Showing papers on "Spanning tree published in 1981"

A fast algorithm for Steiner trees

Abstract: Given an undirected distance graph G=(V, E, d) and a set S, where V is the set of vertices in G, E is the set of edges in G, d is a distance function which maps E into the set of nonnegative numbers and S?V is a subset of the vertices of V, the Steiner tree problem is to find a tree of G that spans S with minimal total distance on its edges. In this paper, we analyze a heuristic algorithm for the Steiner tree problem. The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦ 2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1?1/l) times that of the optimal tree, where l is the number of leaves in the optimal tree.

A data structure for dynamic trees

Abstract: We propose a data structure to maintain a collection of vertex-disjoint trees under a sequence of two kinds of operations: a link operation that combines two trees into one by adding an edge, and a cut operation that divides one tree into two by deleting an edge. Our data structure requires O(log n) time per operation when the time is amortized over a sequence of operations. Using our data structure, we obtain new fast algorithms for the following problems: (1) Computing deepest common ancestors. (2) Solving various network flow problems including finding maximum flows, blocking flows, and acyclic flows. (3) Computing certain kinds of constrained minimum spanning trees. (4) Implementing the network simplex algorithm for the transshipment problem. Our most significant application is (2); we obtain an O(mn log n)-time algorithm to find a maximum flow in a network of n vertices and m edges, beating by a factor of log n the fastest algorithm previously known for sparse graphs.

Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations

Abstract: We consider the problem of routing vehicles stationed at a central facility (depot) to supply customers with known demands, in such a way as to minimize the total distance travelled The problem is referred to as the vehicle routing problem (VRP) and is a generalization of the multiple travelling salesman problem that has many practical applications We present tree search algorithms for the exact solution of the VRP incorporating lower bounds computed from (i) shortest spanning k-degree centre tree (k-DCT), and (ii) q-routes The final algorithms also include problem reduction and dominance tests Computational results are presented for a number of problems derived from the literature The results show that the bounds derived from the q-routes are superior to those from k-DCT and that VRPs of up to about 25 customers can be solved exactly

Minimum cost spanning tree games

Abstract: We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.

Reliability Evaluation in Computer-Communication Networks

Abstract: The network reliability for a computer-communication network is defined and a method based on spanning trees is proposed for its evaluation. The network reliability expression and s-t terminal reliability expression are compared assuming an equal probability of survival for each communication link. Examples illustrate the method.

The Selection of Good Search Terms.

Abstract: This paper tackles the problem of how one might select further search terms, using relevance feedback, given the search terms in the query. These search terms are extracted from a maximum spanning tree connecting all the terms in the index term vocabulary. A number of different spanning trees are generated from a variety of association measures. The retrieval effectiveness for the different spanning trees is shown to be approximately the same. Effectiveness is measured in terms of precision and recall, and the retrieval tests are done on three different test collections.

Fast, Efficient Parallel Algorithms for Some Graph Problems

Abstract: Algorithms for solving graph problems on an unbounded parallel model of computation are considered. Parallel algorithms of time complexity $O(\log ^2 n)$ are described for finding biconnected components, bridges, minimum spanning trees and fundamental cycles. In the algorithms for finding minimum spanning trees, bridges, and fundamental cycles, the number of processors used is small enough that the parallel algorithm is efficient in comparison with the best sequential algorithms for these problems. Several other algorithms are presented which are especially suitable for processing sparse graphs.

An Algorithm for Finding K Minimum Spanning Trees

Abstract: This paper presents an algorithm for finding K minimum spanning trees in an undirected graph. The required time is $O(Km + \min (n^2 ,m\log \log n))$ and the space is $O(K + m)$, where n is the number of vertices and m is the number of edges. The algorithm is based on three subroutines. The first two subroutines are used to obtain the second minimum spanning tree in $O(\min (n^2 ,m\alpha (m,n)))$ steps, where $\alpha (m,n)$ is Tarjan’s inverse of Ackermann’s function [12] which is very slowly growing. The third one obtains the kth minimum spanning tree in $O(m)$ steps when the jth minimum spanning trees for $j = 1,2, \cdots ,k - 1$ are given.

The complexity of designing a network with minimum diameter

Abstract: Given a graph G with edge lengths and costs, we wish to find a subgraph F of G which connects up all the original vertices and minimizes the maximum distance in F, subject to a budget constraint on the sum of the edge costs. In this note we establish NP-hardness for the design problem, even for the simple case where G is a planar graph with maximum degree 3 and the budget restricts the choice to spanning trees. Moreover, the problem of finding a near optimal subgraph F is also NP-hard.

Equiarboreal graphs

Abstract: A graphX is said to beequiarboreal if the number of spanning trees containing a specified edge inX is independent of the choice of edge. We prove that any graph which is a colour class in an association scheme (and thus any distance regular graph) is equiarboreal. We note that a connected equiarboreal graph withM edges andn vertices has edge-connectivity at leastM/(n−1).

A Systolic Data Structure Chip for Connectivity Problems

Abstract: In this paper we present an example of a design for a “data structure chip” and suggest how it can be used for problem solving in a digital system. In particular, we describe a systolic structure which can be used, for a graph, to find the connected components, a spanning tree, or, when used in conjunction with a systolic priority queue, a minimum spanning tree.

A comparison of three algorithms for finding fundamental cycles in a directed graph

Abstract: Given a connected directed graph and a spanning tree, we consider the problem of finding the set of fundamental cycles. In particular, for each cotree arc i and tree arc j, we need to know whether or not i and j are in the same fundamental cycle, and if so, whether or not arcs i and j are oriented in the same direction. This problem has application in primal network flow, longest cycle, and all-cycle algorithms. In this paper, we describe and compare three algorithms for finding fundamental cycles. Computational results are presented on a variety of directed graphs produced by a network generator. Although each of the algorithms has worst case complexity O(kp), where k and p are the number of cotree arcs and nodes, respectively, a variation of a root traceback algorithm is shown to be the fastest in almost all cases.

Algorithm for finding the common spanning trees of two graphs

Abstract: A new algorithm is described which finds the common spanning trees of two graphs efficiently and without duplication. The algorithm can be incorporated in a symbolic circuit analysis program where its speed and storage requirements are such that circuits of considerable complexity can be analysed on small computers.

Minimal Cost-Reliability Ratio Spanning Tree*

Abstract: The problem of finding a minimal cost-reliability ratio spanning tree in a network is considered. The optimal solution to this problem is shown to map into an efficient extreme point of the convex hull of spanning trees in a bicriteria problem of minimizing the negative of the logarithm of reliability and cost. Based on this result a polynomial algorithm consisting of an indirect search (without requiring to find logarithms) in the set of efficient extreme points is given for computing the solution to the cost-reliability ratio problem.

Delay Analysis of Broadcast Routing in Packet-Switching Networks

Abstract: Broadcast addressing is the capability to send a packet from a source node to all other nodes in the network. Store-and-forward, packet-switching networks are not inherently designed to carry broadcast packets, and broadcasting has to be implemented by some sort of routing algorithm. In this paper, the source based forwarding algorithm is considered. With this algorithm, a spanning tree is defined for each node, and broadcast packets are sent along the branches of these trees. Approximation methods are presented to obtain a lower bound and estimates of the mean broadcast time. The accuracy of these methods is evaluated by comparison with simulation.

Finding minimal spanning trees in a euclidean coordinate space

Abstract: The minimal spanning tree problem of a point set in ak-dimensional Euclidean space is considered and a new version of the multifragmentMST-algorithm of Bentley and Friedman is given. The minimal spanning tree is found by repeatedly joining the minimal subtree with the closest subtree. Ak-d tree is used for choosing the connecting edges. Computation time of the algorithm depends on the configuration of the point set: for normally distributed random points the algorithm is very fast. Two extreme cases demandingO(n logn) andO(n 2) operations,n being the cardinality of the point set, are also given.

Complexity of Optimum Undirected Tree Problems: A Survey of Recent Results

Abstract: Networks design is often concerned with the problem of finding optimum weighted spanning trees. This work summarizes some recent results about the computational complexity of these problems with the aim of identifying the borderline between “easy” and “hard” problems. Several tree weight functions and side constraints are considered.

On minimum spanning trees and the intergradation of clusters

Abstract: Cluster analysis groups samples, but does not generally show gradations between clusters. To illustrate these relationships, principal-coordinate analysis and multidimensional scaling can be employed, but these methods may not be appropriate due to structural distortion. A minimum spanning tree (MST) computes a point-to-point path through the original matrix, and clusters are mapped on the MST diagram. The MST linkages, therefore, provide a natural continuum between the clusters, without distortion. Forty-six Silurian Tonoloway Limestone samples were analyzed for 16 sedimentary features indicative of subaerial exposure. Cluster analysis identified nine clusters which were reduced to five final, ordered groups by MST.

A note on weighted minimal cost flows

Abstract: The minimal ratio problem which is treated in the literature for shortest paths [Dantzig/Blattner/Rao;Karp;Lawler, 1966, 1972] and for spanning trees [Chandrasekaran] is considered in a generalized form for network flow problems. The resulting problem of finding a so-calledweighted minimal cost flow can be solved by a negative circuit algorithm or by a shortest augmenting circuit algorithm. The validity of both algorithms follows from a negative circuit theorem which can be proved for weighted minimal cost flows.

Computer Minimization Of Wire Routing For Single Family Dwellings

Abstract: This paper describes a graph theoretic method by which the electrical o utlets in a single family dwelling may be wired in such a manner as to minimize the t otal conductor length from the s ervice panel. A minimal spanning tree algorithm is modified in order to grow a minimally weighted rooted spanning tree that models the general lighting branch circuits. The modified algorithm is implemented in order to obtain a minimal total wire length for a home. The algorithm predicts minimal total wire lenqth, assuming the tree is qrown from preselected arms in conformance with the National Electric Code. An example of the method described is presented.

Minimum spanning tree with normal variates as weights

Abstract: When weights of arcs in a graph are normal variates, we seek a spanning tree maximizing the probability that the sum of wei!:ilts of arcs in the spanning tree is not greater than a given constant. An 0(e 2 n) algorithm for it is given.