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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 non-intersecting 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...

The fastest exact algorithm (in practice) for the rectilinear Steiner tree problem in the plane uses a two-phase 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 lower-bounds 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...

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 best-known 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 three-dimensional rectilinear space the maximum possible degree of a minimum-degree 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...

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 computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas-convex hulls, intersections, searching, proximity, and combinatorial optimizations-are discussed. Seven algorithmic techniques incremental construction, plane-sweep, locus, divide-andconquer, geometric transformation, prune-and-search, and dynamization-are 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.

Motivated by practical VLSI applications, we study the maximum vertex degree in a minimum spanning tree (MST) under arbitrary L p metrics. We show that the maximum vertex degree in a maximum-degree L p MST equals the Hadwiger number of the corresponding unit ball. We then determine the maximum vertex degree in a minimum-degree L p MST; towards this end, we define the MST number, which is closely related to the Hadwiger number. We bound Hadwiger and MST numbers for arbitrary L p metrics, and focus on the L 1 metric, where little was known. We show that the MST number of a diamond is 4, and that for the octahedron the Hadwiger number is 18 and the MST number is either 13 or 14. We also give an exponential lower bound on the MST number for an L p unit ball. Implications to L p minimum spanning trees and related problems are explored.

Abstract. Given n fragments from k>2 genomes, we will show how to find an optimal chain of colinear non-overlapping 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 complex-log 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

Minimum spanning tree problem is a very important problem in VLSI CAD. Given n points in a plane, a min-imum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive ap-proach enumerates edges on all pairs of points and takes at least R(n2) time. More efficient approaches find a min-imum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangu-lation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum span-ning tree construction which is based on a general con-cept 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) sweep-line algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation. 1

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 bounded-degree neighborhood graph (BNG). Based on this framework, we propose an O(n log n) algorithm to construct a 4-BDRMST (RMST with maximum vertex degree ≤ 4). This is the first 4-BDRMST algorithm with such a complexity, and experimental results show that the algorithm is significantly faster than the existing 4-BDRMST algorithms. 1

The Steiner minimum tree problem, which asks for a minimum-length interconnection of a given set of termi-nals in the plane, is one of the fundamental problems in Very Large Scale Integration (VLSI) physical design. Although advances in VLSI manufacturing technologies have introduced additional routing objectives, mini-mum length continues to be the primary objective when routing non-critical nets, since the minimum-length