Showing papers on "Minimum degree spanning tree published in 1985"
01 Sep 1985
TL;DR: A protocol and algorithm are given in which bridges in an extended Local Area Network of arbitrary topology compute, in a distributed fashion, an acyclic spanning subset of the network.
Abstract: A protocol and algorithm are given in which bridges in an extended Local Area Network of arbitrary topology compute, in a distributed fashion, an acyclic spanning subset of the networkThe algorithm converges in time proportional to the diameter of the extended LAN, and requires a very small amount of memory per bridge, and communications bandwidth per LAN, independent of the total number of bridges or the total number of links in the networkAlgorhymeI think that I shall never see A graph more lovely than a treeA tree whose crucial property Is loop-free connectivityA tree which must be sure to span So packets can reach every LANFirst the Root must be selected By ID it is electedLeast cost paths from Root are traced In the tree these paths are placedA mesh is made by folks like me Then bridges find a spanning tree
274 citations
TL;DR: This paper proves that for any e > 0 limn → ∞ Pr(|Ln− ζ(3)/D|) > e) = 0.202, which is the length of the minimum spanning tree in such a graph.
251 citations
TL;DR: A branch and bound algorithm to solve the degree-constrained minimum spanning tree problem and an edge exchange analysis frequently used in the algorithm and three types of heuristic methods are proposed.
76 citations
TL;DR: An algorithm which finds the maximum value of every one of the given paths, and which uses only O(n+e) comparisons in a graph with n vertices and e edges is described.
Abstract: Given a rooted tree with values associated with then vertices and a setA of directed paths (queries), we describe an algorithm which finds the maximum value of every one of the given paths, and which uses only
$$5n + n\log \frac{{\left| A \right| + n}}{n}$$
comparisons. This leads to a spanning tree verification algorithm usingO(n+e) comparisons in a graph withn vertices ande edges. No implementation is offered.
59 citations
TL;DR: In this paper, it was shown that the diameter of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P, as previously claimed in [1] and [2].
Abstract: In this paper it is shown that the diameter D( P ) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P , as previously claimed in [1] and [2]. It is also proved that if P is contained in the disk determined by D( P ) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD( P ) intersects its corresponding edge in the FPVD( P ) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in [1] and [2]. A proof of correctness is given for the minimum spanning circle algorithm proposed in [2] and [3]. Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.
33 citations
TL;DR: It is shown that any pair of edge-disjoint spanning trees can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being a spanning tree.
Abstract: It is well known that any spanning tree of a graph can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being a spanning tree. We consider a variation of this problem concerning pairs of edge-disjoint spanning trees. In particular, it is shown that any pair of edge-disjoint spanning trees can be obtained from any other by a sequence of single edge exchanges in a way that preserves, at each step, the property of being edge-disjoint spanning trees.
32 citations
TL;DR: This chapter presents shrinking algorithm for the spanning circle problem, which has a worst-case bound of 0(n 2 ) and works for any point set.
Abstract: Publisher Summary This chapter discusses an implementation study of two algorithms for the minimum spanning circle problem. The chapter presents shrinking algorithm for the spanning circle problem. This algorithm has a worst-case bound of 0(n 2 ) and works for any point set. There are three major components in an implementation of the Shrink algorithm: (1) the way to find an initial spanning circle; (2) the way to test if the set of contact points fits in a semicircle of the current spanning circle, and, if so, how to compute the separating line; and (3) the way to shrink to a smaller spanning circle. Approximation algorithm to the minimum spanning circle problem is much different. An approximation algorithm computes a convex region that is sure to contain the center of the minimum spanning circle. The area of this region may be made as small as desired allowing the location of the center to be approximated more and more accurately. The accuracy with which the center is located is a parameter of the algorithm and should be chosen with the application in mind.
18 citations
TL;DR: In this article, the authors examined the problem of producing a spanning Eulerian subgraph in an undirected graph and presented polynomial-time algorithms for both the maximization and minimization versions where instances are defined on a restricted class of graphs referred to as series-parallel.
Abstract: In this article, we examine the problem of producing a spanning Eulerian subgraph in an undirected graph. After the ℋ-completeness of the general problem is established, we present polynomial-time algorithms for both the maximization and minimization versions where instances are defined on a restricted class of graphs referred to as series-parallel. Some novelties in the minimization case are discussed, as are heuristic ideas.
16 citations
TL;DR: If the independence number of a k -connected digraph D is at most k, then D has a spanning subgraph which consists of a union of vertex-disjoint directed circuits, and the minimum number of edges required in a k-connected oriented graph is determined.
13 citations
TL;DR: It is shown that for any v, there is an e* such that if, then any nearly balanced graph in ζv, e has more spanning trees than any non-nearly-balanced graph in €1, e.
Abstract: Let di the degree of the ith vertex of a mutigraph and λij be the number of edges between vertex i and vertex j. A multigraph is called nearly balanced if |di − di| ≤1 for all i ≠i′ and |λij −λij| for all i and all j,j′ Let be the collection of all the multigraphs with v vertices and e edges. It is shown that for any v, there is an e* such that if, then any nearly balanced graph in ζv, e has more spanning trees than any non-nearly-balanced graph in ζv, e.
TL;DR: In this article, a sufficient condition for a set to be a feasible tree-diameter set is given and this solves a conjecture by Harary et al. on feasible tree diameter sets.
Abstract: The tree-diameter set of a connected graph G is the set of all diameters of the spanning trees of G , written in the increasing order. A relation between the consecutive elements of this set is obtained and it is shown to be the best possible. A sufficient condition for a set to be a feasible tree-diameter set is given and this solves a conjecture by Harary et al. on feasible tree-diameter sets.
01 Aug 1985
TL;DR: In this paper, it was shown that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases: first, the construction of a minimum weight spanning tree requires, in the worst case, at least $Omega (n^2 )$ messages.
Abstract: In a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in $O(n\log n)$ messages. We show in this work that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases: First we show that the construction of a minimum weight spanning tree requires, in the worst case, at least $\Omega (n^2 )$ messages, and then we show that the construction of a spanning tree where the maximum degree is at most k may require at least $\Omega ({{n^2 } / k})$ messages in the worst case. Actually, in both cases the lower bounds are shown for the number of edges used in the worst case. Moreover, the results are valid for both asynchronous and synchronous networks, and are independent of the lengths of the messages. On the other hand, there are algorithms for the above tasks which achieve these lower bounds, up to a constant factor, and use messages of $O(\log n)$ length.
TL;DR: In this paper, an algorithm is proposed to determine n-1 =card (T) new minimum spanning trees, one for each edge e, of T with a total expense of O(card (N)2) computational operations.
Abstract: Let G = (E N) be an undirected graph and T be a minimum spanning tree. If one edge e of T is removed from G then a new minimum spanning tree Te can be obtained from T recursively. In this paper an algorithm is proposed to determine n— 1 =card (T) new minimum spanning trees Te , one for each edge e, of T. with a total expense of O (card (N)2) computational operations. In addition an application to the symmetric traveling salesman problem is given.