172 research outputs found
Planar bichromatic minimum spanning trees
AbstractGiven a set S of n red and blue points in the plane, a planar bichromatic minimum spanning tree is the shortest possible spanning tree of S, such that every edge connects a red and a blue point, and no two edges intersect. We show that computing this tree is NP-hard in general. For points in convex position, a cubic-time algorithm can be easily designed using dynamic programming. We adapt such an algorithm for the special case where the number of red points (m) is much smaller than the number of blue points (n), resulting in an O(nm2) time algorithm. For the general case, we present a factor O(n) approximation algorithm that runs in O(nlognloglogn) time. Finally, we show that if the number of points in one color is bounded by a constant, the optimal tree can be computed in polynomial time
Diseño de un algoritmo de minerĂa metaheurĂstico para separar puntos de dos colores en un entorno bidimensional
The separation of color points is one of the important issues in computational geometry, which is used in various parts of science; it can be used in facility locating, image processing and clustering. Among these, one of the most widely used computational geometry in the real-world is the problem of covering and separating points with rectangles. In this paper, we intend to consider the problemof separating the two-color points sets, using three rectangles. In fact, our goal is to separate desired blue points from undesired red points by three rectangles, in such a way that these three rectangles contain the most desire points. For this purpose, we provide a metaheuristic algorithm based on the simulated annealing method that could separates blue points from input points, , in time order O (n) with the help of three rectangles. The algorithm is executed with C# and also it has been compared and evaluated with the optimum algorithm results. The results show that our recommended algorithm responses is so close to optimal responses, and also in some cases we obtains the exact optimal response
New results on stabbing segments with a polygon
We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loftier and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard. (C) 2014 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author's final draft
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New Applications of the Nearest-Neighbor Chain Algorithm
The nearest-neighbor chain algorithm was proposed in the eighties as a way to speed up certain hierarchical clustering algorithms. In the first part of the dissertation, we show that its application is not limited to clustering. We apply it to a variety of geometric and combinatorial problems. In each case, we show that the nearest-neighbor chain algorithm finds the same solution as a preexistent greedy algorithm, but often with an improved runtime. We obtain speedups over greedy algorithms for Euclidean TSP, Steiner TSP in planar graphs, straight skeletons, a geometric coverage problem, and three stable matching models. In the second part, we study the stable-matching Voronoi diagram, a type of plane partition which combines properties of stable matchings and Voronoi diagrams. We propose political redistricting as an application. We also show that it is impossible to compute this diagram in an algebraic model of computation, and give three algorithmic approaches to overcome this obstacle. One of them is based on the nearest-neighbor chain algorithm, linking the two parts together
On k-enclosing objects in a coloured point set
We introduce the exact coloured
k
-enclosing object
problem: given a set
P
of
n
points in
R
2
, each of
which has an associated colour in
f
1
;:::;t
g
, and a vec-
tor
c
= (
c
1
;:::;c
t
), where
c
i
2
Z
+
for each 1
i
t
,
nd a region that contains exactly
c
i
points of
P
of
colour
i
for each
i
. We examine the problems of nd-
ing exact coloured
k
-enclosing axis-aligned rectangles,
squares, discs, and two-sided dominating regions in a
t
-coloured point setPostprint (published version
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