent Geometric Multicasting
We present MCFR, a multicasting concurrent face routing algorithm that uses
geometric routing to deliver a message from source to multiple targets. We
describe the algorithm's operation, prove it correct, estimate its performance
bounds and evaluate its performance using simulation. Our estimate shows that
MCFR is the first geometric multicast routing algorithm whose message delivery
latency is independent of network size and only proportional to the distance
between the source and the targets. Our simulation indicates that MCFR has
significantly better reliability than existing algorithms
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