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25
Spanning Trees in Hypergraphs with Applications to Steiner Trees
, 1998
"... This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of ..."
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Cited by 25 (1 self)
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This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of finding a minimumweight spanning tree in a hypergraph. The geometric Steiner tree problem is known to be NPcomplete for the rectilinear metric, and NPhard for the Euclidean metric. The fastest exact algorithms (in practice) for these problems use two phases: First a small but sufficient set of full Steiner trees (FSTs) is generated and then a Steiner minimal tree is constructed from this set. These phases are called FST generation and FST concatenation, respectively, and an overview of each phase is presented. FST concatenation is almost always the most expensive phase, and has traditionally been accomplished via simple backtrack search or dynamic programming.
TwoDimensional Interleaving Schemes with Repetitions: Constructions and Bounds
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2002
"... Twodimensional interleaving schemes with repetitions are considered. These schemes are required for the correction of twodimensional bursts (or clusters) of errors in applications such as optical recording and holographic storage. We assume that a cluster of errors may have an arbitrary shape, and ..."
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Cited by 17 (5 self)
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Twodimensional interleaving schemes with repetitions are considered. These schemes are required for the correction of twodimensional bursts (or clusters) of errors in applications such as optical recording and holographic storage. We assume that a cluster of errors may have an arbitrary shape, and is characterized solely by its area l. Thus, an interleaving scheme A(l, r) of strength l with r repetitions is an (infinite) array of integers defined by the property that every integer appears no more than r times in any connected component of area t. The problem is to minimize, for a given l and r, the interleaving degree deg A(l, r ), which is the total number of distinct integers contained in the array. Optimal interleaving schemes for r = 1 (no repetitions) have been devised in earlier work. Here, we consider interleaving schemes for r &ge; 2. Such schemes reduce the overall redundancy, yet are considerably more difficult to construct and analyze. To this end, we generalize the concept of L_1distance and introduce the notions of tristance, quadristance, and more generally rdispersion. We focus on the special class of interleaving schemes, called lattice interleavers, that is akin to the class of linear codes in coding theory. We construct efficient lattice interleavers for r = 2, 3, 4 and some higher values of r. For r = 2, 3 we show that these lattice interleavers are either optimal for all t or asymptotically optimal for t &rarr; &infin;. We present the results of an extensive computer search that yields the optimal lattice interleavers for r = 2, 3, 4, 5, 6 and l up to about 1000. Finally, we consider an alternative connectivity model for clusters, where two elements in an array are connected if they are adjacent horizontally, vertically, or diagonally. We establish relations between interleavers for this model and interleavers for th...
A Catalog of Hanan Grid Problems
 Networks
, 2000
"... We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including  but not li ..."
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Cited by 10 (2 self)
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We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including  but not limited to  Steiner trees for rectilinear (or isothetic) polygons, obstacleavoiding Steiner trees, group Steiner trees and prizecollecting Steiner trees. Also, the weighted region Steiner tree problem is shown to be solvable on the Hanan grid; this problem has natural applications in VLSI design routing. Finally, we give similar results for other rectilinear problems. 1 Introduction Assume we are given a finite set of points S in the plane. The Hanan grid H(S) of S is obtained by constructing vertical and horizontal lines through each point in S. The main motivation for studying the Hanan grid stems from the fact that it is known to contain a rectilinear Steiner minimum tree (RSMT)...
FORst: A 3Step Heuristic For Obstacleavoiding Rectilinear Steiner Minimal Tree Construction
, 2004
"... Macro cells, IP blocks, and prerouted nets are often regarded as obstacles in VLSI routing phase. Obstacleavoiding rectilinear Steiner minimum tree (OARSMT) algorithms are often used to meet the needs of practical routing applications. However, OARSMT algorithms with multiterminal nets routing st ..."
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Cited by 7 (1 self)
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Macro cells, IP blocks, and prerouted nets are often regarded as obstacles in VLSI routing phase. Obstacleavoiding rectilinear Steiner minimum tree (OARSMT) algorithms are often used to meet the needs of practical routing applications. However, OARSMT algorithms with multiterminal nets routing still can not satisfy the requirements of practical applications. This paper presents a 3step heuristic, named FORst, to tackle the OARSMT problem. In Step1, we partition all terminals into some subsets in the presence of obstacles. Then in Step2, we connect terminals in each connected graph with one or more trees, respectively. In Step3, we connect the forest consisting of trees constructed in Step2 into a completed Steiner tree spanning all terminals while avoiding all obstacles. Two algorithms, called ACORSMT and GFSTRSMT, are proposed to construct OARSMT in a connected graph in Step2, which are suitable for different situations. This algorithm has been implemented and tested on cases with typical obstacles. The experimental results show that FORst is with great efficiency and can get good performance. Moreover, it can tackle large scale nets among complex obstacles, such as a net with 1000 terminals in the presence of 100 rectangular obstacles.
Efficient MaximaFinding Algorithms for Random Planar Samples
 Discrete Mathematics and Theoretical Computer Science (Electronic
, 2003
"... this paper a simple classification of several known algorithms for finding the maxima, together with several new algorithms; among these are two efficient algorithmsone with expected complexity n +O( # nlogn) when the point samples are issued from some planar regions, and another more efficient t ..."
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this paper a simple classification of several known algorithms for finding the maxima, together with several new algorithms; among these are two efficient algorithmsone with expected complexity n +O( # nlogn) when the point samples are issued from some planar regions, and another more efficient than existing ones
An ILP based hierarchical global routing approach for VLSI ASIC design
, 2007
"... The use of integrated circuits in highperformance computing, telecommunications, and consumer electronics has been growing at a very fast pace. The level of integration as measured by the number of logic gates in a chip has been steadily rising due to the rapid progress in processing and intercon ..."
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Cited by 6 (0 self)
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The use of integrated circuits in highperformance computing, telecommunications, and consumer electronics has been growing at a very fast pace. The level of integration as measured by the number of logic gates in a chip has been steadily rising due to the rapid progress in processing and interconnect technology. The interconnect delay in VLSI circuits has become a critical determiner of circuit performance. As a result, circuit layout is starting to play a more important role in today’s chip designs. Global routing is one of the key subproblems of circuit layout which involves finding an approximate path for the wires connecting the elements of the circuit without violating resource constraints. In this paper, several integer programming (ILP) based global routing models are fully investigated and explored. The resulting ILP problem is relaxed and solved as a linear programming (LP) problem followed by a rounding heuristic to obtain an integer solution. Experimental results obtained show that the proposed combined WVEM (wirelength, via, edge capacity) model can optimize several global routing objectives simultaneously and effectively. In addition, several hierarchical methods are combined with the proposed flat ILP based global router to reduce the CPU time by about 66 % on average for edge capacity model (ECM).
The Coming of Age of (Academic) Global Routing
 ISPD'08
, 2008
"... Wire routing, an important step in modern VLSI design, is increasingly responsible for timing closure and manufacturability. The CAD community has witnessed remarkable improvements in speed and quality of global routing algorithms in response to the inaugural ISPD 2007 Global Routing Contest, where ..."
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Cited by 6 (0 self)
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Wire routing, an important step in modern VLSI design, is increasingly responsible for timing closure and manufacturability. The CAD community has witnessed remarkable improvements in speed and quality of global routing algorithms in response to the inaugural ISPD 2007 Global Routing Contest, where prizes were awarded for best results on a new set of large industry benchmarks. In this paper, we review the state of the art in global routing and identify several critical techniques that distinguish top routing algorithms. We also discuss open challenges and offer predictions regarding the future of routing research.
BerryEsseen bounds for the number of maxima in planar regions
 ELECTRONIC JOURNAL OF PROBABILITY
, 2003
"... We derive the optimal convergence rate O(n 1/4 ) in the central limit theorem for the number of maxima in random samples chosen uniformly at random from the right triangle of the shape #. A local limit theorem with rate is also derived. The result is then applied to the number of maxima in general ..."
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Cited by 5 (2 self)
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We derive the optimal convergence rate O(n 1/4 ) in the central limit theorem for the number of maxima in random samples chosen uniformly at random from the right triangle of the shape #. A local limit theorem with rate is also derived. The result is then applied to the number of maxima in general planar regions (upperbounded by some smooth decreasing curves) for which a nearoptimal convergence rate to the normal distribution is established.
Efficient MultiLayer ObstacleAvoiding Rectilinear Steiner Tree Construction
"... Given a set of pins and a set of obstacles on routing layers, a multilayer obstacleavoiding rectilinear Steiner minimal tree (MLOARSMT) connects these pins by rectilinear edges within layers and vias between layers, and avoids running through any obstacle to construct a Steiner tree with a minim ..."
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Cited by 4 (0 self)
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Given a set of pins and a set of obstacles on routing layers, a multilayer obstacleavoiding rectilinear Steiner minimal tree (MLOARSMT) connects these pins by rectilinear edges within layers and vias between layers, and avoids running through any obstacle to construct a Steiner tree with a minimal total cost. The MLOARSMT problem is very important for many VLSI designs with pins being located in multiple routing layers that contain numerous routing obstacles incurred from IP blocks, power networks, prerouted nets, etc. Therefore, it is desired to develop an effective algorithm for the MLOARSMT problem. However, there is no existing work on this MLOARSMT problem. In this paper, we first formulate the MLOARSMT problem and identify key different properties of the problem from its singlelayer counterpart. Based on the multilayer obstacleavoiding spanning graph (MLOASG), we present the first algorithm to solve the MLOARSMT problem. Our algorithm can guarantee an optimal solution for any 2pin net and many higherpin nets. Experiments show that our algorithm results in 33 % smaller total costs on average than a constructionbycorrection heuristic which is widely used for Steinertree construction in the recent literature.
Delayrelated Secondary Objectives for Rectilinear Steiner Minimum Trees
 Discrete Applied Mathematics
, 2001
"... The rectilinear Steiner tree problem in the plane is to construct a minimumlength tree interconnecting a set of points (called terminals) consisting of horizontal and vertical line segments only. Rectilinear Steiner minimum trees (RSMTs) can today be computed quickly for realistic instances occ ..."
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Cited by 4 (1 self)
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The rectilinear Steiner tree problem in the plane is to construct a minimumlength tree interconnecting a set of points (called terminals) consisting of horizontal and vertical line segments only. Rectilinear Steiner minimum trees (RSMTs) can today be computed quickly for realistic instances occurring in VLSI design. However, interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the length of paths or direct delay measures should be taken into account when constructing rectilinear Steiner trees. We consider the problem of nding an RSMT that  as a secondary objective  minimizes a signal delay related objective. Given a source (one of the terminals) we give some structural properties of RSMTs for which the weighted sum of path lengths from the source to the other terminals is minimized. Also, we present exact and heuristic algorithms for constructing RSMTs with weighted sum of path lengths or Elmore delays secondary objectives. Computational results for industrial designs are presented. Research Institute for Discrete Mathematics, University of Bonn, Germany; email: peyer@or.unibonn.de. y Department of Computer Science, University of Copenhagen, Denmark; email: martinz@diku.dk. z Department of Computer Science, University of Copenhagen, Denmark; email: david@diku.dk. 1 1