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Minimum Weight Euclidean t-spanner is NP-Hard
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In this paper, it was shown that the problem of deciding whether there exists an Euclidean t-spanner, for a given set of points in the plane, of weight at most w is NP-hard for every real constant t > 1.Abstract:
Given a set P of points in the plane, an Euclidean t-spanner for P is a geometric graph that preserves the Euclidean distances between every pair of points in P up to a constant factor t. The weight of a geometric graph refers to the total length of its edges. In this paper we show that the problem of deciding whether there exists an Euclidean t-spanner, for a given set of points in the plane, of weight at most w is NP-hard for every real constant t > 1, both whether planarity of the t-spanner is required or not.read more
Citations
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Proceedings ArticleDOI
Computing a Minimal Set of t-Spanning Motion Primitives for Lattice Planners
TL;DR: This paper provides a proof that the minimal t–spanning control set problem for a lattice defined over an arbitrary robot configuration space is NP-complete, and presents a compact mixed integer linear programming formulation to compute an optimal t-spanner.
Proceedings ArticleDOI
Online Euclidean Spanners
Sujoy Bhore,Csaba D. Tóth +1 more
TL;DR: It is proved that any online spanner algorithm for a sequence of n points in ℝ^d under the L₂ norm has competitive ratio Ω(f(n), where lim_{n → ∞} f(n) = ∞.
Journal ArticleDOI
Minimizing the Sum of Distances to a Server in a Constraint Network
TL;DR: A bi-criteria approximation algorithm is presented, that approximates both the weight and the cost of the network with respect to an optimal network, in ( | V | / e) O ( h / e ) .
References
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Book ChapterDOI
Reducibility Among Combinatorial Problems
TL;DR: The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible.
Book
Geometric Spanner Networks
Giri Narasimhan,Michiel Smid +1 more
TL;DR: In this paper, the authors present rigorous descriptions of the main algorithms and their analyses for different variations of the Geometric Spanner Network Problem, and present several basic principles and results that are used throughout the book.
Journal ArticleDOI
Tree Spanners
Leizhen Cai,Derek G. Corneil +1 more
TL;DR: It is shown that a tree 1-spanner, if it exists, in a weighted graph with $m$ edges and $n$ vertices is a minimum spanning tree and can be found in $O(m \log \beta(m, n)$ time, and the problem of determining the existence of a tree $t$- spanner in a Weighted graph is proven to be NP-complete.
Journal ArticleDOI
A fast algorithm for constructing sparse euclidean spanners
Gautam Das,Giri Narasimhan +1 more
TL;DR: An O(n log2 n) time algorithm which, given a set V of n points in k-dimensional space (for any fixed k), and any real constant t > 1, produces a t-spanner of the complete Euclidean graph of V, which is similar to the size and weight of spanners constructed by the greedy algorithm.
Journal ArticleDOI
New sparseness results on graph spanners
TL;DR: This paper shows that for an arbitrary positive edge-weighted graph G, for any t>1, and any ∈>0, a t-spanner of G with weight can be constructed in polynomial time, and shows that (log2 n)-spanners of weight O(1) · wt(MST) can be constructing.