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13
Rectilinear shortest paths through polygonal obstacles in O(n(log n)^{3/2}) time
, 1987
"... The problem of finding a rectilinear shortest path amongst obstacles may be stated as follows: Given a set of obstacles in the plane find a shortest rectilinear (L 1 ) path from a point s to a point t which avoids all obstacles. The path may touch an obstacle but may not cross an obstacle. We study ..."
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Cited by 35 (0 self)
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The problem of finding a rectilinear shortest path amongst obstacles may be stated as follows: Given a set of obstacles in the plane find a shortest rectilinear (L 1 ) path from a point s to a point t which avoids all obstacles. The path may touch an obstacle but may not cross an obstacle. We study the rectilinear shortest path problem for the case where the obstacles are nonintersecting simple polygons, and present an O(n (logn) 2 ) algorithm for finding such a path, where n is the number of vertices of the obstacles. This algorithm requires O(nlogn) space. Another algorithm is given that requires O(n(logn) 3/2 ) time and space. We also study the case of rectilinear obstacles in three dimensions, and show that L 1 shortest paths can be found in O(n 2 (log n) 3 ) time. 1. Introduction In this paper we consider the problem of finding rectilinear (L 1 ) shortest paths between points when there may be obstacles present. The problem may be formulated as follows: Given a set o...
Rectilinear Full Steiner Tree Generation
 NETWORKS
, 1997
"... The fastest exact algorithm (in practice) for the rectilinear Steiner tree problem in the plane uses a twophase scheme: First a small but sufficient set of full Steiner trees (FSTs) is generated and then a Steiner minimum tree is constructed from this set by using simple backtrack search, dynamic p ..."
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Cited by 27 (5 self)
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The fastest exact algorithm (in practice) for the rectilinear Steiner tree problem in the plane uses a twophase scheme: First a small but sufficient set of full Steiner trees (FSTs) is generated and then a Steiner minimum tree is constructed from this set by using simple backtrack search, dynamic programming or an integer programming formulation. FST generation methods can be seen as problem reduction algorithms and are also useful as a first step in providing good upper and lowerbounds for large instances. Currently, the time needed to generate FSTs poses a significant overhead for FST based exact algorithms. In this paper we present a very efficient algorithm for the rectilinear FST generation problem which removes this overhead completely. Based on information obtained in a preprocessing phase, the new algorithm "grows" FSTs while applying several new and important optimality conditions. For randomly generated instances approximately 4n FSTs are generated (where n is the number o...
FLUTE: Fast Lookup Table Based Wirelength Estimation Technique
 In In Proceedings of the IEEE/ACM International Conference on ComputerAided Design
, 2004
"... Wirelength estimation is an important tool to guide the design optimization process in early design stages. In this paper, we present a novel wirelength estimation technique called FLUTE. Our technique is based on precomputed lookup table to make wirelength estimation very fast and very accurate fo ..."
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Cited by 26 (8 self)
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Wirelength estimation is an important tool to guide the design optimization process in early design stages. In this paper, we present a novel wirelength estimation technique called FLUTE. Our technique is based on precomputed lookup table to make wirelength estimation very fast and very accurate for low degree 1 nets. We show experimentally that for FLUTE, RMST, and HPWL, the average error in wirelength are 0.72%, 4.23%, and8.71%, respectively, and the normalized runtime are 1, 1.24, and 0.16, respectively. 1
Efficient steiner tree construction based on spanning graphs
 IEEE Transactions ComputerAided Design
, 2004
"... Abstract—The Steiner Minimal Tree (SMT) problem is a very important problem in very large scale integrated computeraided design. Given points on a plane, an SMT connects these points through some extra points (called Steiner points) to achieve a minimal total length. Even though there exist many he ..."
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Cited by 23 (3 self)
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Abstract—The Steiner Minimal Tree (SMT) problem is a very important problem in very large scale integrated computeraided design. Given points on a plane, an SMT connects these points through some extra points (called Steiner points) to achieve a minimal total length. Even though there exist many heuristic algorithms for this problem, they have either poor performances or expensive running time. This paper records an implementation of an efficient SMT algorithm that has a worst case running time of ( log) and a performance close to that of the Iterated 1Steiner algorithm. The algorithm efficiently combines Borah et al.’s edge substitute concept with Zhou et al.’s spanning graph. Extensive experimental studies are conducted to compare it with other programs. Index Terms—Graph algorithms, routing, Steiner tree. I.
LowDegree Minimum Spanning Trees
 Discrete Comput. Geom
, 1999
"... Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where ..."
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Cited by 20 (1 self)
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Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the Lp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the bestknown bounds for the maximum MST degree for arbitrary Lp metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane there exists an MST with maximum degree of at most 4, and for threedimensional rectilinear space the maximum possible degree of a minimumdegree MST is either 13 or 14. 1 Introduction Minimum spanning tree (MST) construction is a classic optimization problem for which several efficient algorithms are known [9] [15] [19]. Solutions of many other problems hinge on the construction of an MST as an intermediary step [4], with th...
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 19 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
Efficient minimum spanning tree construction without Delaunay triangulation
 UNI 4.0 Security Addendum, ATM Forum BTDSIGSEC
, 2001
"... Minimum spanning tree problem is a very important problem in VLSI CAD. Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least R(n2) time. More eff ..."
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Cited by 7 (0 self)
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Minimum spanning tree problem is a very important problem in VLSI CAD. Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least R(n2) time. More efficient approaches find a minimum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangulation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum spanning tree construction which is based on a general concept of spanning graphs. A spanning graph is a natural definition and not necessarily a Delaunay triangulation. Based on this framework, we then design an O(n log n) sweepline algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation. 1
Multiple genome alignment: Chaining algorithms revisited
 In Proceedings of the 14th Annual Symposium on Combinatorial Pattern Matching, LNCS 2676
, 2003
"... Abstract. Given n fragments from k>2 genomes, we will show how to find an optimal chain of colinear nonoverlapping fragments in time O(n log k−2 n log log n) and space O(n log k−2 n). Our result solves an open problem posed by Myers and Miller because it reduces the time complexlog ity of their al ..."
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Cited by 5 (3 self)
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Abstract. Given n fragments from k>2 genomes, we will show how to find an optimal chain of colinear nonoverlapping fragments in time O(n log k−2 n log log n) and space O(n log k−2 n). Our result solves an open problem posed by Myers and Miller because it reduces the time complexlog ity of their algorithm by a factor 2 n and the space complexity by log log n a factor log n. Fork = 2 genomes, our algorithm takes O(n log n) time and O(n) space. 1
An efficient lowdegree RMST algorithm for VLSI/ULSI physical design
 in Lecture Notes in Computer Science (LNCS) 3254—Integrated Circuit and System Design
, 2004
"... Abstract. Motivated by very/ultra large scale integrated circuit (VLSI/ULSI) physical design applications, we study the construction of rectilinear minimum spanning tree (RMST) with its maximum vertex degree as the constraint. Given a collection of n points in the plane, we firstly construct a graph ..."
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Cited by 1 (1 self)
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Abstract. Motivated by very/ultra large scale integrated circuit (VLSI/ULSI) physical design applications, we study the construction of rectilinear minimum spanning tree (RMST) with its maximum vertex degree as the constraint. Given a collection of n points in the plane, we firstly construct a graph named the boundeddegree neighborhood graph (BNG). Based on this framework, we propose an O(n log n) algorithm to construct a 4BDRMST (RMST with maximum vertex degree ≤ 4). This is the first 4BDRMST algorithm with such a complexity, and experimental results show that the algorithm is significantly faster than the existing 4BDRMST algorithms. 1