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21
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 47 (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...
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 37 (7 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 30 (5 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.
Highly scalable algorithms for rectilinear and octilinear Steiner trees
 In Proc. Asian and South Pacific Design Automation Conf
, 2003
"... problem, which asks for a minimumlength interconnection of a given set of terminals in the rectilinear plane, is one of the fundamental problems in electronic design automation. Recently there has been renewed interest in this problem due to the need for highly scalable algorithms able to handle ne ..."
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Cited by 29 (3 self)
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problem, which asks for a minimumlength interconnection of a given set of terminals in the rectilinear plane, is one of the fundamental problems in electronic design automation. Recently there has been renewed interest in this problem due to the need for highly scalable algorithms able to handle nets with tens of thousands of terminals. In this paper we give a practical � heuristic for computing nearoptimal rectilinear Steiner trees based on a batched version of the greedy triple contraction algorithm of Zelikovsky [21]. Experiments conducted on both random and industry testcases show that our heuristic matches or exceeds the quality of best known RSMT heuristics, e.g., on random instances with more than 100 terminals our heuristic improves over the rectilinear minimum spanning tree by an average of 11%. Moreover, our heuristic has very well scaling runtime, e.g., it can route a 34kterminals net extracted from a real design in less than 25 seconds compared to over 86 minutes needed by the edgebased heuristic of Borah, Owens, and Irwin [3]. Since our heuristic is graphbased, it can be easily modified to handle practical considerations such as routing obstacles, preferred directions, via costs, and octilinear routing – indeed, experimental results show only a small factor increase in runtime when switching from rectilinear to octilinear routing. I.
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 28 (4 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.
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 26 (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...
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 24 (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...
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 effic ..."
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Cited by 8 (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.
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 ..."
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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