Results 1  10
of
20
Object Location Using Path Separators
, 2006
"... We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpat ..."
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Cited by 43 (11 self)
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We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpath separable graphs can be used to solve several object location problems: (1) a smallworldization with an average polylogarithmic number of hops; (2) an (1 + ε)approximate distance labeling scheme with O(log n) space labels; (3) a stretch(1 + ε) compact routing scheme with tables of polylogarithmic space; (4) an (1+ε)approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to boundeddimension isometric sparable graphs.
Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
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Cited by 30 (12 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p
Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs
, 2002
"... We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running t ..."
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Cited by 27 (2 self)
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We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running time bounded by an exponential with a sublinear exponent. For precision disk graphs of bounded radius ratio, we show that the problem is xed parameter tractable with respect to parameter k.
A Simple and Fast Approach for Solving Problems on Planar Graphs
 In Proc. 21st STACS, volume 2996 of LNCS
, 2004
"... It is well known that the celebrated LiptonTarjan planar separation theorem, in a combination with a divideandconquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2 , where n is th ..."
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Cited by 17 (2 self)
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It is well known that the celebrated LiptonTarjan planar separation theorem, in a combination with a divideandconquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2 , where n is the number of vertices. However, the constants hidden in bigOh, usually are too large to claim the algorithms to be practical even on graphs of moderate size. Here we introduce a new algorithm design paradigm for solving problems on planar graphs. The paradigm is so simple that it can be explained in any textbook on graph algorithms: Compute tree or branch decomposition of a planar graph and do dynamic programming. Surprisingly such a simple approach provides faster algorithms for many problems. For example, Independent Set on planar graphs can be solved in time O(2 ) and Dominating Set in time O(2 ). In addition, significantly broader class of problems can be attacked by this method. Thus with our approach, Longest cycle on planar graphs is solved in time O(2 and Bisection is solved in time O(2 ). The proof of these results is based on complicated combinatorial arguments that make strong use of results derived by the Graph Minors Theory. In particular we prove that branchwidth of a planar graph is at most 2.122 # n. In addition we observe how a similar approach can be used for solving di#erent fixed parameter problems on planar graphs. We prove that our method provides the best so far exponential speedup for fundamental problems on planar graphs like Vertex Cover, (Weighted) Dominating Set, and many others.
Engineering Planar Separator Algorithms
, 2009
"... We consider classical lineartime planar separator algorithms, determining for a given planar graph a small subset of its nodes whose removal divides the graph into two components of similar size. These algorithms are based on planar separator theorems, which guarantee separators of size O ( √ n) a ..."
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Cited by 8 (3 self)
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We consider classical lineartime planar separator algorithms, determining for a given planar graph a small subset of its nodes whose removal divides the graph into two components of similar size. These algorithms are based on planar separator theorems, which guarantee separators of size O ( √ n) and remaining components of size at most 2n/3 (where n denotes the number of nodes in the graph). In this article, we present a comprehensive experimental study of the classical algorithms applied to a large variety of graphs, where our main goal is to find separators that do not only satisfy upper bounds, but also possess other desirable characteristics with respect to separator size and component balance. We achieve this by investigating a number of specific alternatives for the concrete implementation and finetuning of certain parts of the classical algorithms. It is also shown that the choice of several parameters influences the separation quality considerably. Moreover, we propose as planar separators the usage of fundamental cycles, whose size is at most twice the diameter of the graph: For graphs of small diameter, the guaranteed bound is better than the O ( √ n) bounds, and it turns out that this simple strategy almost always outperforms the other
Theory and application of width bounded geometric separator
 In 23 rd Annual Symposium on Theoretical Aspects of Computer Science (STACS
, 2006
"... Abstract. We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any ddimensional Euclidean space. The separator is applied in obtaining 2 O( √ n) time exact algorithms for a class of NPcomplete geometric probl ..."
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Cited by 7 (2 self)
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Abstract. We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any ddimensional Euclidean space. The separator is applied in obtaining 2 O( √ n) time exact algorithms for a class of NPcomplete geometric problems, whose previous algorithms take n O( √ n) time [2,5,1]. One of those problems is the well known disk covering problem, which seeks to determine the minimal number of fixed size disks to cover n points on a plane [10]. They also include some NPhard problems on disk graphs such as the maximum independent set problem, the vertex cover problem, and the minimum dominating set problem. 1
New Upper Bounds on the Decomposability of Planar Graphs and Fixed Parameter Algorithms
 J. Graph Theory
, 2002
"... It is known that a planar graph on n vertices has branchwidth/treewidth bounded by ff n. In many algorithmic applications it is useful to have a small bound on the constant ff. We give a proof of the best, so far, upper bound for the constant ff. In particular, for the case of treewidth, ff ..."
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Cited by 5 (2 self)
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It is known that a planar graph on n vertices has branchwidth/treewidth bounded by ff n. In many algorithmic applications it is useful to have a small bound on the constant ff. We give a proof of the best, so far, upper bound for the constant ff. In particular, for the case of treewidth, ff ! 3:182 and for the case of branchwidth, ff ! 2:122. Our proof is based on the planar separation theorem of Alon, Seymour & Thomas and some minmax theorem of the graph minors series. Based on these bounds we introduce a new method for solving different fixed parameter problems on planar graphs. We prove that our method provides the best so far exponential speedup for fundamental problems on planar graphs like Vertex Cover, Dominating Set, Independent Set and many others.
Counting Triangulations Approximately
"... We consider the problem of counting straightedge triangulations of a given set P of n points in the plane. Until very recently it was not known whether the exact number of triangulations of P can be computed asymptotically faster than by enumerating all triangulations. We now know that the number o ..."
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Cited by 2 (1 self)
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We consider the problem of counting straightedge triangulations of a given set P of n points in the plane. Until very recently it was not known whether the exact number of triangulations of P can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of P can be computed in O ∗ (2n) time [2], which is less than the lower bound of Ω(2.43n) on the number of triangulations of any point set [11]. In this paper we address the question of whether one can approximately count triangulations in subexponential time. We present an algorithm with subexponential running time and subexponential approximation ratio, that is, if we denote by Λ the output of our algorithm, and by cn the exact number of triangulations of P, for some positive constant c, we prove that cn ≤ Λ ≤ cn · 2o(n). This is the first algorithm that in subexponential time computes a (1 + o(1))approximation of the base of the number of triangulations, more precisely, c ≤ Λ 1 n ≤ (1 + o(1))c. Our algorithm can be adapted to approximately count other crossingfree structures on P, keeping the quality of approximation and running time intact. Our algorithm may be useful in guessing, through experiments, the right constants c1 and c2 such that the number of triangulations of any set of n points is between c n 1 and c n 2. Currently there is a large gap between c1 and c2. We know that c1 ≥ 2.43 and c2 ≤ 30. 1
Sublinear time widthbounded separators and their applications to the protein sidechain packing problem
 JOURNAL OF COMBINATORIAL OPTIMIZATION
, 2007
"... Given d > 2 and a set of n grid points Q in ℜ d, we design a randomized algorithm that finds a wwide separator, which is determined by a hyperplane, in O(n 2 d log n) sublinear time such that Q has at most ( d + o(1))n points one either side of the hyperplane, and at most d+1 cdwn d−1 d points ..."
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Cited by 1 (1 self)
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Given d > 2 and a set of n grid points Q in ℜ d, we design a randomized algorithm that finds a wwide separator, which is determined by a hyperplane, in O(n 2 d log n) sublinear time such that Q has at most ( d + o(1))n points one either side of the hyperplane, and at most d+1 cdwn d−1 d points within w 2 distance to the hyperplane, where cd is a constant for fixed d. In particular, c3 = 1.209. To our best knowledge, this is the first sublinear time algorithm for finding geometric separators. Our 3D separator is applied to derive an algorithm for the protein sidechain packing problem, which improves and simplifies the previous algorithm of Xu [26].