Showing papers on "Spanning tree published in 1993"

Power system observability with minimal phasor measurement placement

Abstract: The placement of a minimal set of phasor measurement units (PMUs) so as to make the system measurement model observable, and thereby linear, is investigated. A PMU placed at a bus measures the voltage as well as all the current phasors at that bus, requiring the extension of the topological observability theory. In particular, the concept of spanning tree is extended to that of spanning measurement subgraph with an actual or a pseudomeasurement assigned to each of its branches. The minimal PMU set is found through a dual search algorithm which uses both a modified bisecting search and a simulated-annealing-based method. The former fixes the number of PMUs while the latter looks for a placement set that leads to an observable network for a fixed number of PMUs. In order to accelerate the procedure, an initial PMU placement is provided by a graph-theoretic procedure which builds a spanning measurement subgraph according to a depth-first search. From computer simulation results for various test systems it appears that only one fourth to one third of the system buses need to be provided with PMUs in order to make the system observable. >

How bad is naive multicast routing

Abstract: In previous approaches to routing multicast connections in networks, the emphasis has been on the source transmitting to a fixed set of destinations (the multicast group). There are some applications where destinations will join and leave the multicast group. Under these conditions, computing an 'optimal' spanning tree after each modification may not be the best way to proceed. An alternative is to make modest alterations to an existing spanning tree to derive a new one. An extreme, though nonoptimal, variation of this is to use minimal cost source to destination routing for each destination, effectively ignoring the existing multicast tree. The authors examine just how nonoptimal these trees are in random general topology networks and conclude that they are worse by only a small factor. The factor is reduced still further if a hierarchy is imposed on the random network to give a more realistic model. >

Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings Via Transfer-Impedances

Abstract: Let $G$ be a finite graph or an infinite graph on which $\mathbb{Z}^d$ acts with finite fundamental domain. If $G$ is finite, let $\mathbf{T}$ be a random spanning tree chosen uniformly from all spanning trees of $G$; if $G$ is infinite, methods from Pemantle show that this still makes sense, producing a random essential spanning forest of $G$. A method for calculating local characteristics (i.e., finite-dimensional marginals) of $\mathbf{T}$ from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix ($n$-dimensional minors) are needed to compute small ($n$-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for $G$, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when $G$ is an infinite graph admitting a $Z^d$-action with finite quotient. The same computation also gives the entropy of the law of $\mathbf{T}$. These results are applied to the problem of tiling certain lattices by dominos--the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to tree-valued random variables and criteria for these generalized moments to determine a distribution.

Many birds with one stone: multi-objective approximation algorithms

Abstract: We study network-design problems with multiple design objectives. In particular, we look at two cost measures to be minimized simultaneously: the total cost of the network and the maximum degree of any node in the network. Our main result can be roughly stated as follows: given an integer $b$, we present approximation algorithms for a variety of network-design problems on an $n$-node graph in which the degree of the output network is $O(b \log (\frac{n}{b}))$ and the cost of this network is $O(\log n)$ times that of the minimum-cost degree-$b$-bounded network. These algorithms can handle costs on nodes as well as edges. Moreover, we can construct such networks so as to satisfy a variety of connectivity specifications including spanning trees, Steiner trees and generalized Steiner forests. The performance guarantee on the cost of the output network is nearly best-possible unless $NP = \tilde{P}$. We also address the special case in which the costs obey the triangle inequality. In this case, we obtain a polynomial-time approximation algorithm with a stronger performance guarantee. Given a bound $b$ on the degree, the algorithm finds a degree-$b$-bounded network of cost at most a constant time optimum. There is no algorithm that does as well in the absence of triangle inequality unless $P = NP$. We also show that in the case of spanning networks, we can simultaneously approximate within a constant factor yet another objective: the maximum cost of any edge in the network, also called the bottleneck cost of the network. We extend our algorithms to find TSP tours and $k$-connected spanning networks for any fixed $k$ that simultaneously approximate all these three cost measures.

Graph types

Abstract: Recursive data structures are abstractions of simple records and pointers. They impose a shape invariant, which is verified at compile-time and exploited to automatically generate code for building, copying, comparing, and traversing values without loss of efficiency. However, such values are always tree shaped, which is a major obstacle to practical use.We propose a notion of graph types, which allow common shapes, such as doubly-linked lists or threaded trees, to be expressed concisely and efficiently. We define regular languages of routing expressions to specify relative addresses of extra pointers in a canonical spanning tree. An efficient algorithm for computing such addresses is developed. We employ a second-order monadic logic to decide well-formedness of graph type specifications. This logic can also be used for automated reasoning about pointer structures.

Optimally sparse spanners in 3-dimensional Euclidean space

Abstract: Let V be a set of n points in 3-dimensional Euclidean space A subgraph of the complete Euclidean graph is a t-spanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most ttimes d(u, v) We show that for any t > 1, a greedy algorithm produces a t-spanner with O(n) edges, and total edge weight O(1)wt(MST), where MST is a minimum spanning tree of V

A sub-linear time distributed algorithm for minimum-weight spanning trees

Abstract: This paper considers the question of identifying the parameters governing the behavior of fundamental global network problems. Many papers on distributed network algorithms consider the task of optimizing the running time successful when an O(n) bound is achieved on an n-vertex network. We propose that a more sensitive parameter is the network's diameter Diam. This is demonstrated in the paper by providing a distributed minimum-weight spanning tree algorithm whose time complexity is sub-linear in n, but linear in Diam (specifically, O(Diam+n/sup 0.614/)). Our result is achieved through the application of graph decomposition and edge elimination techniques that may be of independent interest. >

Time Optimal Self-Stabilizing Spanning Tree Algorithms

Abstract: In this paper we present time-optimal self-stabilizing algorithms for asynchronous distributed spanning tree computation in networks. We present both a randomized algorithm for anonymous networks as well as a deterministic version for ID-based networks. Our protocols are the first to be time-optimal (i.e. stabilize in time O(diameter)) without any prior knowledge of the network size or diameter, assuming we are allowed messages of size O(ID). Both results are achieved through a new technique of symmetry breaking that may be of independent interest.

An optimal algorithm for selection in a min-heap

Abstract: An O(k)-time algorithm is presented for selecting the kth smallest element in a binary min-heap of size n⪢k. The approach uses recursively defined data structures that impose a hierarchical grouping on certain elements in the heap. The result establishes a further example of a partial order for which the kth smallest element can be determined in time proportional to the information theory lower bound. Two applications, to a resource allocation problem and to the enumeration of the k smallest spanning trees, are identified.

Applications of a new space-partitioning technique

Abstract: We present several applications of a recent space-partitioning technique of Chazelle, Sharir, and Welzl (Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, pp. 23---33). Our results include efficient algorithms for output-sensitive hidden surface removal, for ray shooting in two and three dimensions, and for constructing spanning trees with low stabbing number.

Scan-first search and sparse certificates: an improved parallel algorithm for k -vertex connectivity

Abstract: Given a graph $G = (V,E)$, a certificate of k-vertex connectivity is an edge subset $E' \subset E$ such that the subgraph $(V,E')$ is k-vertex connected if and only if G is k-vertex connected. Let n and m denote the number of vertices and edges. A certificate is called sparse if it contains $O(kn)$ edges.For undirected graphs, this paper introduces a graph search called the scan-first search, and shows that a certificate with at most $k(n - 1)$ edges can be computed by executing scan-first search k times in sequence on subgraphs of G. For each of the parallel, distributed, and sequential models of computation, the complexity of scan-first search matches the best complexity of any graph search on that model. In particular, the parallel scan-first search runs in $O(\log n)$ time using $C(n,m)$ processors on a CRCW PRAM, where $C(n,m)$ is the number of processors needed to find a spanning tree in each connected component in $O(\log n)$ time, and the parallel certificate algorithm runs in $O(k\log n)$ time us...

A direct combination of the Prim and Dijkstra constructions for improved performance-driven global routing

Abstract: Motivated by analysis of distributed RC delay in routing trees, a new tree construction is proposed for performance-driven global routing which directly trades off between Prim's minimum spanning tree algorithm and Dijkstra's shortest path tree algorithm. This direct combination of two objective functions and their corresponding optimal algorithms contrasts with the more indirect 'shallow-light' methods. The authors' method achieves routing trees which satisfy a given routing tree radius bound while using less wire than previous methods. Detailed simulations show that these wirelength savings translate into significantly improved delay over standard MST routing in both IC and multichip module (MCM) interconnect technologies. >

Method of deleting and adding nodes in a spanning tree network by collating replies from other nodes

Abstract: A computer forms a node in a network, consisting of computer nodes linked together into a minimum spanning tree topology. When a computer receives a message from a first node linked to it, it forwards the message to other nodes linked to that computer, as well as storing information about the message. As replies are received from the other computers, they are stored and collated together, to allow the computer to send just a single reply message back to the originating node based on the collated replies. This single reply is in turn collated at the next node. The message requests deletion of a particular node from the network. No node deletes the node from the network until replies have been received from all the nodes to which the message was forwarded.

Topological design of interconnected LAN/MAN networks

Abstract: The authors describe a methodology for designing interconnected LAN/MAN networks with the objective of minimizing the average network delay. They consider IEEE 802 standard LANs interconnected by transparent bridges. These bridges are required to form a spanning tree topology. The authors propose a simulated annealing-based algorithm for designing minimum delay spanning tree topologies. In order to measure the quality of the solutions, a lower bound for the average network delay is found. The algorithm is extended to design the overall LAN/MAN topology consisting of a MAN or high-speed data service interconnecting several clusters of bridged LANs. Comparison with the lower bound and several other measures show that the solutions are not very far from the global minimum. >

Minimax open shortest path first routing algorithms in networks supporting the SMDS service

Abstract: Two quasi-static minimax open shortest path first (OSPF) routing algorithms in networks supporting the Switched Multi-megabit Data Service (SMDS) are presented and compared. In OSPF routing, the network is modeled as a graph and each link is associated with a nonnegative arc weight. A shortest path spanning tree is calculated for each origin to carry both the individually addressed and the group addressed (multicast) traffic. The OSPF routing protocol is adopted as a major part of the default inter-switching system interface (ISSI) routing algorithm for SMDS networks where arc weights are inversely proportional to the aggregate link set capacities. The problem of choosing a set of link set metrics is considered so that the maximum link utilization factor is minimized in an SMDS network. The problem is formulated as a nonlinear mixed integer programming problem. >

An Implementation of the Dual Affine Scaling Algorithm for Minimum-Cost Flow on Bipartite Uncapacitated Networks

Abstract: This paper describes an implementation of the dual affine scaling algorithm for linear programming specialized to solve minimum-cost flow problems on bipartite uncapacitated networks. This implementation uses a preconditioned conjugate gradient algorithm to solve the system of linear equations that determines the search direction at each iteration of the interior point algorithm. Two preconditioners are considered: a diagonal preconditioner and a preconditioner based on the incidence matrix of an approximate maximum weighted spanning tree of the network. Under dual nondegeneracy this spanning tree allows for early identification of the optimal solution. By applying an $\epsilon $-perturbation to the cost vector, an optimal extreme-point primal solution is produced in the presence of dual degeneracy. The implementation is tested by solving several large instances of randomly generated assignment problems, comparing solution times with the network simplex code NETFLO and the relaxation algorithm code RELAX....

Multicasting in a linear lightwave network

Abstract: Dynamic routing algorithms are proposed for setting up multicast connections in a linear lightwave network (LLN). The problem of finding a physical path for the multicast connection so as to satisfy all the constraints in the LLN is shown to be NP-complete, and a heuristic approach is presented. An algorithm is presented that decomposes the LLN into edge disjoint trees with at least one spanning tree. A multicast call is allocated a physical path on one of the trees, using the smallest component tree (SCT) or the minimum interference tree (MIT) criterion. Finally, the call is allocated the least used channel from among channels that can be allocated to it. The best performance (low blocking probability) is obtained when the LLN is decomposed into many spanning trees, each of them having a small diameter. It is also found that the selection of trees for each call using the MIT criterion exhibits better performance than with the SCT criterion. >

Graph—theoretical approach to colour picture segmentation and contour classification

Abstract: The segmentation of colour pictures in a graph-theoretical context is considered. The procedure aims at identifying, extracting and classifying visually important features on the image plane, such as regions of homogeneous colour and chromatic transitions. Well established principles of colour theory and graph theory are combined to obtain a unified representation of a colour picture. The picture is represented by means of a weighted graph, constructed so as to reflect the specification of the colour space employed as well as important relationships between picture elements. A spanning tree of the graph is obtained by iteratively minimising a specific picture distortion measure. This tree structure describes a hierarchy of partitions on the image plane. Each partition comprises disjoint regions containing elements with similar attribute. Due to the fact that region identification and edge detection form dual problems from the graph-theoretical viewpoint, region contours defined by such partitions form a hierarchy. To avoid artificial contouring, a specific type of artefact introduced by the segmentation algorithm, the use of higher level information, is considered. It is shown that, when texture (which is taken into account at an intermediate stage of picture segmentation) is combined with colour as joint similarity attributes of regions, an improved hierarchical description of contours is possible. This facilitates the progressive elimination of the undesirable contours and leads to the visual enhancement of the segmentation obtained.

Balancing minimum spanning and shortest path trees

Abstract: Efficient algorithms are known for computing a minimum spann.ing tree, or a shortest path. tree (with a fixed vertex as the root). The weight of a shortest path tree can be much more than the weight of a minimum spa,nning tree. Conversely, the distance bet,ween the root, and any vertex in a minimum spanning tree may be much more than the distance bet#ween the two vertices in the graph. Consider the problem of balancing between the two kinds of trees: Does every graph contain a tree that is “light” (at most a constant times heavier than the minimum spanning t,ree), such that the distance from the root to any vertex in t,he tree is no more than a constant times the true distance? This paper answers the question in the affirmative. It is shown that there is a continuous tradeoff between the two parameters. For every y > 0, there is a tree in the graph whose total weight is at most 1 + $? times the weight of a minimum spanning tree, such that the di&nce in the tree between the root, and any vertex is at, most 1 + &y times the true distance. Efficient sequential and parallel algorithms achieving these factors are provided. The algorithms are shown to be optimal in two ways. First, it is shown that no algorithm can achieve better factors in all graphs, because there a.re graphs that do not have better trees. Second, it is shown that even on a per-graph basis, finding trees that achieve better factors is NP-hard.

Boundary edge elements and spanning tree technique in three‐dimensional electromagnetic field computation

Abstract: The boundary discretization of the curl-free vector variable such as the magnetic field h and the divergencefree variable such as the surface current density k is discussed. Instead of introducing the scalar variables, the vector variables h and k are discretized by the curl-conform and the div-conform triangular edge elements, respectively. The degrees of freedom are associated with the boundary edges. In order to ensure the null curl of h and the null divergence of k, a spanning tree technique is used to identify the independent edges. The triangular edge elements contain the first-order nodal elements when expressing h or k by the scalar variables

A branch and bound algorithm for the capacitated minimum spanning tree problem

Abstract: Given an undirected graph G = (N, E) with a cost associated with each edge C: E → R+ and a demand associated with each node A: N → R+. A special node is designated as the center. The capacitated minimum spanning tree (CMST) problem is to find a minimum spanning tree for graph G such that the sum of demands on each branch stem from the center does not exceed a given capacity. The CMST problem has many applications in network design, centralized telecommunications, and vehicle routing. In this paper, we present a new formulation and a full optimization algorithm by branch and bound. The lower bounds are generated by Lagrangean relaxation with tightening constraints. Computational results based upon the methodology presented are shown. © 1993 by John Wiley & Sons, Inc.

Scattering and gathering messages in networks of processors

Abstract: The operations of scattering and gathering in a network of processors involve one processor of the network (P/sub 0/) communicating with all other processors. In scattering, P/sub 0/ sends distinct messages to P/sub 0/. The authors consider networks that are trees of processors. Algorithms for scattering messages from and gathering messages to the processor that resides at the root of the tree are presented. The algorithms are quite general, in that the messages transmitted can differ arbitrarily in length; quite strong, in that they send messages along noncolliding paths, and hence do not require any buffering or queueing mechanisms in the processors; and quite efficient in that algorithms for scattering in general trees are optimal, the algorithm for gathering in a path is optimal and the algorithms for gathering in general trees are nearly optimal. The algorithms can easily be converted using spanning trees to efficient algorithms for scattering and gathering in networks of arbitrary topologies. >

The complexity of detecting crossingfree configurations in the plane

Abstract: The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.

Balancing minimum spanning trees and shortest-path trees

Abstract: We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and aź>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+ź2ź times the shortest-path distance, and yet the total weight of the tree is at most 1+ź2/ź times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.

Long non-crossing configurations in the plane

Abstract: We study some geometric maximization problems in the Euclidean plane under the non-crossing constraint. Given a set V of 2n points in general position in the plane, we investigate the following geometric configurations using straight-line segments and the Euclidean norm: (i) longest non-crossing matching, (ii) longest non-crossing hamiltonian path, (iii) longest non-crossing spanning tree. We propose simple and efficient algorithms to approximate these structures within a constant factor of optimality. Somewhat surprisingly, we also show that our bounds are within a constant factor of optimality even without the non-crossing constraint. For instance, we give an algorithm to compute a non-crossing matching whose total length is at least 2/π of the longest (possibly crossing) matching, and show that the ratio 2/π between the non-crossing and crossing matching is the best possible. Perhaps due to their utter simplicity, our methods also seem more general and amenable to applications in other similar contexts.