Improved upper bounds for the Steiner ratio

Discrete & Computational Geometry, 1993

Suppose we are given a sequence of n points in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph, (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known on-line algorithms whose competitive ratio is O(log n) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(log n/ log log n) for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well.

Mathematics

In 1989, Chartrand, Oellermann, Tian and Zou introduced the Steiner distance for graphs. This is a natural generalization of the classical graph distance concept. Let Γ be a connected graph of order at least 2, and S\V(Γ). Then, the minimum size among all the connected subgraphs whose vertex sets contain S is the Steiner distancedΓ(S) among the vertices of S. The Steiner k-eccentricity ek(v) of a vertex v of Γ is defined by ek(v)=max{dΓ(S)|S\V(Γ),|S|=k,andv∈S}, where n and k are two integers, with 2≤k≤n, and the Steiner k-diameter of Γ is defined by sdiamk(Γ)=max{ek(v)|v∈V(Γ)}. In this paper, we present an algorithm to derive the Steiner distance of a graph; in addition, we obtain a relationship between the Steiner k-diameter of a graph and its line graph. We study various properties of the Steiner diameter through a combinatorial approach. Moreover, we characterize graph Γ when sdiamk(Γ) is given, and we determine sdiamk(Γ) for some special graphs. We also discuss some of the appli...

Algorithmica, 2009

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(n log n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worst-case bounds in both cases.

1994

The rectilinear Steiner tree problem requires a shortest tree spanning a given vertex subset in the plane with rectilinear distance. It was proved that the output length of Zelikovsky's 25] and Berman/Ramaiyer 3] heuristics is at most 1.375 and 97 72 1:347 of the optimal length, respectively. It was claimed that these bounds are not tight. Here we improve these bounds to 1.3125 and 61 48 1:271, respectively, and give e cient algorithms to nd approximations of such quality. We also prove that for Zelikovsky's heuristic this bound cannot be less than 1.3.

SIAM Journal on Computing, 2011

Given a set S of vertices in a connected graph G, the classic Steiner tree problem asks for the minimum number of edges of a connected subgraph of G that contains S. We study this problem in the hypercube. Given a set S of vertices in the n-dimensional hypercube Q n , the Steiner cost of S, denoted by cost(S), is the minimum number of edges among all connected subgraphs of Q n that contain S. We obtain the following results on cost(S). Let be any given small, positive constant, and set k = |S|.

Given two sets of points in the plane, $P$ of $n$ terminals and $S$ of $m$ Steiner points, a Steiner tree of $P$ is a tree spanning all points of $P$ and some (or none or all) points of $S$. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, $P$ and $S$, and a positive integer $k \le m$, find a bottleneck Steiner tree of $P$ with at most $k$ Steiner points. The problem has application in the design of wireless communication networks. We first show that the problem is NP-hard and cannot be approximated within factor $\sqrt{2}$, unless $P=NP$. Then, we present a polynomial-time approximation algorithm with performance ratio 2.

2005

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. They conjectured that the lower bound is not tight. We use new ideas, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10 −11 )π/2.

Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011

We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s, t ∈ P , and an integer k > 0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(n log 2 n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. , who gave an O(n 2 log n)-time algorithm. We also study the dual version of the problem, where a value λ > 0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most λ.

Proceedings of the twenty-seventh annual symposium on Computational geometry, 2011

Given two sets in the plane, R of n (terminal) points and S of m (Steiner) points, a full Steiner tree is a Steiner tree in which all points of R are leaves. In the bottleneck full Steiner tree (BFST) problem, one has to find a full Steiner tree T (with any number of Steiner points from S), such that the length of the longest edge in T is minimized, and, in the k-BFST problem, has to find a full Steiner tree T with at most k ≤ m Steiner points from S such that the length of the longest edge in T is minimized. The problems are motivated by wireless network design. In this paper, we present an exact algorithm of O((n + m) log 2 m) time to solve the BFST problem. Moreover, we show that the k-BFST problem is NP-hard and that there exists a polynomial-time approximation algorithm for the problem with performance ratio 4.

arXiv (Cornell University), 2008

We give a 1.25 approximation algorithm for the Steiner Tree Problem with distances one and two, improving on the best known bound for that problem. We give a new approximation algorithm for the problem of finding a minimum Steiner tree for metric spaces with distances one and two. It improves over the best known approximation factor for that problem of 1.279 . Moreover, unlike the result of Robins and Zelikovsky, our methods yields a single algorithm, whereas gives an approximation scheme. A metric with distances 1 and 2 can be represented as a graph, so edges are pairs in distance 1 and non-edges are pairs in distance 2. We will denote by STP[1,2] the Steiner Tree Problem restricted to such metrics. The problem instance of STP[1,2] is a graph G = (V, E) that defines a metric in this way, and a set R ⊂ V of terminal nodes. A valid solution is a set unordered node pairs T such that R is contained in a connected component of (V, E). We minimize |T ∩ E| + 2|T -E|.