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Showing papers on "Spanning tree published in 2006"


Journal ArticleDOI
TL;DR: This work studies the basic properties of twenty 1-square-mile samples of street patterns of different world cities and finds that cities of the same class, e.g., grid-iron or medieval, exhibit roughly similar properties.
Abstract: Recent theoretical and empirical studies have focused on the structural properties of complex relational networks in social, biological, and technological systems. Here we study the basic properties of twenty 1-square-mile samples of street patterns of different world cities. Samples are turned into spatial valued graphs. In such graphs, the nodes are embedded in the two-dimensional plane and represent street intersections, the edges represent streets, and the edge values are equal to the street lengths. We evaluate the local properties of the graphs by measuring the meshedness coefficient and counting short cycles (of three, four, and five edges), and the global properties by measuring global efficiency and cost. We also consider, as extreme cases, minimal spanning trees (MST) and greedy triangulations (GT) induced by the same spatial distribution of nodes. The measures found in the real and the artificial networks are then compared. Surprisingly, cities of the same class, e.g., grid-iron or medieval, exhibit roughly similar properties. The correlation between a priori known classes and statistical properties is illustrated in a plot of relative efficiency vs cost.

347 citations


Journal ArticleDOI
Feng Xiao1, Long Wang1
TL;DR: In this article, the authors investigated state consensus problems for discrete-time multi-agent systems with changing communication topologies and bounded time-varying communication delays. But their analysis was based on the properties of non-negative matrices.
Abstract: In this paper, we investigate state consensus problems for discrete-time multi-agent systems with changing communications topologies and bounded time-varying communication delays. The analysis in this paper is based on the properties of non-negative matrices. We first extend the model of networks of dynamic agents to the case with multiple time-delays and prove that if the communication topology, time-delays, and weighting factors are time-invariant, then the necessary and sufficient condition that the multi-agent system solves a consensus problem is that the communication topology, represented by a directed graph, has spanning trees. Then we allow for dynamically changing communication topologies and bounded time-varying communication delays, and present some sufficient conditions for state consensus of system. Finally, as a special case of our model, the problem of asynchronous information exchange is also discussed.

340 citations


Journal ArticleDOI
TL;DR: The results show that the proposed method for regionalization combines performance and quality, and it is a good alternative to other regionalization methods found in the literature.
Abstract: Regionalization is a classification procedure applied to spatial objects with an areal representation, which groups them into homogeneous contiguous regions This paper presents an efficient method for regionalization The first step creates a connectivity graph that captures the neighbourhood relationship between the spatial objects The cost of each edge in the graph is inversely proportional to the similarity between the regions it joins We summarize the neighbourhood structure by a minimum spanning tree (MST), which is a connected tree with no circuits We partition the MST by successive removal of edges that link dissimilar regions The result is the division of the spatial objects into connected regions that have maximum internal homogeneity Since the MST partitioning problem is NP‐hard, we propose a heuristic to speed up the tree partitioning significantly Our results show that our proposed method combines performance and quality, and it is a good alternative to other regionalization methods fou

280 citations


Proceedings ArticleDOI
13 Nov 2006
TL;DR: This paper proposes two minimum spanning tree based clustering algorithms that partitions a point set into a group of clusters by maximizing the overall standard deviation reduction, without a given k value.
Abstract: The minimum spanning tree clustering algorithm is known to be capable of detecting clusters with irregular boundaries. In this paper, we propose two minimum spanning tree based clustering algorithms. The first algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of the point set and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. The second algorithm partitions a point set into a group of clusters by maximizing the overall standard deviation reduction, without a given k value. We present our experimental results comparing our proposed algorithms to k-means and EM. We also apply our algorithms to image color clustering and compare our algorithms to the standard minimum spanning tree clustering algorithm.

223 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called a skeleton, a special type of spanning tree based on the edge betweenness centrality.
Abstract: We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called a skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the property of the critical branching tree. The original fractal networks are viewed as a fractal skeleton dressed with local shortcuts. An in silico model with both the fractal scaling and the scale-invariance properties is also constructed. The framework of fractal networks is useful in understanding the utility and the redundancy in networked systems.

204 citations


Proceedings Article
08 May 2006
TL;DR: A new geographic routing algorithm, Greedy Distributed Spanning Tree Routing (GDSTR), that finds shorter routes and generates less maintenance traffic than previous algorithms, and requires an order of magnitude less bandwidth to maintain its trees than CLDP.
Abstract: We present a new geographic routing algorithm, Greedy Distributed Spanning Tree Routing (GDSTR), that finds shorter routes and generates less maintenance traffic than previous algorithms. While geographic routing potentially scales well, it faces the problem of what to do at local dead ends where greedy forwarding fails. Existing geographic routing algorithms handle dead ends by planarizing the node connectivity graph and then using the right-hand rule to route around the resulting faces. GDSTR handles this situation differently by switching instead to routing on a spanning tree until it reaches a point where greedy forwarding can again make progress. In order to choose a direction on the tree that is most likely to make progress towards the destination, each GDSTR node maintains a summary of the area covered by the subtree below each of its tree neighbors. While GDSTR requires only one tree for correctness, it uses two for robustness and to give it an additional forwarding choice. Our simulations show that GDSTR finds shorter routes than geographic face routing algorithms: GDSTR's stretch is up to 20% less than the best existing algorithm in situations where dead ends are common. In addition, we show that GDSTR requires an order of magnitude less bandwidth to maintain its trees than CLDP, the only distributed planarization algorithm that is known to work with practical radio networks.

198 citations


Proceedings ArticleDOI
21 Oct 2006
TL;DR: The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree.
Abstract: We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k+2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to l-edge-connected subgraphs is briefly discussed.

193 citations


Journal Article
TL;DR: It is found that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called a skeleton, a special type of spanning tree based on the edge betweenness centrality.
Abstract: We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called a skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the property of the critical branching tree. The original fractal networks are viewed as a fractal skeleton dressed with local shortcuts. An in silico model with both the fractal scaling and the scale-invariance properties is also constructed. The framework of fractal networks is useful in understanding the utility and the redundancy in networked systems.

180 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the propagator of the real Euclidean ϕ4 model on the four-dimensional Moyal plane is renormalizable to all orders.
Abstract: In this paper we give a much more efficient proof that the real Euclidean ϕ4-model on the four-dimensional Moyal plane is renormalisable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular role because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph.

164 citations


Proceedings ArticleDOI
17 Jul 2006
TL;DR: The Wireless Autonomous Spanning Tree Protocol (WASP) as mentioned in this paper uses crosslayer techniques to achieve efficient distributed coordination of the separated wireless links, where traffic in the network is controlled by setting up a spanning tree and broadcasting scheme messages over it that are used both by the parent and the children of each node in the tree.
Abstract: Wireless body area networks (WBANs) have gained a lot of interest in the world of medical monitoring. Current implementations generally use a large single hop network to connect all sensors to a personal server. However recent research pointed out that multihop networks are more energy-efficient and even necessary when applied near the human body with inherent severe propagation loss. In this paper we present a slotted multihop approach to medium access control and routing in wireless body area networks, the Wireless Autonomous Spanning tree Protocol or WASP. It uses crosslayer techniques to achieve efficient distributed coordination of the separated wireless links. Traffic in the network is controlled by setting up a spanning tree and by broadcasting scheme messages over it that are used both by the parent and the children of each node in the tree. We analyze the performance of WASP and show the simulation results.

154 citations


Journal ArticleDOI
TL;DR: An extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices is developed and it is shown how oriented spanning trees can be used to explicitly and systematically construct optimal networks under network topological constraints.

Journal ArticleDOI
TL;DR: Almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, StrongConnectivity, Minimum Spanning Tree, and Single Source Shortest Paths are given.
Abstract: Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the complexity of Connectivity is also in $\Theta(n^{3/2})$ in the matrix model but in $\Theta(n)$ in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

Journal ArticleDOI
TL;DR: It is possible that a single-objective problem can be solved more efficiently via a generalized multi-objectives model of the problem, and this is proved by investigating the computation of minimum spanning trees.
Abstract: Many real-world problems are multi-objective optimization problems and evolutionary algorithms are quite successful on such problems. Since the task is to compute or approximate the Pareto front, multi-objective optimization problems are considered as more difficult than single-objective problems. One should not forget that the fitness vector with respect to more than one objective contains more information that in principle can direct the search of evolutionary algorithms. Therefore, it is possible that a single-objective problem can be solved more efficiently via a generalized multi-objective model of the problem. That this is indeed the case is proved by investigating the computation of minimum spanning trees.

Journal ArticleDOI
TL;DR: An analytical approximation for the probability of edge and vertex detection is derived that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph.

Journal ArticleDOI
TL;DR: This paper shows strong unconditional lower bounds on the time-approximation trade-off of the distributed minimum spanning tree problem, and shows some of its variants.
Abstract: The design of distributed approximation protocols is a relatively new and rapidly developing area of research. However, so far, little progress has been made in the study of the hardness of distributed approximation. In this paper we initiate the systematic study of this subject and show strong unconditional lower bounds on the time-approximation trade-off of the distributed minimum spanning tree problem, and show some of its variants.

Journal ArticleDOI
Michael Elkin1
TL;DR: A protocol that constructs the minimum-weight spanning tree (MST) in Õ(μ(G,μ) = w + √n) rounds, where μ( G, μ) is the

Proceedings ArticleDOI
15 May 2006
TL;DR: A polynomial-time tree construction algorithm is described that dramatically improves the coverage time even when used as a basis for a simple, inefficient, coverage algorithm.
Abstract: This paper discusses the problem of building efficient coverage paths for a team of robots. An efficient multirobot coverage algorithm should result in a coverage path for every robot, such that the union of all paths generates a full coverage of the terrain and the total coverage time is minimized. A method, underlying several coverage algorithms, suggests the use of spanning trees as base for creating coverage paths. Current studies assume that the spanning tree is given, and try to make the most out of the given configuration. However, overall performance of the coverage is heavily dependent on the given spanning tree. This paper tackles the open challenge of constructing a coverage spanning tree that minimizes the time to complete coverage. We argue that the choice of the initial spanning tree has far reaching consequences concerning the coverage time, and if the tree is constructed appropriately, it could considerably reduce the coverage time of the terrain. Therefore the problem studied here is finding spanning trees that would decrease the coverage time of the terrain when used as base for multi-robot coverage algorithms. The main contributions of this paper are twofold. First, it provides initial sound discussion and results concerning the construction of the tree as a crucial base for any efficient coverage algorithm. Second, it describes a polynomial-time tree construction algorithm that, as shown in extensive simulations, dramatically improves the coverage time even when used as a basis for a simple, inefficient, coverage algorithm

Journal ArticleDOI
TL;DR: The algorithm extends to a (2 + ɛ)-approximation algorithm for the minimum tour that visits any k vertices, provided the edge costs satisfy the triangle inequality.
Abstract: For any ɛ > 0 we give a (2 + ɛ)-approximation algorithm for the problem of finding a minimum tree spanning any k vertices in a graph (k-MST), improving a 3-approximation algorithm by Garg [10]. As in [10] the algorithm extends to a (2 + ɛ)-approximation algorithm for the minimum tour that visits any k vertices, provided the edge costs satisfy the triangle inequality.

Journal ArticleDOI
TL;DR: In this article, the authors discuss the properties and the dynamic origin of the scale-invariant structure of river patterns and its relation to optimal selection, and conclude that nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms.
Abstract: This paper reviews theoretical and observational material on form and function of natural networks appeared in somewhat disparate contexts from physics to biology, whose study is related to hydrologic research. Moving from the exact result that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution, we discuss the properties and the dynamic origin of the scale-invariant structure of river patterns and its relation to optimal selection. We argue that at least in the fluvial landscape, nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms. We also show that optimal networks are spanning loopless configurations only under precise physical requirements that arise under the constraints imposed by continuity. In the case of rivers, every spanning tree proves a local minimum of total energy dissipation. This is stated in a theorem form applicable to generic networks, suggesting that other branching structures occurring in nature (e.g., scale-free and looping) may possibly arise through optimality to different selective pressures. We thus conclude that one recurrent self-organized mechanism for the dynamic origin of fractal forms is the robust strive for imperfect optimality that we see embedded in many natural patterns, chief and foremost hydrologic ones. Copyright 2006 by the American Geophysical Union.

Journal ArticleDOI
TL;DR: This work exploits the structure of these graphs with a two-level approach to drawing, where the graph is decomposed into a tree of biconnected components, and performs an order of magnitude faster than the best previous approaches.
Abstract: Quasi-trees, namely graphs with tree-like structure, appear in many application domains, including bioinformatics and computer networks. Our new SPF approach exploits the structure of these graphs with a two-level approach to drawing, where the graph is decomposed into a tree of biconnected components. The low-level biconnected components are drawn with a force-directed approach that uses a spanning tree skeleton as a starting point for the layout. The higher-level structure of the graph is a true tree with meta-nodes of variable size that contain each biconnected component. That tree is drawn with a new area-aware variant of a tree drawing algorithm that handles high-degree nodes gracefully, at the cost of allowing edge-node overlaps. SPF performs an order of magnitude faster than the best previous approaches, while producing drawings of commensurate or improved quality

Journal ArticleDOI
TL;DR: The spanning ratio for Gabriel graphs is shown, and it is proved that there exist point sets whose spanning ratio is at least $\left( \frac{1}{2} - o(1) \right) \sqrt{n} $.
Abstract: The spanning ratio of a graph defined on n points in the Euclidean plane is the maximum ratio over all pairs of data points (u,v) of the minimum graph distance between u and v divided by the Euclidean distance between u and v A connected graph is said to be an S-spanner if the spanning ratio does not exceed S For example, for any S there exists a point set whose minimum spanning tree is not an S-spanner At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 242-spanner [J M Keil and C A Gutwin, Discrete Comput Geom, 7 (1992), pp 13-28] For proximity graphs between these two extremes, such as Gabriel graphs [K R Gabriel and R R Sokal, Systematic Zoology, 18 (1969), pp 259-278], relative neighborhood graphs [G T Toussaint, Pattern Recognition, 12 (1980), pp 261-268], and $\beta$-skeletons [D G Kirkpatrick and J D Radke, Comput Geom, G T Toussaint, ed, Elsevier, Amsterdam, 1985, pp 217-248] with $\beta$ in [0,2] some interesting questions arise We show that the spanning ratio for Gabriel graphs (which are $\beta$-skeletons with $\beta$ = 1) is $\Theta ( \sqrt{n})$ in the worst case For all $\beta$-skeletons with $\beta$ in [0,1], we prove that the spanning ratio is at most $O(n^\gamma)$, where $\gamma = (1-\log_2(1+\sqrt{1-\beta^2}))/2$ For all $\beta$-skeletons with $\beta$ in [1,2], we prove that there exist point sets whose spanning ratio is at least $\left( \frac{1}{2} - o(1) \right) \sqrt{n} $ For relative neighborhood graphs [G T Toussaint, Pattern Recognition, 12 (1980), pp 261-268] (skeletons with $\beta$ = 2), we show that there exist point sets where the spanning ratio is $\Omega(n)$ For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all $\beta$-skeletons with $\beta$ in [1,2] tends to $\infty$ in probability as $\sqrt{\log n / \log \log n}$

Patent
26 Apr 2006
TL;DR: In this article, a method of implementing a spanning tree protocol for a wireless network conforming to IEEE 802.1 standard is presented. But the protocol is not suitable for wireless networks.
Abstract: A method of implementing a spanning tree protocol for a wireless network conforming to a wireless network standard, the spanning tree protocol substantially conforming to the IEEE 802.1 standard, including a first wireless bridging node wirelessly transmitting BPDU information to other wireless bridging nodes of the network or wirelessly receiving BPDU information from other wireless bridging nodes, the BPDU information encapsulated in one or more control/management frames, e.g., beacon or probe response frames of the wireless network standard, the BPDU information relating to a spanning tree topology containing the first and other wireless bridging nodes.

Journal Article
TL;DR: In this paper, the authors considered self-stabilization in a model of anonymous, asynchronous interacting agents deployed in a network of unknown size, and proposed protocols for leader election in rings, local addressing in degree-bounded graphs, and establishing consistent global direction in an undirected ring.
Abstract: Self-stabilization in a model of anonymous, asynchronous interacting agents deployed in a network of unknown size is considered. Dijkstra-style round-robin token circulation can be done deterministically with constant space per node in this model. Constant-space protocols are given for leader election in rings, local-addressing in degree-bounded graphs, and establishing consistent global direction in an undirected ring. A protocol to construct a spanning tree in regular graphs using O(log D) memory is also given, where D is the diameter of the graph. A general method for eliminating nondeterministic transitions from the self-stabilizing implementation of a large family of behaviors is used to simplify the constructions, and general conditions under which protocol composition preserves behavior are used in proving their correctness.

Journal ArticleDOI
TL;DR: In this article, a general model of weighted networks via an optimization principle is proposed, and the topology of the optimal network turns out to be a spanning tree that minimizes a combination of topological and metric quantities.
Abstract: Inspired by studies on the airports' network and the physical Internet, we propose a general model of weighted networks via an optimization principle. The topology of the optimal network turns out to be a spanning tree that minimizes a combination of topological and metric quantities. It is characterized by strongly heterogeneous traffic, non-trivial correlations between distance and traffic and a broadly distributed centrality. A clear spatial hierarchical organization, with local hubs distributing traffic in smaller regions, emerges as a result of the optimization. Varying the parameters of the cost function, different classes of trees are recovered, including in particular the minimum spanning tree and the shortest path tree. These results suggest that a variational approach represents an alternative and possibly very meaningful path to the study of the structure of complex weighted networks.

Journal ArticleDOI
TL;DR: This paper describes a new exact method, based on Benders decomposition, for the robust spanning tree problem with interval data, which is shown to be very fast on all the benchmarks considered, and in particular on those that were harder to solve for the methods previously known.

Journal ArticleDOI
TL;DR: This paper presents an $O(|V|^3)$ algorithm for finding four independent spanning trees in a 4-connected graph that makes use of chain decompositions of 4- connected graphs.
Abstract: Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an $O(|V|^2)$ algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an $O(|V|^3)$ algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.

Journal ArticleDOI
TL;DR: This paper presents an adaptive spanning tree routing mechanism, using real-time reinforcement learning strategies, and demonstrates via simulation that without additional control packets for tree maintenance, adaptive spanning trees can maintain the "best" connectivity to the base station.
Abstract: One of the most common communication patterns in sensor networks is routing data to a base station, while the base station can be either static or mobile Even in static cases, a static spanning tree may not survive for a long time due to failures of sensor nodes In this paper, we present an adaptive spanning tree routing mechanism, using real-time reinforcement learning strategies We demonstrate via simulation that without additional control packets for tree maintenance, adaptive spanning trees can maintain the "best" connectivity to the base station, in spite of node failures or mobility of the base station And by using a general constraint-based routing specification, one can apply the same strategy to achieve load balancing and to control network congestion effectively in real time

Journal ArticleDOI
TL;DR: In this paper, the number of spanning trees on the generalized Sierpinski gasket with dimension 2, 3, 4 and 4 was shown to be 2, 2 and 3.
Abstract: We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are also obtained.

Journal ArticleDOI
TL;DR: This paper presents a comprehensive mathematical model for evaluating the overall performance of a bridge network based on probability analyses of network connectivity, user satisfaction, and structural reliability of the critical bridges in the network.
Abstract: This paper presents a comprehensive mathematical model for evaluating the overall performance of a bridge network based on probability analyses of network connectivity, user satisfaction, and structural reliability of the critical bridges in the network. A bridge network consists of all nodes of interest in a geographical region. These nodes of interest are connected to each other through multiple paths. The network performance evaluation in terms of connectivity is formulated by using an event tree technique. The network performance measure of user satisfaction deals with traffic demand and capacity of each link in the network. Moreover, the shortest paths in terms of total traffic costs are identified by network optimization algorithms for each pair of the origin and destination nodes of interest under the specified traffic demands. Using this information, the minimum-weight spanning tree (MST) that consists of the identified shortest paths is constructed. The bridges associated with MST are defined as the critical bridges in the network. The network performance in terms of structural reliability of the critical bridges can be computed from system reliabilities of the critical bridges by using a series-parallel system model. Finally, by combining the above three criteria, a single numerical measure is proposed to evaluate the overall performance of the bridge network. This novel approach is illustrated on a group of fourteen existing bridges with different reliability profiles located in Colorado. This study provides the basis of a network-level bridge management system where lifetime reliability and life-cycle costs are the key considerations for optimal bridge maintenance strategies.

Patent
22 Feb 2006
TL;DR: In this paper, the authors proposed an improved unicast routing, multicast routing and unicast load sharing for IEEE 802.1Q networks, where each bridge is the root of its own multiple spanning tree instance (MSTI).
Abstract: The present invention provides improved unicast routing, multicast routing and unicast load sharing as compared with conventional methods. Preferred implementations of the invention provide improvements to IEEE 802.1Q. According to preferred aspects of the invention, each bridge is the root of its own multiple spanning tree instance ('MSTI'). Preferred implementations of the invention require no learning of media access control ('MAC') addresses on the backbone of a network. Some methods of the invention can resolve spanning tree asymmetries. Preferred implementations of the invention require a very low computational load for control protocols.