Showing papers on "Spanning tree published in 1997"

The longest edge of the random minimal spanning tree

Abstract: For n points placed uniformly at random on the unit square, suppose $M_n$ (respectively, $M'_n$) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of $n \pi M_n^2 - \log n$ converges weakly to the double exponential; we give a new proof of this. We show that $P[M'_n = M_n] \to 1$, so that the same weak convergence holds for $M'_n$ .

H3: laying out large directed graphs in 3D hyperbolic space

Abstract: We present the H3 layout technique for drawing large directed graphs as node-link diagrams in 3D hyperbolic space. We can lay out much larger structures than can be handled using traditional techniques for drawing general graphs because we assume a hierarchical nature of the data. We impose a hierarchy on the graph by using domain-specific knowledge to find an appropriate spanning tree. Links which are not part of the spanning tree do not influence the layout but can be selectively drawn by user request. The volume of hyperbolic 3-space increases exponentially, as opposed to the familiar geometric increase of euclidean 3-space. We exploit this exponential amount of room by computing the layout according to the hyperbolic metric. We optimize the cone tree layout algorithm for 3D hyperbolic space by placing children on a hemisphere around the cone mouth instead of on its perimeter. Hyperbolic navigation affords a Focus+Context view of the structure with minimal visual clutter. We have successfully laid out hierarchies of over 20,000 nodes. Our implementation accommodates navigation through graphs too large to be rendered interactively by allowing the user to explicitly prune or expand subtrees.

Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees

Abstract: Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the $k$ smallest spanning trees of a weighted undirected graph in $O(m \log \beta (m,n) + \min \{ k^{3/2}, km^{1/2} \} )$ time, where $m$ is the number of edges and $n$ the number of vertices in the graph. The techniques are extended to find the $k$ smallest spanning trees in an embedded planar graph in $O(n + k (\log n)^3 )$ time. Ambivalent data structures are also used to dynamically maintain 2-edge-connectivity information. Edges and vertices can be inserted or deleted in $O(m^{1/2})$ time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in $O(\log n)$ time, where $m$ and $n$ are understood to be the current number of edges and vertices, respectively.

Nesting of reducible and irreducible loops

Abstract: Recognizing and transforming loops are essential steps in any attempt to improve the running time of a program. Aggressive restructuring techniques have been developed for single-entry (reducible) loops, but restructurers and the dataflow and dependence analysis they rely on often give up in the presence of multientry (irreducible) loops. Thus one irreducible loop can prevent the improvement of all loops in a procedure. This article give an algorithm to build a loop nesting tree for a procedure with arbitrary control flow. The algorithm uses definitions of reducible and irreducible loops which allow either kind of loop to be nested in the other. The tree construction algorithm, an extension of Tarjan's algorithm for testing reducibility, runs in almost linear time. In the presence of irreducible loops, the loop nesting tree can depend on the depth-first spanning tree used to build it. In particular, the header node representing a reducible loop in one version of the loop nesting tree can be the representative of an irreducible loop in another. We give a normalization method that maximizes the set of reducible loops discovered, independent of the depth-first spanning tree used. The normalization require the insertion of at most one node and one edge per reducible loop.

An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs

Abstract: Let G be an undirected graph with V vertices and E edges. Many algorithms have been developed for enumerating all spanning trees in G. Most of the early algorithms use a technique called "backtracking." Recently, several algorithms using a different technique have been proposed by Kapoor and Ramesh (1992), Matsui (1993), and Shioura and Tamura (1993). They find a new spanning tree by exchanging one edge of a current one. This technique has the merit of enabling us to compress the whole output of all spanning trees by outputting only relative changes of edges. Kapoor and Ramesh first proposed an O(N + V + E)-time algorithm by adopting such a "compact" output, where N is the number of spanning trees. Another algorithm with the same time complexity was constructed by Shioura and Tamura. These are optimal in the sense of time complexity but not in terms of space complexity because they take O(VE) space. We refine Shioura and Tamura's algorithm and decrease the space complexity from O(VE) to O(V + E) while preserving the time complexity. Therefore, our algorithm is optimal in the sense of both time and space complexities.

Destination-driven routing for low-cost multicast

Abstract: We present a destination-driven algorithm that optimizes for applications, such as group video or teleconferencing, that require multicast trees with low total cost. The destination-driven algorithm uses a greedy strategy based on shortest-path trees and minimal spanning trees but biases routes through destinations. The performance of the algorithm is analyzed through extensive simulation and compared with several Steiner tree heuristics and the popular shortest-path tree (SPT) method. The algorithm is found to produce trees with significantly lower overall cost than the SPT while maintaining reasonable per-destination performance. Its performance also compares well with other known Steiner heuristics. Moreover, the algorithm does not suffer from high complexity common to most Steiner tree heuristics and builds a route by querying only incident links for cost information.

Maintaining Minimum Spanning Trees in Dynamic Graphs

Abstract: We present the first fully dynamic algorithm for maintaining a minimum spanning tree in time o(√n) per operation. To be precise, the algorithm uses O(n 1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic. An immediate consequence is the first fully dynamic deterministic algorithm for maintaining connectivity and, bipartiteness in amortized time O(n 1/3 log n) per update, with O(1) worst case time per query.

Euclidean Steiner minimum trees: An improved exact algorithm

Abstract: The Euclidean Steiner tree problem asks for a shortest network interconnecting a set of terminals in the plane. Over the last decade, the maximum problem size solvable within 1 h (for randomly generated problem instances) has increased from 10 to approximately 50 terminals. We present a new exact algorithm, called geosteiner96. It has several algorithmic modifications which improve both the generation and the concatenation of full Steiner trees. On average, geosteiner96 solves randomly generated problem instances with 50 terminals in less than 2 min and problem instances with 100 terminals in less than 8 min. In addition to computational results for randomly generated problem instances, we present computational results for (perturbed) regular lattice instances and public library instances. © 1997 John Wiley & Sons, Inc. Networks 30:149–166, 1997

On the complexity of testing membership in the core of min-cost spanning tree games

Abstract: LetN = {1,...,n} be a finite set of players andKN the complete graph on the node setN∪{0}. Assume that the edges ofKN have nonnegative weights and associate with each coalitionS∪N of players as costc(S) the weight of a minimal spanning tree on the node setS∪{0}.

Proximity problems on moving points

Abstract: A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dlmensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably efficient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree.

A note on genetic algorithms for degree‐constrained spanning tree problems

Abstract: The degree-constrained spanning tree problem is of high practical importance. Up to now, there are few effective algorithms to solve this problem because of its NP-hard complexity. In this paper, we present a new approach to solve this problem by using genetic algorithms and computational results to demonstrate the effectiveness of the proposed approach. © 1997 John Wiley & Sons, Inc. Networks 30: 91–95, 1997

A simpler minimum spanning tree verification algorithm

Abstract: The problem considered here is that of determining whether a given spanning tree is a minimal spanning tree. In 1984 Komlos presented an algorithm which required only a linear number of comparisons, but nonlinear overhead to determine which comparisons to make. We simplify his algorithm and give a linear-time procedure for its implementation in the unit cost RAM model. The procedure uses table lookup of a few simple functions, which we precompute in time linear in the size of the tree.

Spanning trees with many or few colors in edge-colored graphs

Abstract: Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard.

The delay-constrained minimum spanning tree problem

Abstract: We formulate the problem of constructing broadcast trees for real-time traffic with delay constraints in networks with asymmetric link loads as a delay-constrained minimum spanning tree (DCMST) problem in directed networks. Then, we prove that this problem is NP-complete, and we propose an efficient heuristic to solve the problem based on Prim's algorithm for the unconstrained minimum spanning tree problem. Simulation results under realistic networking conditions show that our heuristic performance is close to optimal. Delay-constrained minimum Steiner tree heuristics can be used to solve the DCMST problem. Simulation results indicate that the fastest delay-constrained minimum Steiner tree heuristic, DMCT is not as efficient as the heuristic we propose, while the most efficient delay-constrained minimum Steiner tree heuristic, BSMA, is much slower than our proposed heuristic and does not construct delay-constrained broadcast trees of lower cost.

A fast recursive shortest spanning tree for image segmentation and edge detection

Abstract: This correspondence presents a fast recursive shortest spanning tree algorithm for image segmentation and edge detection. The conventional algorithm requires a complexity of o(n/sup 2/) for an image of n pixels, while the complexity of our approach is bounded by O(n), which is a new lower bound for algorithms of this kind. The total memory requirement of our fast algorithm is 20% smaller.

A tabu search algorithm for the capacitated shortest spanning tree problem.

Abstract: The Capacitated Shortest Spanning Tree Problem consists of determining a shortest spanning tree in a vertex weighted graph such that the weight of every subtree linked to the root by an edge does not exceed a prescribed capacity. We propose a tabu search heuristic for this problem, as well as dynamic data structures developed to speed up the algorithm. Computational results on new randomly generated instances and on instances taken from the literature indicate that the proposed approach produces high-quality solutions within reasonable computing times. © 1997 John Wiley & Sons, Inc. Networks 29: 161–171, 1997

Reliable broadcasting and secure distributing in channel networks

Abstract: Let T/sub 1/, /spl middot//spl middot//spl middot/, T/sub n/ be n spanning trees rooted at node r of graph G. If for any node /spl nu/ of G, among the n paths from r to /spl nu/, each path in each spanning tree of T/sub 1/, /spl middot//spl middot//spl middot/, T/sub n/, there are k (k/spl les/n) internally disjoint paths, then T/sub 1/, /spl middot//spl middot//spl middot/, T/sub n/, are said to be (k, n)-independent spanning trees rooted at r. A graph is called an (k, n)-channel graph if G has (k, n)-independent spanning trees rooted at each node of G. We study two fault-tolerant communication tasks in (k, n)-channel graphs. The first task is reliable broadcasting. We analyze the relation between the reliability and the efficiency of broadcasting. The second task is secure message distributing. It is required that each message should be received by its destination node and that we should keep the message secret from the nodes called adversaries. We give two message distribution schemes. The first scheme uses secret sharing, and it can tolerate t+k-n listening adversaries for any t

Minimizing Diameters of Dynamic Trees

Abstract: In this paper we consider an on-line problem related to minimizing the diameter of a dynamic tree T. A new edge f is added, and our task is to delete the edge e of the induced cycle so as to minimize the diameter of the resulting tree TU {f}{e}. Starting with a tree with n nodes, we show how each such best swap can be found in worst-case O(log2n) time. The problem was raised by Italiano and Ramaswami at ICALP'94 together with a related problem for edge deletions. Italiano and Ramaswami solved both problems in O(n) time per operation.

Video distribution on multicast networks

Abstract: The Internet is being used to distribute video programming to small, widely distributed audiences over a multicast backbone (Mbone). This type of service can be used to deliver specialized programming to the general population. As the service becomes more widely used, conserving the required transmission facilities becomes more important. We define the problem of designing multicast networks for video distribution. The problem is related to the Steiner tree problem, with an interesting twist. The line costs are not constant. An heuristic is developed to solve the problem and the new heuristic is related to and compared with heuristics, developed for the Steiner tree problem and algorithms to design minimum depth and minimum spanning trees.

Improved bounds on planar k-sets and k-levels

Abstract: We prove an O(nk/sup 1/3/) upper bound for planar k-sets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k-levels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general.

Complexity of the minimum base game on matroids

Abstract: This paper studies the complexity of computing solution concepts for a cooperative game, called the minimum base game (MBG) (E, c), where its characteristic function c : 2E ↦ ℜ is defined as c(S) = (the weight w(B) of a minimum weighted base B ⊆ S), for a given matroid M = (E, ℐ) and a weight function w : E ↦ ℜ. The minimum base game contains, as a special case, the minimum spanning tree game (MSTG) in an edge-weighted graph in which players are located on the edges. By interpreting solution concepts of games (such as core, τ-value and Shapley value) in terms of matroid theory, we obtain: The core of MBG is nonempty if and only if the matroid M has no circuit consisting only of edges with negative weights; checking the concavity and subadditivity of an MBG can be done in oracle-polynomial time; the τ-value of an MBG exists if and only if the core is not empty, the τ-value of MSTG can be computed in polynomial time while there is no oracle-polynomial algorithm for a general MBG; computing the Shapley value...

NP-Completness Results for Minimum Planar Spanners

Abstract: For any fixed parameter t ≥ 1, a t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the NP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t ≥ 3, and for planar unweighted graphs and digraphs if t ≥ 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum planar t-spanners and establish its NP-hardness for similar fixed values of t.