Approximated algorithms for the minimum dilation triangulation problem

2014 ◽  
Vol 20 (2) ◽  
pp. 189-209 ◽  
Author(s):  
Maria Gisela Dorzán ◽  
Mario Guillermo Leguizamón ◽  
Efrén Mezura-Montes ◽  
Gregorio Hernández-Peñalver
Keyword(s):  
Minimum Dilation

Computing Minimum Dilation Spanning Trees in Geometric Graphs

2015 ◽  
pp. 297-309
Author(s):  
Aléx F. Brandt ◽  
Miguel F. A. de Gaiowski ◽  
Pedro J. de Rezende ◽  
Cid C. de Souza
Keyword(s):  
Spanning Trees ◽  
Geometric Graphs ◽  
Minimum Dilation

Embedding of Hypercubes into l-Sibling Trees

2013 ◽  
Vol 14 (04) ◽  
pp. 1350018
Author(s):  
R. SUNDARA RAJAN
Keyword(s):  
Fault Tolerance ◽  
Parallel Computation ◽  
Interconnection Networks ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Structural Regularity ◽  
Minimum Dilation ◽  
High Degree

The hypercube is one of the most popular interconnection networks due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. In this paper, we introduce a tree called l-sibling trees and the main results obtained in this paper are: (1) For r ≥ 1, the minimum wirelength of embedding r-dimensional hypercube into r-dimensional l-sibling tree. (2) For r ≥ 1, embedding of r-dimensional extended l-sibling tree into caterpillar with minimum dilation. (3) Based on the proof of (1), we provide an O(r)-linear time algorithm to compute the minimum wirelength of embedding r-dimensional hypercube into r-dimensional l-sibling tree.

scholarly journals Computing a minimum-dilation spanning tree is NP-hard

2008 ◽  
Vol 41 (3) ◽  
pp. 188-205 ◽  
Author(s):  
Otfried Cheong ◽  
Herman Haverkort ◽  
Mira Lee
Keyword(s):  
Spanning Tree ◽  
Np Hard ◽  
Minimum Dilation

scholarly journals COMPUTING GEOMETRIC MINIMUM-DILATION GRAPHS IS NP-HARD

2010 ◽  
Vol 20 (02) ◽  
pp. 147-173 ◽  
Author(s):  
PANOS GIANNOPOULOS ◽  
ROLF KLEIN ◽  
CHRISTIAN KNAUER ◽  
MARTIN KUTZ ◽  
DÁNIEL MARX
Keyword(s):  
Hard Problem ◽  
Np Hard ◽  
Np Hard Problem ◽  
Minimum Dilation ◽  
Set Of Points ◽  
Edge Crossings

We prove that computing a geometric minimum-dilation graph on a given set of points in the plane, using not more than a given number of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. We also show that the problem remains NP-hard even when a minimum-dilation tour or path is sought; not even an FPTAS exists in this case.

scholarly journals Minimum dilation stars

2007 ◽  
Vol 37 (1) ◽  
pp. 27-37 ◽  
Author(s):  
David Eppstein ◽  
Kevin A. Wortman
Keyword(s):  
Minimum Dilation

An exact algorithm for the minimum dilation triangulation problem

2017 ◽  
Vol 69 (2) ◽  
pp. 343-367 ◽  
Author(s):  
Sattar Sattari ◽  
Mohammad Izadi
Keyword(s):  
Exact Algorithm ◽  
Minimum Dilation

scholarly journals On Perturbation Resilience of Non-uniform k-Center

Algorithmica ◽  
2021 ◽  
Author(s):  
Sayan Bandyapadhyay
Keyword(s):  
Approximation Algorithm ◽  
Metric Space ◽  
The Other ◽  
Negative Numbers ◽  
Clustering Problem ◽  
Fixed Set ◽  
Tree Metrics ◽  
Perturbation Resilience ◽  
Minimum Dilation ◽  
Special Case

AbstractThe Non-Uniform k-center (NUkC) problem has recently been formulated by Chakrabarty et al. [ICALP, 2016; ACM Trans Algorithms 16(4):46:1–46:19, 2020] as a generalization of the classical k-center clustering problem. In NUkC, given a set of n points P in a metric space and non-negative numbers $$r_1, r_2, \ldots , r_k$$ r 1 , r 2 , … , r k , the goal is to find the minimum dilation $$\alpha $$ α and to choose k balls centered at the points of P with radius $$\alpha \cdot r_i$$ α · r i for $$1\le i\le k$$ 1 ≤ i ≤ k , such that all points of P are contained in the union of the chosen balls. They showed that the problem is $$\mathsf {NP}$$ NP -hard to approximate within any factor even in tree metrics. On the other hand, they designed a “bi-criteria” constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the $$r_i$$ r i ’s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial (Comb Probab Comput 21(5):643–660, 2012). We show that the problem under 2-perturbation resilience is polynomial time solvable when the $$r_i$$ r i ’s belong to a constant-sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any “good” approximation for the problem.

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