Efficient construction of a bounded-degree spanner with low weight

LetS be a set ofn points in ℝ ^d and lett>1 be a real number. At-spanner forS is a graph having the points ofS as its vertices such that for any pairp, q of points there is a path between them of length at mostt times the Euclidean distance betweenp andq. An efficient implementation of a greedy algorithm is given that constructs at-spanner having bounded degree such that the total length of all its edges is bounded byO (logn) times the length of a minimum spanning tree forS. The algorithm has running timeO (n log ^d n). Applying recent results of Das, Narasimhan, and Salowe to thist-spanner gives anO(n log ^d n)-time algorithm for constructing at-spanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree forS. Previously, noo(n ²)-time algorithms were known for constructing at-spanner of bounded degree. In the final part of the paper, an application to the problem of distance enumeration is given. This work was supported by the ESPRIT Basic Research Actions Program, under Contract No. 7141 (Project ALCOM II).     

Minimum Weight Convex Steiner Partitions

New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n/log log n) times longer than a Euclidean minimum spanning tree (EMST), and this bound is the best possible. Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudo-triangle. We also show that the minimum length convex Steiner partition of n points along a pseudo-triangle is at most O(log log n) times longer than an EMST, and this bound is also the best possible. Our methods are constructive and lead to O(nlog n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worst-case bounds in both cases.     

AnO(logk)-approximation algorithm for thek minimum spanning tree problem in the plane

Givenn points in the Euclidean plane, we consider the problem of finding the minimum tree spanning anyk points. The problem isNP-hard and we give anO(logk)-approximation algorithm.     

Rectilinear steiner tree heuristics and minimum spanning tree algorithms using geographic nearest neighbors

We study the application of the geographic nearest neighbor approach to two problems. The first problem is the construction of an approximately minimum length rectilinear Steiner tree for a set ofn points in the plane. For this problem, we introduce a variation of a subgraph of sizeO(n) used by YaO [31] for constructing minimum spanning trees. Using this subgraph, we improve the running times of the heuristics discussed by Bern [6] fromO(n ² log n) toO(n log² n). The second problem is the construction of a rectilinear minimum spanning tree for a set ofn noncrossing line segments in the plane. We present an optimalO(n logn) algorithm for this problem. The rectilinear minimum spanning tree for a set of points can thus be computed optimally without using the Voronoi diagram. This algorithm can also be extended to obtain a rectilinear minimum spanning tree for a set of nonintersecting simple polygons.The results in this paper are a part of Y. C. Yee's Ph.D. thesis done at SUNY at Albany. He was supported in part by NSF Grants IRI-8703430 and CCR-8805782. S. S. Ravi was supported in part by NSF Grants DCI-86-03318 and CCR-89-05296.     

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p ₁ , p ₂ , \ldots, p _n } be a set of n {\lilsf terminal points} in the Euclidean plane, where point p _i has a {\lilsf service request of grade} g(p _i ) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ⋅s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points p _i and p _j there is a path whose minimum grade of service is at least as large as \min(g(p _i ), g(p _j )) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+ε) -approximation to the minimum cost network under a given topology or its degeneracies in O(n ^1.5 (log n + log (1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.     

An approximation algorithm for a bottleneck k-Steiner tree problem in the Euclidean plane

We study a bottleneck Steiner tree problem: given a set P={p₁,p₂,…,p_n} of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the problem.     

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E _r ^k ) whereE _r ^k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE _r ¹⁷ . We also prove that ¦E _r ^k ¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n ²) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)^0.5 m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n ^1.5 log^0.5 n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n ² +n ^1.5 log^0.5 n).This research was partially supported by a grant from the National Science Council of the Republic of China under Grant NSC-78-0408-E-007-05.     

Geometric complexity of some location problems

Given a set ofn demand points with weightW _i ,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An (n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.Supported in part by the National Science Foundation under Grants ECS 83-40031 and DCR 84-20814.     

An O(n log n) Average Time Algorithm for Computing the Shortest Network under a Given Topology

In 1992 F. K. Hwang and J. F. Weng published an O(n ² ) time algorithm for computing the shortest network under a given full Steiner topology interconnecting n fixed points in the Euclidean plane. The Hwang—Weng algorithm can be used to improve substantially existing algorithms for the Steiner minimum tree problem because it reduces the number of different Steiner topologies to be considered dramatically. In this paper we present an improved Hwang—Weng algorithm. While the worst-case time complexity of our algorithm is still O(n ² ) , its average time complexity over all the full Steiner topologies interconnecting n fixed points is O (n log n ). Received August 24, 1996; revised February 10, 1997.     

New primal and dual matching heuristics

We describe a new heuristic for constructing a minimum-cost perfect matching designed for problems on complete graphs whose cost functions satisfy the triangle inequality (e.g., Euclidean problems). The running time for ann node problem is O(n logn) after a minimum-cost spanning tree is constructed. We also describe a procedure which, added to Kruskal's algorithm, produces a lower bound on the size of any perfect matching. This bound is based on a dual problem which has the following geometric interpretation for Euclidean problems: Pack nonoverlapping disks centered at the nodes and moats surrounding odd sets of nodes so as to maximize the sum of the disk radii and moat widths.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Fast heuristic algorithms for rectilinear steiner trees

A fundamental problem in circuit design is how to connectn points in the plane, to make them electrically common using the least amount of wire. The tree formed, a Steiner tree, is usually constructed with respect to the rectilinear metric. The problem is known to be NP-complete; an extensive review of proposed heuristics is given. An early algorithm by Hanan is shown to have anO(n logn) time implementation using computational geometry techniques. The algorithm can be modified to do sequential searching inO(n ²) total time. However, it is shown that the latter approach runs inO(n ^3/2) expected time, forn points selected from anm×m grid. Empirical results are presented for problems up to 10,000 points.     

Exact Algorithms for the Bottleneck Steiner Tree Problem

Given n points, called terminals, in the plane ℝ² and a positive integer k, the bottleneck Steiner tree problem is to find k Steiner points from ℝ² and a spanning tree on the n+k points that minimizes its longest edge length. Edge length is measured by an underlying distance function on ℝ², usually, the Euclidean or the L ₁ metric. This problem is known to be NP-hard. In this paper, we study this problem in the L _p metric for any 1≤p≤∞, and aim to find an exact algorithm which is efficient for small fixed k. We present the first fixed-parameter tractable algorithm running in f(k)⋅nlog ² n time for the L ₁ and the L _∞ metrics, and the first exact algorithm for the L _p metric for any fixed rational p with 1<p<∞ whose time complexity is f(k)⋅(n ^k +nlog n), where f(k) is a function dependent only on k. Note that prior to this paper there was no known exact algorithm even for the L ₂ metric.     

A Tight Lower Bound for Computing the Diameter of a 3D Convex Polytope

The diameter of a set P of n points in ℝ ^d is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3-dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(nlog n) time in the algebraic computation tree model. It shows that the O(nlog n) time algorithm of Ramos for computing the diameter of a point set in ℝ³ is optimal for computing the diameter of a 3-polytope. We also give a linear time reduction from Hopcroft’s problem of finding an incidence between points and lines in ℝ² to the diameter problem for a point set in ℝ⁷.     

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Let S be a set ofn points in the plane. For an arbitrary positive rationalr, we construct a planar straight-line graph onS that approximates the complete Euclidean graph onS within the factor (1 + 1/r)[2/3 cos(/6)], and it has length bounded by 2r + 1 times the length of a minimum Euclidean spanning tree onS. Given the Deiaunay triangulation ofS, the graph can be constructed in linear time.     

Geodesic regression on spheres from a numerical optimization viewpoint

In the geodesic regression problem it is given a set of data points at known times and the goal is to find a geodesic that best fits the data. This problem corresponds to the generalization of the classical linear regression problem to curved spaces. Here we are interested in the geodesic regression problem on Euclidean spheres. Contrary to the Euclidean situation, the normal equations turn out to be highly nonlinear. To overcome this difficulty, we look at the geodesic regression problem in the unit n-sphere as an optimization problem in the Euclidean space ?ⁿ⁺¹ and use the MATLAB optimization toolbox to solve it numerically.     

Computing euclidean maximum spanning trees

An algorithm is presented for finding a maximum-weight spanning tree of a set ofn points in the Euclidean plane, where the weight of an edge (p _i ,p _j ) equals the Euclidean distance between the pointsp _i andp _j . The algorithm runs inO(n logh) time and requiresO(n) space;h denotes the number of points on the convex hull of the given set. If the points are vertices of a convex polygon (given in order along the boundary), then our algorithm requires only a linear amount of time and space. These bounds are the best possible in the algebraic computation-tree model. We also establish various properties of maximum spanning trees that can be exploited to solve other geometric problems.     

A linear time algorithm for max-min length triangulation of a convex polygon

We consider the following planar max-min length triangulation problem: given a set of n points in the Euclidean plane, find a triangulation such that the length of the shortest edge in the triangulation is maximized. In this paper, a linear time algorithm is proposed for computing the max-min length triangulation of a set of points in convex position. In addition, an O(nlogn) time algorithm is proposed for computing the max-min length k-set triangulation of a set of points in convex position, where we are to compute a set of k vertices such that the max-min length triangulation on them is minimized over all possible k-set. We further show that the graph version of max-min length triangulation is NP-complete, and some common heuristics such as greedy algorithm are in general not able to give a bounded-ratio approximation to the max-min length triangulation.     

Improved computation of plane Steiner Minimal Trees

ASteiner Minimal Tree (SMT) for a given setA = {a ₁,...,a _n } in the plane is a tree which interconnects these points and whose total length, i.e., the sum of lengths of the branches, is minimum. To achieve the minimum, the tree may contain other points (Steiner points) besidesa ₁,...,a _n . Various improvements are presented to an earlier computer program of the authors for plane SMTs. These changes have radically reduced machine times. The existing program was limited in application to aboutn = 30, while the innovations have facilitated solution of many randomly generated 100-point problems in reasonable processing times.This work was supported by the Canadian Natural Sciences and Engineering Council under Grant Numbers A-7544 and A-7558.     

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Let S be a set ofn points in the plane. For an arbitrary positive rationalr, we construct a planar straight-line graph onS that approximates the complete Euclidean graph onS within the factor (1 + 1/r)[2π/3 cos(π/6)], and it has length bounded by 2r + 1 times the length of a minimum Euclidean spanning tree onS. Given the Deiaunay triangulation ofS, the graph can be constructed in linear time.     

Orthogonal linear regression algorithm based on augmented matrix formulation

The problem of n-dimensional orthogonal linear regression is a problem of finding an n-dimensional hyperplane minimising the sum of Euclidean distances between this hyperplane and a given set of m points, where m n. This nonlinear programming problem has been re-cast in an augmented matrix form and solved as a sequence of iteratively re-weighted least square problems. The proposed algorithm is seen as an alternative to the recently published algorithm by Cavalier and Melloy [1].