共查询到20条相似文献,搜索用时 24 毫秒
1.
Tight Bounds for Connecting Sites Across Barriers 总被引:1,自引:0,他引:1
David Krumme Eynat Rafalin Diane L. Souvaine Csaba D. Tóth 《Discrete and Computational Geometry》2008,40(3):377-394
Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight line minimum cost spanning tree on the sites, where the cost
is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines,
it is known that there is a spanning tree such that every barrier is crossed by
tree edges, and this bound is asymptotically optimal. Asano et al. showed that if the barriers are pairwise disjoint line
segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Lower bound constructions are known with 3 crossings per barrier and 2n total cost.
We obtain tight bounds on the minimum cost of spanning trees in the special case where the barriers are interior disjoint
line segments that form a convex subdivision of the plane and there is a point in every cell of the subdivision. In particular,
we show that there is a spanning tree such that every barrier crosses at most 2 tree edges, and there is a spanning tree of
total cost 5n/3. Both bounds are the best possible.
Work by Eynat Rafalin and Diane Souvaine was supported by the National Science Foundation under Grant #CCF-0431027.
E. Rafalin’s research conducted while at Tufts University. 相似文献
2.
It is shown that a set of $n$ disjoint line segments in the plane can
always be illuminated by $\lfloor (n+1)/2\rfloor$ light sources,
improving an earlier bound of $\lfloor 2n/3\rfloor$ due to Czyzowicz
et al. It is also shown that $\lfloor 4(n+1)/5 \rfloor$ light sources
are always sufficient and sometimes necessary to illuminate the free
space and both sides of $n$ disjoint line segments for every $n\geq
2$. 相似文献
3.
Every Set of Disjoint Line Segments Admits a Binary Tree 总被引:1,自引:0,他引:1
Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments
such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree
is 3. Furthermore, there exist configurations of line segments where any such tree requires degree 3. We provide an O(nlog n) time algorithm for constructing such a tree, and show that this is optimal.
Received September 14, 1999, and in revised form January 17, 2001. Online publication August 29, 2001. 相似文献
4.
The interior Dirichlet problem for Laplace's equation on a plane polygonal region $\Omega$ with boundary $\Gamma$ may be reformulated as a second kind integral equation on $\Gamma$. This equation may be solved by the Nyström method using the composite trapezoidal rule. It is known that if the mesh has $O(n)$ points and is graded appropriately, then $O(1/n^2)$ convergence is obtained for the solution of the integral equation and the associated solution to the Dirichlet problem at any $x\in \Omega$. We present a simple extrapolation scheme which increases these rates of convergence to $O(1/n^4)$ . 相似文献
5.
Adrian Dumitrescu Joseph S. B. Mitchell Micha Sharir 《Discrete and Computational Geometry》2004,31(2):207-227
We provide a variety of new upper and lower bounds
and simpler proof techniques for the efficient construction of
binary space partitions (BSPs) of axis-parallel rectangles
of various dimensions.
(a) We construct a set of $n$ disjoint axis-parallel segments
in the plane such that any binary space auto-partition has size
at least $2n-o(n)$, almost matching an upper bound of dAmore and
Franciosa.
(b) We establish a similar lower bound of
$7n/3-o(n)$ for disjoint rectangles in the plane.
(c) We simplify and improve BSP constructions of Paterson and Yao for
disjoint segments in $\reals^d$ and disjoint rectangles in $\reals^3$.
(d) We derive a worst-case bound of $\Theta(n^{5/3})$ for the size of
BSPs of disjoint $2$-rectangles in $4$-space.
(e) For disjoint $k$-rectangles in $d$-space, we prove the worst-case
bound $\Theta(n^{d/(d-k)})$, for any $k<d/2$; this bound holds for all
$k<d$ if the rectangles are allowed to intersect. 相似文献
6.
A spanning tree with no more than 3 leaves is called a spanning 3-ended tree.In this paper, we prove that if G is a k-connected(k ≥ 2) almost claw-free graph of order n and σ_(k+3)(G) ≥ n + k + 2, then G contains a spanning 3-ended tree, where σk(G) =min{∑_(v∈S)deg(v) : S is an independent set of G with |S| = k}. 相似文献
7.
Eynat Rafalin Diane L. Souvaine Csaba D. Tóth 《Discrete and Computational Geometry》2010,43(2):221-241
We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈?, a $\frac{1}{r}We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈ℕ, a
\frac1r\frac{1}{r}
-cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in ℝ3 and a parameter r∈ℕ, we construct a
\frac1r\frac{1}{r}
-cutting of size O(r
2). For n axis-aligned rectangles in ℝ3, we construct a
\frac1r\frac{1}{r}
-cutting of size O(r
3/2).
As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across
barriers. Given n points and a finite set of disjoint disk barriers in ℝ3, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most
O(?n)O(\sqrt{n})
edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n
1/3) edges. Both bounds are best possible. 相似文献
8.
Guo-Rong Wang & Sen-Quan Lu 《计算数学(英文版)》1988,6(4):348-354
The parallel arithmetic complexities for computing generalized inverse $A^+$, computing the minimum-norm least-squares solution of $Ax=b$, computing order $m+n-r$ determinants and finding the characteristic polynomials of order $m+n-r$ matrices are shown to have the same grawth rate. Algorithms are given that compute $A^+$ and $A_{MN}^+$ in $O(\log r\dot \log n+\log m)$ and $O(\log^2n+\log m)$ steps using a number of processors which is a polynomial in $m, \ n$ and $r$ $(A\in B_r^{m\times n},r=rank \ A)$. 相似文献
9.
Given a nonconvex simple polygon $P$ with $n$ vertices, is it possible to construct a data structure which after preprocessing can answer halfspace area queries (i.e., given a line, determine the area of the portion of the polygon above the line) in $o(n)$ time? We answer negatively, proving an $\Omega(n)$ lower bound on the query time of any data structure performing this task. We then consider the offline version of the same problem: given a polygon $P$ with $n$ vertices, and $k$ query lines, we present an algorithm that computes the area of $P$ on both sides of each line in $O(k^{2/3}n^{2/3+\varepsilon}+(n+k)\polylog{n})$ time. Variants of this method allow the query of a collection of weighted polygons with or without holes, and solve several other related problems within the same time bounds. 相似文献
10.
Lie-Heng Wang 《计算数学(英文版)》1983,1(2):99-105
In this paper, we estimate the error of the linear finite element solutions of the obstacle problem and the unilateral problem with monotone operator. We obtained $O(h)$ error bound for the obstacle problem and $O(h^{3/4})$ error bound for the unilateral problem. And if the solution $u^*$ of the unilateral problem possesses more smoothness, then $O(h)$ error bound can be obtained in the same way as [2]. 相似文献
11.
Csaba D. Tth 《Computational Geometry》2002,21(3):527-204
What is the minimal number of light sources which is always sufficient to illuminate the plane in the presence of n disjoint opaque line segments? For n5, O'Rourke proved that 2n/3 light sources are always sufficient and sometimes necessary, if light sources can be placed on the line segments and thus they can illuminate both sides of a segment.
We prove that 2(n+1)/3 light sources are always sufficient and sometimes necessary, if light sources cannot be placed on the line segments. An O(nlogn) time algorithm is presented which allocates at most 2(n+1)/3 light sources collectively illuminating the plane. 相似文献
12.
Sándor P. Fekete Marco E. Lübbecke Henk Meijer 《Discrete and Computational Geometry》2008,40(4):595-621
The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected
by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum
stabbing number for a given set of vertices. The complexity of finding a spanning tree of minimum stabbing number is one of
the original 30 questions on “The Open Problems Project” list of outstanding problems in computational geometry by Demaine,
Mitchell, and O’Rourke.
We show
-hardness of stabbing problems by means of a general proof technique. For matchings, this also implies a nontrivial lower
bound on the approximability. On the positive side, we propose a cut-based integer programming formulation for minimizing
the stabbing number of matchings and spanning trees. From the corresponding linear programming relaxation we obtain polynomial-time
lower bounds and show that there always is an optimal fractional solution that contains an edge of at least constant weight.
We conjecture that the resulting iterated rounding scheme constitutes a constant-factor approximation algorithm.
An extended abstract appeared in the Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms [11].
M.E. Lübbecke visits to Kingston and Stony Brook were supported by a DFG travel grant.
H. Meijer partially supported by NSERC while visiting Braunschweig in 2002. 相似文献
13.
Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth 《Discrete and Computational Geometry》2013,49(1):89-131
We prove that for every set of n pairwise disjoint line segments in the plane in general position, where n is even, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzer et al. (Comput. Geom. 42:617–626, 2009). The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees. 相似文献
14.
A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth 5 proved in any proper edge coloring of the complete graph K2n(n > 2) with 2n ? 1 colors, there are two edge‐disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a1,…, ak) is a color distribution for the complete graph Kn, n ≥ 5, such that , then there exist two edge‐disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph Kn with the above distribution if T is a non‐star multicolored spanning tree of Kn, then there exists a multicolored spanning tree T' of Kn such that T and T' are edge‐disjoint. Also it is shown that if Kn, n ≥ 6, is edge colored with k colors and , then there exist two edge‐disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221–232, 2007 相似文献
15.
Jeroen Kuipers 《International Journal of Game Theory》1997,26(3):367-377
In this paper we consider a generalization of the minimum cost spanning tree game. The generalized model allows for more than one supplier, where each supplier offers a different type of service to the customers and each customer specifies a non-empty subset of these suppliers to which he wishes to be connected. We show that the core of such a game may be empty, but that it is always non-empty if there is at least one customer who wants to be connected to all suppliers. Furthermore, the core is always non-empty if there are at most two suppliers. 相似文献
16.
We address the asymptotic behaviour of the vibrations of a body occupying a domain $\Omega\subset\mathbb{R}^n, n=2,3$. The density, which depends on a small parameter $\varepsilon$\nopagenumbers\end , is of the order $O(1)$\nopagenumbers\end out of certain regions where it is $O(\varepsilon^{‐m})$\nopagenumbers\end with $m>2$\nopagenumbers\end . These regions, the concentrated masses with diameter $O(\varepsilon)$\nopagenumbers\end , are located near the boundary, at mutual distances $O(\eta)$\nopagenumbers\end , with $\eta=\eta(\varepsilon)\rightarrow 0$\nopagenumbers\end . We impose Dirichlet (resp. Neumann) conditions at the points of $\partial\Omega$\nopagenumbers\end in contact with (resp. out of) the masses. We look at the asymptotic behaviour, as $\varepsilon\rightarrow 0$\nopagenumbers\end , of the eigenvalues of order $O(1)$\nopagenumbers\end , the high frequencies, of the corresponding eigenvalue problem. We show that they accumulate on the whole positive real axis and characterize those giving rise to global vibrations of the whole system. We use the fact that the corresponding eigenfunctions, microscopically, present a skin effect in the concentrated masses. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
17.
Let G be a graph with n(G) vertices and m(G) be its matching number.The nullity of G,denoted by η(G),is the multiplicity of the eigenvalue zero of adjacency matrix of G.It is well known that if G is a tree,then η(G) = n(G)-2m(G).Guo et al.[Jiming GUO,Weigen YAN,Yeongnan YEH.On the nullity and the matching number of unicyclic graphs.Linear Alg.Appl.,2009,431:1293 1301]proved that if G is a unicyclic graph,then η(G)equals n(G)-2m(G)-1,n(G)-2m(G),or n(G)-2m(G) +2.In this paper,we prove that if G is a bicyclic graph,then η(G) equals n(G)-2m(G),n(G)-2m(G)±1,n(G)-2m(G)±2or n(G)-2m(G) + 4.We also give a characterization of these six types of bicyclic graphs corresponding to each nullity. 相似文献
18.
设$(M,\,T)$是一个带有光滑对合$T$的光滑闭流形,
$T$在$M$上的不动点集为 $F=\{x\,|\,T(x)=x,\,x\in M\}$,
则$F$为$M$的闭子流形的不交并. 本文证明了: 当$F=P(2m,\,2l+1)\sqcup
P(2m,\,2n+1)$时,其中$n>l\geq m,\,m\neq1,\,3$, $(M,\,T)$协边于零. 相似文献
19.
For any 2-coloring of the segments determined by n points in general position in the plane, at least one of the color classes contains a non-self-intersecting spanning tree.
Under the same assumptions, we also prove that there exist pairwise disjoint segments of the same color, and this bound is tight. The above theorems were conjectured by Bialostocki
and Dierker. Furthermore, improving an earlier result of Larman et al., we construct a family of m segments in the plane, which has no more than members that are either pairwise disjoint or pairwise crossing. Finally, we discuss some related problems and generalizations.
Received October 17, 1995, and in revised form February 10, 1996. 相似文献
20.
A finite set $N \subset \R^d$ is a {\em weak $\eps$-net} for an
$n$-point set $X\subset \R^d$ (with respect to convex sets) if $N$
intersects every convex set $K$ with $|K\,\cap\,X|\geq \eps n$. We
give an alternative, and arguably simpler, proof of the fact, first
shown by Chazelle et al., that every
point set $X$ in $\R^d$ admits a weak $\eps$-net of cardinality
$O(\eps^{-d}\polylog(1/\eps))$. Moreover, for a number of special
point sets (e.g., for points on the moment curve), our method gives
substantially better bounds. The construction
yields an algorithm to construct such weak
$\eps$-nets in time $O(n\ln(1/\eps))$. 相似文献