Showing papers on "Spanning tree published in 2002"

Message-optimal connected dominating sets in mobile ad hoc networks

Abstract: A connected dominating set (CDS) for a graph G(V,E) is a subset V1 of V, such that each node in V--V1 is adjacent to some node in V1, and V1 induces a connected subgraph. A CDS has been proposed as a virtual backbone for routing in wireless ad hoc networks. However, it is NP-hard to find a minimum connected dominating set (MCDS). Approximation algorithms for MCDS have been proposed in the literature. Most of these algorithms suffer from a very poor approximation ratio, and from high time complexity and message complexity. Recently, new distributed heuristics for constructing a CDS were developed, with constant approximation ratio of 8. These new heuristics are based on a construction of a spanning tree, which makes it very costly in terms of communication overhead to maintain the CDS in the case of mobility and topology changes.In this paper, we propose the first distributed approximation algorithm to construct a MCDS for the unit-disk-graph with a emph constant approximation ratio, and emph linear time and emph linear message complexity. This algorithm is fully localized, and does not depend on the spanning tree. Thus, the maintenance of the CDS after changes of topology guarantees the maintenance of the same approximation ratio. In this algorithm each node requires knowledge of its single-hop neighbors, and only a constant number of two-hop and three-hop neighbors. The message length is O( log n) bits.

Clustering gene expression data using a graph-theoretic approach: an application of minimum spanning trees.

Abstract: Motivation: Gene expression data clustering provides a powerful tool for studying functional relationships of genes in a biological process. Identifying correlated expression patterns of genes represents the basic challenge in this clustering problem. Results: This paper describes a new framework for representing a set of multi-dimensional gene expression data as a Minimum Spanning Tree (MST), a concept from the graph theory. A key property of this representation is that each cluster of the expression data corresponds to one subtree of the MST, which rigorously converts a multi-dimensional clustering problem to a tree partitioning problem. We have demonstrated that though the inter-data relationship is greatly simplified in the MST representation, no essential information is lost for the purpose of clustering. Two key advantages in representing a set of multi-dimensional data as an MST are: (1) the simple structure of a tree facilitates efficient implementations of rigorous clustering algorithms, which otherwise are highly computationally challenging; and (2) as an MST-based clustering does not depend on detailed geometric shape of a cluster, it can overcome many of the problems faced by classical clustering algorithms. Based on the MST representation, we have developed a number of rigorous and efficient clustering algorithms, including two with guaranteed global optimality. We have implemented these algorithms as a computer software EXpression data Clustering Analysis and VisualizATiOn Resource (EXCAVATOR). To demonstrate its effectiveness, we have tested it on three data sets, i.e. expression data from yeast Saccharomyces cerevisiae, expression data in response of human fibroblasts to serum, and Arabidopsis expression data in response to chitin elicitation. The test results are highly encouraging. Availability: EXCAVATOR is available on request from the authors.

An optimal minimum spanning tree algorithm

Abstract: We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T*(m,n)) where T* is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine.Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T* are T*(m,n) = Ω(m) and T*(m,n) = O(m ∙ α(m,n)), where α is a certain natural inverse of Ackermann's function.Even under the assumption that T* is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edge-exposure martingale for Gn,p.

Distributed heuristics for connected dominating sets in wireless ad hoc networks

Abstract: A connected dominating set (CDS) for a graph G(V, E) is a subset V' of V, such that each node in V — V' is adjacent to some node in V', and V' induces a connected subgraph. CDSs have been proposed as a virtual backbone for routing in wireless ad hoc networks. However, it is NP-hard to find a minimum connected dominating set (MCDS). An approximation algorithm for MCDS in general graphs has been proposed in the literature with performance guarantee of 3 + In Δ where Δ is the maximal nodal degree [1]. This algorithm has been implemented in distributed manner in wireless networks [2]–[4]. This distributed implementation suffers from high time and message complexity, and the performance ratio remains 3 + In Δ. Another distributed algorithm has been developed in [5], with performance ratio of Θ(n). Both algorithms require two-hop neighborhood knowledge and a message length of Ω (Δ). On the other hand, wireless ad hoc networks have a unique geometric nature, which can be modeled as a unit-disk graph (UDG), and thus admits heuristics with better performance guarantee. In this paper we propose two destributed heuristics with constant performance ratios. The time and message complexity for any of these algorithms is O(n), and O(n log n), respectively. Both of these algorithms require only single-hop neighborhood knowledge, and a message length of O (1).

The Laplacian and Dirac operators on critical planar graphs

Abstract: On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and Dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of the Dirac operator and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete holomorphic functions which, via convolutions, gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs.

The Steiner Tree Problem: A Tour through Graphs, Algorithms, and Complexity

Abstract: 1 Basics I: Graphs.- 1.1 Introduction to graph theory.- 1.2 Excursion: Random graphs.- 2 Basics II: Algorithms.- 2.1 Introduction to algorithms.- 2.2 Excursion: Fibonacci heaps and amortized time.- 3 Basics III: Complexity.- 3.1 Introduction to complexity theory.- 3.2 Excursion: More NP-complete problems.- 4 Special Terminal Sets.- 4.1 The shortest path problem.- 4.2 The minimum spanning tree problem.- 4.3 Excursion: Matroids and the greedy algorithm.- 5 Exact Algorithms.- 5.1 The enumeration algorithm.- 5.2 The Dreyfus-Wagner algorithm.- 5.3 Excursion: Dynamic programming.- 6 Approximation Algorithms.- 6.1 A simple algorithm with performance ratio 2.- 6.2 Improving the time complexity.- 6.3 Excursion: Machine scheduling.- 7 More on Approximation Algorithms.- 7.1 Minimum spanning trees in hypergraphs.- 7.2 Improving the performance ratio I.- 7.3 Excursion: The complexity of optimization problems.- 8 Randomness Helps.- 8.1 Probabilistic complexity classes.- 8.2 Improving the performance ratio II.- 8.3 An almost always optimal algorithm.- 8.4 Excursion: Primality and cryptography.- 9 Limits of Approximability.- 9.1 Reducing optimization problems.- 9.2 APX-completeness.- 9.3 Excursion: Probabilistically checkable proofs.- 10 Geometric Steiner Problems.- 10.1 A characterization of rectilinear Steiner minimum trees.- 10.2 The Steiner ratios.- 10.3 An almost linear time approximation scheme.- 10.4 Excursion: The Euclidean Steiner problem.- Symbol Index.

A Matter of Degree: Improved Approximation Algorithms for Degree-Bounded Minimum Spanning Trees

Abstract: In this paper, we present a new bicriteria approximation algorithm for the degree-bounded minimum spanning tree problem. In this problem, we are given an undirected graph, a nonnegative cost function on the edges, and a positive integer B*, and the goal is to find a minimum-cost spanning tree T with maximum degree at most B*. In an n-node graph, our algorithm finds a spanning tree with maximum degree O(B*+logn) and cost O(optB*), where optB* is the minimum cost of any spanning tree whose maximum degree is at most B*. Our algorithm uses ideas from Lagrangean duality. We show how a set of optimum Lagrangean multipliers yields bounds on both the degree and the cost of the computed solution.

Spiral-STC: an on-line coverage algorithm of grid environments by a mobile robot

Abstract: We describe an on-line sensor based algorithm for covering planar areas by a square-shaped tool attached to a mobile robot. Let D be the tool size. The algorithm, called Spiral-STC, incrementally subdivides the planar work-area into disjoint D-size cells, while following a spanning tree of the resulting grid. The algorithm covers general grid environments using a path whose length is at most (n + m)D, where n is the number of D-size cells and m /spl les/ n is the number of boundary cells, defined as cells that share at least one point with the grid boundary. We also report that any on-line coverage algorithm generates a covering path whose length is at least (2 - /spl epsiv/)l/sub opt/ in the worst case, where l/sub opt/ is the length of the optimal covering path. Since (n + m)D /spl les/ 2l/sub opt/, Spiral-STC is worst-case optimal. Moreover, m << n in practical environments, and the algorithm generates close-to-optimal covering paths in such environments. Simulation results demonstrate the spiral-like covering patterns typical to the algorithm.

A Functional Approach to External Graph Algorithms

Abstract: We present a new approach for designing external graph algorithms and use it to design simple, deterministic and randomized external algorithms for computing connected components, minimum spanning forests, bottleneck minimum spanning forests, maximal independent sets (randomized only), and maximal matchings in undirected graphs. Our I/ O bounds compete with those of previous approaches. We also introduce a semi-external model, in which the vertex set but not the edge set of a graph fits in main memory. In this model we give an improved connected components algorithm, using new results for external grouping and sorting with duplicates. Unlike previous approaches, ours is purely functional—without side effects—and is thus amenable to standard checkpointing and programming language optimization techniques. This is an important practical consideration for applications that may take hours to run.

Efficient integration of multi-hop wireless and wired networks with QoS constraints

Abstract: This work considers the problem of designing an efficient and low-cost infrastructure for connecting static multi-hop wireless networks with wired backbone, while ensuring QoS requirements such as bandwidth and delay. This infrastructure is useful for designing low cost and fast deployed access networks in rural and suburban areas. It may also be used for providing access to sensor networks or for efficient facility placement in wireless networks. In these networks some nodes are chosen as access points and function as gateways to access a wired backbone. Each access point serves a cluster of its nearby user and a spanning tree rooted at the access point is used for message delivery. The work addresses both the design optimization and the operation aspects of the system. From the design perspective, we seek for a partition of the network nodes into minimal number of disjoint clusters that satisfy multiple constraints; Each cluster is required to be a connected graph with an upper bound on its radius. We assume that each node has a weight (representing its bandwidth requirement) and the total weight of all cluster nodes is also bounded. We show that these clustering requirements can be formulated as an instance of the Capacitated Facility Location problem (CFLP) with additional constraints. By breaking the problem into two sub-problems and solving each one separately, we propose polynomial time approximation algorithms that calculate solutions within a constant factor of the optimal ones. From the operation viewpoint, we introduce an adaptive delivery mechanism that maximizes the throughput of each cluster without violating the QoS constraints.

The Laplacian and $\bar\partial$ operators on critical planar graphs

Abstract: On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.

Lattice structure for orientations of graphs

Abstract: Earlier researchers have studied the set of orientations of a connected finite graph $G$, and have shown that any two such orientations having the same flow-difference around all closed loops can be obtained from one another by a succession of local moves of a simple type. Here I show that the set of orientations of $G$ having the same flow-differences around all closed loops can be given the structure of a distributive lattice. The construction generalizes partial orderings that arise in the study of alternating sign matrices. It also gives rise to lattices for the set of degree-constrained factors of a bipartite planar graph; as special cases, one obtains lattices that arise in the study of plane partitions and domino tilings. Lastly, the theory gives a lattice structure to the set of spanning trees of a planar graph.

Dynamic Generators of Topologically Embedded Graphs

Abstract: We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)^3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log^4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-genus low-diameter graphs.

On the Spanning Ratio of Gabriel Graphs and beta-skeletons

Abstract: The spanning ratio of a graph defined on n points in the Euclidean plane is the maximal ratio over all pairs of data points (u, v), of the minimum graph distance between u and v, over the Euclidean distance between u and v. A connected graph is said to be a k-spanner if the spanning ratio does not exceed k. For example, for any k, there exists a point set whose minimum spanning tree is not a k-spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42- spanner[11]. For proximity graphs inbetween these two extremes, such as Gabriel graphs[8], relative neighborhood graphs[16] and s-skeletons[12] with s ? [0, 2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are s-skeletons with s = 1) is ?(?n) in the worst case. For all s-skeletons with s ? [0, 1], we prove that the spanning ratio is at most O(n?) where ? = (1 - log2(1 +?1 - s2))/2. For all s-skeletons with s ? [1, 2), we prove that there exist point sets whose spanning ratio is at least (1/2- o(1) ?n. For relative neighborhood graphs[16] (skeletons with s = 2), we show that there exist point sets where the spanning ratio is ?(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 s-skeletons with s ? [1, 2] tends to ? in probability as ?log n/ log log n.

Application of network calculus to general topologies using turn-prohibition

Abstract: Network calculus is known to apply in general only to feedforward routing networks, i.e., networks where routes do not create cycles of interdependent packet flows. We address the problem of using network calculus in networks of arbitrary topology. For this purpose, we introduce a novel algorithm, called turn-prohibition (TP), that breaks all the cycles in a network and thus prevents any interdependence between flows. We prove that the TP-algorithm prohibits the use of at most 1/3 of the total number turns in a network, for any network topology. Using analysis and simulation, we show that the TP-algorithm significantly outperforms other approaches for breaking cycles, such as the spanning tree and up/down routing algorithms, in terms of network utilization and delay bounds. Our simulation results also show that the network utilization achieved with the TP-algorithm is within a factor of two of the maximum theoretical network utilization, for networks of up to 50 nodes of degree four. Thus, in many practical cases, the restriction of network calculus to feed-forward routing networks may not represent a significant limitation.

Topological properties of OTIS-networks

Abstract: We conduct a general study of the topological properties of optical transpose interconnection systems (OTIS). We first obtain their basic topological metrics of size, degree, shortest distance and diameter, and then we obtain results related to the recursive structure and efficient embedding of meshes, cubes, spanning trees and cycles. We also present minimal one-to-one routing and optimal broadcasting algorithms, and we show how to construct node-disjoint paths between any two nodes of an OTIS network. Recent studies have addressed only particular members of the general class of OTIS networks. In this paper, we present unified tools for obtaining the topological properties of an arbitrary OTIS network based on the properties of the corresponding factor network.

Capturing term dependencies using a language model based on sentence trees

Abstract: We describe a new probabilistic Sentence Tree Language Modeling approach that captures term dependency patterns in Topic Detection and Tracking's (TDT) Story Link Detection task. New features of the approach include modeling the syntactic structure of sentences in documents by a sentence-bin approach and a computationally efficient algorithm for capturing the most significant sentence-level term dependencies using a Maximum Spanning Tree approach, similar to Van Rijsbergen's modeling of document-level term dependencies.The new model is a good discriminator of on-topic and off-topic story pairs providing evidence that sentence-level term dependencies contain significant information about relevance. Although runs on a subset of the TDT2 corpus show that the model is outperformed by the unigram language model, a mixture of the unigram and the Sentence Tree models is shown to improve on the best performance especially in the regions of low false alarms.

Balancing Minimum Spanning and Shortest Path Trees

Abstract: This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 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+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.

Restoration of IP networks using precalculated restoration routing tables

Abstract: Fast restoration of an IP network in the event of a link failure is achieved, in advance of the conventional recalculation scheme which remains as a second phase of restoration, by pre-calculating for the network a minimum number of spanning trees arranged to provide alternative paths in the event that communication between two routers is determined to have failed, by providing routing tables based upon the spanning trees and by switching to the new routing tables when a link failure is detected using high speed detection at the physical layer to avoid delay. An algorithm is provided to calculate the spanning trees which are kept to a low number of two or three which is practical for storage in the memory of the routers.

STAR: a transparent spanning tree bridge protocol with alternate routing

Abstract: With increasing demand for multimedia applications, local area network (LAN) technologies are rapidly being upgraded to provide support for quality of service (QoS). In a network that consists of an interconnection of multiple LANs via bridges, the QoS of a flow depends on the length of an end-to-end forwarding path. In the IEEE 802.1D standard for bridges, a spanning tree is built among the bridges for loop-free frame forwarding. Albeit simple, this approach does not support all-pair shortest paths. In this paper, we present a novel bridge protocol, the Spanning Tree Alternate Routing (STAR) Bridge Protocol, that attempts to find and forward frames over alternate paths that are provably shorter than their corresponding tree paths. Being backward compatible to IEEE 802.1D, our bridge protocol allows cost-effective performance enhancement of an existing extended LAN by incrementally replacing a few bridges in the extended LAN by the new STAR bridges. We develop a strategy to ascertain bridge locations for maximum performance gain. Our study shows that we can significantly improve the end-to-end performance when deploying our bridge protocol.

Completely Independent Spanning Trees in Maximal Planar Graphs

Abstract: Let G be a graph. Let T1, T2, . . . , Tk be spanning trees in G. If for any two vertices u, v in G, the paths from u to v in T1, T2, . . . , Tk are pairwise openly disjoint, then we say that T1, T2, . . . , Tk are completely independent spanning trees in G. In this paper, we show that there are two completely independent spanning trees in any 4-connected maximal planar graph. Our proof induces a linear-time algorithm for finding such trees. Besides, we show that given a graph G, the problem of deciding whether there exist two completely independent spanning trees in G is NP-complete.

A Randomized Time-Work Optimal Parallel Algorithm for Finding a Minimum Spanning Forest

Abstract: We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write (EREW) PRAM. This result is optimal w.r. t. both work and parallel time, and is the first provably optimal parallel algorithm for this problem under both measures. We also give a simple, general processor allocation scheme for tree-like computations.

Improved algorithms for constructing fault-tolerant spanners

Abstract: Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k-fault-tolerant spanners for S. If in such a spanner at most k vertices and/or edges are removed, then each pair of points in the remaining graph is still connected by a "short" path. First, an algorithm is given that transforms an arbitrary spanner into a k-fault-tolerant spanner. For the Euclidean metric in Rd, this leads to an O (n log n + c(k) n)-time algorithm that constructs a k-fault-tolerant spanner of degree O(c(k)), whose total edge length is O(c(k)) times the weight of a minimum spanning tree of S, for some constant c. For constant values of k, this result is optimal, In the second part of the paper, algorithms are presented for the Euclidean metric in Rd. These algorithms construct (i) in O(n log n + k(2)n) time, a k-fault-tolerant spanner with O (k(2)n) edges, and (ii) in O(kn log n) time, such a spanner with O(kn log n) edges. (Less)