Given a set S of n points in the plane and a parameter ε>0, a Euclidean (1+ε)
spanner is a geometric graph G=(S,E) that contains a path of weight at most (1+ε)∥pq∥2
for all p,q∈S . We show that the minimum weight of a Euclidean (1+ε)spanner for n points in the unit square [0,1]2 is O(ε−3/2n−−√), and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yaographs. It improves upon the baseline O(ε−2n−−√), obtained by combining a tight bound for the weight of an MST and a tight bound for the lightness of Euclidean (1+ε)spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to dspace for all d∈N
: The minimum weight of a Euclidean (1+ε)spanner for n points in the unit cube [0,1]d
is Od(ε(1−d2)/dn(d−1)/d), and this bound is the best possible. For the n×n section of the integer lattice, we show that the minimum weight of a Euclidean (1+ε)spanner is between Ω(ε−3/4n2) and O(ε−1log(ε−1)n2). These bounds become Ω(ε−3/4n−−√) and O(ε−1log(ε−1)n−−√) when scaled to a grid of n points in [0,1]2.
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Computing βStretch Paths in Drawings of Graphs
Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π = f(P), is βstretch if π is a simple (nonselfintersecting) curve, and for every pair of distinct points p ∈ P and q ∈ P , the length of the subcurve of π connecting f(p) with f(q) is at most β∥f(p) − f(q)∥, where ∥.∥ denotes the Euclidean distance. We introduce and study the βstretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a βstretch path P connecting s and t. We also output P if it exists. The βSP quantifies a notion of “near straightness” for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a βstretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NPcomplete. We complement this result by giving a quasipolynomial time algorithm, that for a given ε > 0, β ∈ O(poly(log V (G))), and s, t ∈ V (G), outputs a βstretch path between s and t, if a (1 − ε)βstretch path between s and t exists in the drawing.
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 NSFPAR ID:
 10179493
 Date Published:
 Journal Name:
 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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