395 research outputs found
Covering Paths and Trees for Planar Grids
Given a set of points in the plane, a covering path is a polygonal path that
visits all the points. In this paper we consider covering paths of the vertices
of an n x m grid. We show that the minimal number of segments of such a path is
except when we allow crossings and , in which case the
minimal number of segments of such a path is , i.e., in this case
we can save one segment. In fact we show that these are true even if we
consider covering trees instead of paths.
These results extend previous works on axis-aligned covering paths of n x m
grids and complement the recent study of covering paths for points in general
position, in which case the problem becomes significantly harder and is still
open
SOLVING THE 106 YEARS OLD 3^k POINTS PROBLEM WITH THE CLOCKWISE-ALGORITHM
In this paper, we present the clockwise-algorithm that solves the extension in -dimensions of the infamous nine-dot problem, the well known two-dimensional thinking outside the box puzzle. We describe a general strategy that constructively produces minimum length covering trails, for any ∈ N−{0}, solving the NP-complete (3×3×⋯×3)-points problem inside a 3×3×⋯×3 hypercube. In particular, using our algorithm, we explicitly draw different covering trails of minimal length h() = (3^ − 1)/2, for = 3, 4, 5. Furthermore, we conjecture that, for every ≥ 1, it is possible to solve the 3^-points problem with h() lines starting from any of the 3^ nodes, except from the central one. Finally, we cover 3×3×3 points with a tree of size 12
Solving the years old points problem with the clockwise-algorithm
In this paper, we present the clockwise-algorithm that solves the extension
in -dimensions of the infamous nine-dot problem, the well-known
two-dimensional thinking outside the box puzzle. We describe a general strategy
that constructively produces minimum length covering trails, for any , solving the NP-complete -point problem inside hypercubes. In
particular, using our algorithm, we explicitly draw different covering trails
of minimal length , for . Furthermore, we
conjecture that, for every , it is possible to solve the -point
problem with lines starting from any of the nodes, except from the
central one. Finally, we cover a grid with a tree of size
.Comment: 17 pages, 12 figures. A video animation of the solution from 1 to 4
dimensions can be found on YouTube
(https://www.youtube.com/watch?v=SSL9R0hQRKM
On Euclidean Steiner (1+?)-Spanners
Lightness and sparsity are two natural parameters for Euclidean (1+?)-spanners. Classical results show that, when the dimension d ? ? and ? > 0 are constant, every set S of n points in d-space admits an (1+?)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ? > 0 for constant d ? ? have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+?)-spanner. They gave upper bounds of O?(?^{-(d+1)/2}) for the minimum lightness in dimensions d ? 3, and O?(?^{-(d-1))/2}) for the minimum sparsity in d-space for all d ? 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+?)-spanners of lightness O(?^{-1}log?) in the plane, where ? ? ?(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points.
In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+?)-spanners. Using a new geometric analysis, we establish lower bounds of ?(?^{-d/2}) for the lightness and ?(?^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ? 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+?)-spanners of lightness O(?^{-1}log n) for n points in Euclidean plane
Light Euclidean Steiner Spanners in the Plane
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio
of the spanner weight to the weight of the minimum spanning tree of a finite
set of points in . In a recent breakthrough, Le and Solomon
(2019) established the precise dependencies on and of the minimum lightness of -spanners, and
observed that additional Steiner points can substantially improve the
lightness. Le and Solomon (2020) constructed Steiner -spanners
of lightness in the plane, where is the \emph{spread} of the point set, defined as the ratio
between the maximum and minimum distance between a pair of points. They also
constructed spanners of lightness in
dimensions . Recently, Bhore and T\'{o}th (2020) established a lower
bound of for the lightness of Steiner
-spanners in , for . The central open
problem in this area is to close the gap between the lower and upper bounds in
all dimensions .
In this work, we show that for every finite set of points in the plane and
every , there exists a Euclidean Steiner
-spanner of lightness ; this matches the
lower bound for . We generalize the notion of shallow light trees, which
may be of independent interest, and use directional spanners and a modified
window partitioning scheme to achieve a tight weight analysis.Comment: 29 pages, 14 figures. A 17-page extended abstract will appear in the
Proceedings of the 37th International Symposium on Computational Geometr
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