On the number of minimal 1-Steiner trees

Discrete Optimization, 2008

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Discrete Mathematics, 1994

Let Q(n) be the n-dimensional hypercube, and X a set of points in Q(n). The Steiner problem for the hypercube is to find the smallest number L(n,X) of edges in any subtree of Q(n) which spans X. Let W(k,n) be the set of points in Q(n) having weight k, where we normalize k+1 ≤ n 2

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.

Algorithmica, 1988

In recent years, researchers have proven many theorems of the following form: given points distributed according to a Poisson process with intensity parameter N on the unit square, the length of the shortest spanning tree (rectilinear Steiner tree, traveling salesman tour, or some other functional) on these points is, with probability one, asymptotic to/3-fN for some constant/3 (which is presumably different for different functionals). Though these theorems are well understood, very little is known about the constants /3. In this paper we prove that the constants in the cases of rectilinear spanning tree and rectilinear Steiner tree do, indeed, differ. This proof is constructive in the sense that we give a polynomial-time heuristic algorithm that produces a Steiner tree of expected length some fraction shorter than a minimum spanning tree. We continue the analysis to prove a second result: the expected value of the minimum number of Steiner points in a shortest rectilinear Steiner tree grows linearly with N.

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].

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|.

2006

We show that the α-weight of an MST over n points in a metric space with upper box dimension d has a bound independent of n if α < d and does not have one if α > d.

cs.uoi.gr

The Kn-complement of a graph G, denoted by Kn -G, is defined as the graph obtained from the complete graph Kn by removing a set of edges that span G; if G has n vertices, then Kn -G coincides with the complement G of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K m n ± G, where K m n is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K m n ; the graph n by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K m n by adding and removing edges of multigraphs spanned by sets of edges of the graph K m n . We also prove closed formulas for the number of spanning tree of graphs of the form K m n ± G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.

Applied Mathematics Letters, 2010

1994

We study the problem of nding small trees. Classical network design problems are considered with the additional constraint that only a speci ed number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2 p k for the general edge-weighted case and O(k 1=4 ) for the case of points in the plane.

Journal of Discrete Algorithms, 2012

We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in O(B 2 tw+2 · tw · |V |) time, where tw is the graph's treewidth and the Bell number B k is the number of partitions of a k-element set. This is a linear time algorithm for graphs with fixed treewidth and a polynomial algorithm for tw = O(log |V |/ log log |V |). While being faster than the previously known algorithms, our thereby used coloring scheme can be extended to give new, improved algorithms for the prize-collecting Steiner tree as well as the k-cardinality tree problems.

Discrete and Computational Geometry, 2003

We consider the problem of finding low-cost spanning trees for sets of n points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of m barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost O(min(m 2 , m √ n)); (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost O(m); (iii) if the barriers are disjoint convex objects, then there is always a spanning tree of cost O(n + m).

Discrete & Computational Geometry, 2010

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T ) = O(k · n 1/k ) · w(M ST (M )), and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(M ST (M )) and unweighted diameter O(k · n 1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.

1997

Let G = (V , E) be a simple graph with n vertices, e edges and d 1 be the highest degree. Further let λ i , i = 1, 2,. .. , n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ 2 = λ 3 =. .. = λ n−1 if and only if G is a complete graph or a star graph or a (d 1 , d 1) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d 1 only: t (G) ≤ 2e − d 1 − 1 n − 2 n−2 .

Algorithmica, 2006

We present some fundamental structural properties for minimum length networks (known as Steiner minimum trees) interconnecting a given set of points in an environment in which edge segments are restricted to £ ¥