Two probabilistic results on rectilinear steiner trees (original) (raw)
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Approaching the 5/4���approximation for rectilinear Steiner trees
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.
Lecture Notes in Computer Science, 2012
In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph and a set of terminal nodes. The goal is to compute a min-cost tree S which spans all terminals. In this paper we consider the min-power version of the problem (a.k.a. symmetric multicast), which is better suited for wireless applications. Here, the goal is to minimize the total power consumption of nodes, where the power of a node v is the maximum cost of any edge of S incident to v. Intuitively, nodes are antennas (part of which are terminals that we need to connect) and edge costs define the power to connect their endpoints via bidirectional links (so as to support protocols with ack messages). Observe that we do not require that edge costs reflect Euclidean distances between nodes: this way we can model obstacles, limited transmitting power, non-omnidirectional antennas etc. Differently from its min-cost counterpart, min-power Steiner tree is NP-hard even in the spanning tree case (a.k.a. symmetric connectivity), i.e. when all nodes are terminals. Since the power of any tree is within once and twice its cost, computing a ρst ≤ ln(4) + ε [Byrka et al.'10] approximate min-cost Steiner tree provides a 2ρst < 2.78 approximation for the problem. For min-power spanning tree the same approach provides a 2 approximation, which was improved to 5/3 + ε with a non-trivial approach in [Althaus et al.'06].
Near Optimal Bounds for Steiner Trees in the Hypercube
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|.
On-line steiner trees in the Euclidean plane
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.
On Approximation of the Power-p and Bottleneck Steiner Trees
Combinatorial Optimization, 2000
Many VLSI routing applications, as well as the facility location problem involve computation of Steiner trees with non-linear cost measures. We consider two most frequent versions of this problem. In the power-p Steiner problem the cost is de ned as the sum of the edge lengths where each length is raised to the power p > 1. In the bottleneck Steiner problem the objective cost is the maximum of the edge lengths. We show that the power-p Steiner problem is MAX SNP-hard and that one cannot guarantee to nd a bottleneck Steiner tree within a factor less than 2, unless P = NP. We prove that in any metric space the minimum spanning tree is at most a constant times worse than the optimal power-p Steiner tree. In particular, for p = 2, we show that the minimum spanning tree is at most 23.3 times worse than the optimum and we construct an instance for which it is 17.2 times worse. We also present a better approximation algorithm for the bottleneck Steiner problem with performance guarantee log 2 n, where n is the number of terminals (the minimum spanning tree can be 2 log 2 n times worse than the optimum).
An improved LP-based approximation for steiner tree
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10, 2010
The Steiner tree problem is one of the most fundamental AEÈ-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from ¾ to the current best ½ [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than ¾ [Vazirani,Rajagopalan-SODA'99].
On the number of minimal 1-Steiner trees
Discrete & Computational Geometry, 1994
We count the number of nonisomorphic geometric minimum spanning trees formed by adding a single point to an n-point set in d-dimensional space, by relating it to a family of convex decompositions of space. The O(n d log 2d2-d n) bound that we obtain significantly improves previously known bounds and is tight to within a polylogarithmic factor.
Analysis of Steiner subtrees of Random Trees for Traceroute Algorithms
2007
We consider in this paper the problem of discovering, via a traceroute algorithm, the topology of a network, whose graph is spanned by an infinite branching process. A subset of nodes is selected according to some criterion. As a measure of efficiency of the algorithm, the Steiner distance of the selected nodes, i.e. the size of the spanning sub-tree of these nodes, is investigated. For the selection of nodes, two criteria are considered: A node is randomly selected with a probability, which is either independent of the depth of the node (uniform model) or else in the depth biased model, is exponentially decaying with respect to its depth. The limiting behavior the size of the discovered subtree is investigated for both models.
A near linear time approximation scheme for Steiner tree among obstacles in the plane
Computational Geometry, 2010
We present a polynomial-time approximation scheme (PTAS) for the Steiner tree problem with polygonal obstacles in the plane with running time O(n log 2 n), where n denotes the number of terminals plus obstacle vertices. To this end, we show how a planar spanner of size O(n log n) can be constructed that contains a (1 + ǫ)-approximation of the optimal tree. Then one can find an approximately optimal Steiner tree in the spanner using the algorithm of for the Steiner tree problem in planar graphs. We prove this result for the Euclidean metric and also for all uniform orientation metrics, i.e. particularly the rectilinear and octilinear metrics.