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.     

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.     

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.     

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.     

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.     

Two new criteria for finding Steiner hulls in Steiner tree problems

The Steiner tree problem considered in this paper is that of finding a network of minimum length connecting a given setK of terminals in a regionR of the Euclidean plane. ASteiner hull forK inR is any subregion ofR known to contain a Steiner tree forK inR. Two new criteria are given for finding Steiner hulls—one for the Steiner tree problem on plane graphs and one for the rectilinear Steiner tree problem—which strengthen known polynomial-time methods of finding Steiner hulls.This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-84-0140. Reproduction in whole or part is permitted for any purpose of the United States Government.     

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.     

Steiner Trees in General Nonuniform Orientations

In this paper we consider Steiner minimum trees (SMT) in the plane, where the connections can only be along a given set of fixed but arbitrary (not necessarily uniform) orientations. The orientations define a metric, called the general orientation metric, A _σ, where σ is the number of orientations. We prove that in A _σ metric, there exists an SMT whose Steiner points belong to an (n−2)-level grid. This result generalizes a result by Lee and Shen [11], and a result by Du and Hwang [5]. In the former case, the same result was obtained for the special case when all orientations are uniform, while in the latter case the same result was proven for the special case when there are only three arbitrary orientations. We then modify the proof used in the main result for the special case when σ=3, i.e., only three arbitrary orientations are considered, and obtain a better result, which states that there exists an SMT whose Steiner points belong to an -level grid. The result has also been obtained by Lin and Xue [9] using a different approach. Received September 27, 1999; revised August 14, 2000     

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.     

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.     

Two new criteria for finding Steiner hulls in Steiner tree problems

The Steiner tree problem considered in this paper is that of finding a network of minimum length connecting a given setK of terminals in a regionR of the Euclidean plane. ASteiner hull forK inR is any subregion ofR known to contain a Steiner tree forK inR. Two new criteria are given for finding Steiner hulls—one for the Steiner tree problem on plane graphs and one for the rectilinear Steiner tree problem—which strengthen known polynomial-time methods of finding Steiner hulls.     

Small Strictly Convex Quadrilateral Meshes of Point Sets

In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3\lfloorn/2\rfloor internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n\geq 4, there are point sets for which \lceil(n–3)/2\rceil–1 Steiner points are necessary for a convex quadrilateral mesh.     

Small Strictly Convex Quadrilateral Meshes of Point Sets

In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3\lfloorn/2\rfloor internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n\geq 4, there are point sets for which \lceil(n–3)/2\rceil–1 Steiner points are necessary for a convex quadrilateral mesh.     

A linear-time algorithm to construct a rectilinear Steiner minimal tree fork-extremal point sets

Ak-extremal point set is a point set on the boundary of ak-sided rectilinear convex hull. Given ak-extremal point set of sizen, we present an algorithm that computes a rectilinear Steiner minimal tree in timeO(k ⁴ n). For constantk, this algorithm runs inO(n) time and is asymptotically optimal and, for arbitraryk, the algorithm is the fastest known for this problem.     

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).     

Direct dominance of points

Several transitive relations of geometrical objects (like inclusion of intervals on a line or polygons in the plain), which are important in VLSI design applications, can be translated into the dominance relation a dominates b iff (a ≠ b and a _j ≧ b _j for j = 1,…d) of points a = (a ₁,...,a _d ),b = (b ₁,…b _d ) in R ^d by representing the objects as points in a suitable way. If only the transitive reduction (see [7]) of the given relation is required and not all the implications by transitivity, one can restrict oneself to the direct dominances in the corresponding point set N; here a dominates b directly means that a dominates b and there is no—with respect to dominance—intermediate c in N (see [5]). To estimate the advantage of this restriction, information about the numbers of dominant and directly dominant pairs in a set of n points is required, both numbers essentially depending upon how the points are distributed in R ^d . In this paper we assume the n points in question to be identically and independently distributed; then we can expect q·n·(n–1) dominance pairs. For a certain class of distributions including the uniform distribution we prove the theorem, that the expected number of direct dominance pairs is asymptotically equal to 1/(d?1)! · n1n ^{d ? 1}(n). Hence algorithms which compute only the direct dominances instead of all dominances are worth to be considered.     

A proof of the Gilbert-Pollak conjecture on the Steiner ratio

LetP be a set ofn points on the euclidean plane. LetL _s(P) andL _m (P) denote the lengths of the Steiner minimum tree and the minimum spanning tree onP, respectively. In 1968, Gilbert and Pollak conjectured that for anyP,L _s (P)(3/2)L _m (P). We provide a proof for their conjecture in this paper.supported by NSF under grant STC88-09648.supported in part by the National Natural Science Foundation of China.     

On a circle-cover minimization problem

Given a set of n circular-arcs A₁, A₂,...,A_n, we consider the problem of finding a minimum number of circular-arcs whose union covers the circle. Specifically, if the endpoints of these n arcs are given, we show that O(n log n) is both sufficient and necessary for solving the minimum cover problem. Furthermore, with O(n log n) preprocessing time we find the minimum cover in O(n) time.     

Lower bounds on space complexity for contextfree recognition

Summary Using methods from linear algebra and crossing-sequence arguments it is shown that logarithmic space is necessary for the recognition of all context-free nonregular subsets of {a₁}^* ... {a_n}^*, where {a₁,...,a_n} is some alphabet. It then follows that log n is a lower bound on the space complexity for the recognition of any bounded or deterministic non-regular context-free language.     

A proof of the Gilbert-Pollak conjecture on the Steiner ratio

LetP be a set ofn points on the euclidean plane. LetL _s(P) andL _m (P) denote the lengths of the Steiner minimum tree and the minimum spanning tree onP, respectively. In 1968, Gilbert and Pollak conjectured that for anyP,L _s (P)≥(√3/2)L _m (P). We provide a proof for their conjecture in this paper.