Showing papers on "Prim's algorithm published in 1996"

The Constrained Minimum Spanning Tree Problem

Abstract: Given an undirected graph with two different nonnegative costs associated with every edge e (say, w e for the weight and l e for the length of edge e) and a budget L, consider the problem of finding a spanning tree of total edge length at most L and minimum total weight under this restriction This constrained minimum spanning tree problem is weakly NP-hard We present a polynomial-time approximation scheme for this problem This algorithm always produces a spanning tree of total length at most (1 + e)L and of total weight at most that of any spanning tree of total length at most L, for any fixed e >0 The algorithm uses Lagrangean relaxation, and exploits adjacency relations for matroids

Minimum cuts in near-linear time

Abstract: We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a random- ized (Monte Carlo) algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log 3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n 2 log n) time. This variant has an optimal 51# parallelization. Both

Balanced spanning trees in complete and incomplete star graphs

Abstract: Efficiently solving the personalized broadcast problem in an interconnection network typically relies on finding an appropriate spanning tree in the network. In this paper, we show how to construct in a complete star graph an asymptotically balanced spanning tree, and in an incomplete star graph a near-balanced spanning tree. In both cases, the tree is shown to have the minimum height. In the literature, this problem has only been considered for the complete star graph, and the constructed tree is about 4/3 times taller than the one proposed in this paper.

Comparison of Heuristic Algorithms for the Degree Constrained Minimum Spanning Tree

Abstract: The degree constrained minimum spanning tree on a graph is the problem of generating a minimum spanning tree with constraints on the number of arcs that can be incident to vertices of the graph. In this paper we develop several heuristics, including simulated annealing, neural networks, greedy and greedy random algorithms for constructing such trees. We compare the computational performance of all of these approaches against the performance of an exact solution approach, using standard problems taken from the literature.

An efficient delay-constrained minimum spanning tree heuristic

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. This is the first heuristic designed specifically for solving the DCMST problem. Simulation results under realistic networking conditions show that our heuristic’s performance is close to optimal when the link loads are symmetric as well as when asymmetric link loads are used. 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 [1], is not as efficient as the heuristic we propose, while the most efficient delay-constrained minimum Steiner tree heuristic, BSMA [2], is much slower than our proposed heuristic and does not construct cheaper delay-constrained broadcast trees.

Using Sparsification for Parametric Minimum Spanning Tree Problems

Abstract: Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs.

An Improved Algorithm for the Constrained Bottleneck Spanning Tree Problem

Abstract: We propose an algorithm to solve the bottleneck spanning tree problem with an additional linear constraint. Our algorithm has an improved worst case performance over the best known algorithm for this problem. In a graph with n nodes and m edges such that m ≥ O(n log n log log*n), where log* n is the iterative logarithm of n, our algorithm runs in O(m) time and hence is the best possible in that case. For a large class of graphs, the proposed algorithm has almost the same complexity as that of computing just one minimum spanning tree.

A Near-linear-time Approximation Algorithm for Maximum-leaf Spanning Tree

Abstract: Given an undirected graph $G$, finding a spanning tree of $G$ with maximum number of leaves is not only NP-complete but also MAX~SNP-complete. The approximation ratio of the previously best known approximation algorithm for maximum leaf spanning tree is three. However, the high-order running time required by the previous algorithm makes it impractical. In this paper we give a new factor-three approximation algorithm for the same problem. The running time required by our algorithm is almost linear in the size of $G$. This improves the previous algorithm by a factor of $\tilde{\Omega}(mn^3)$, where $n$ is the number of nodes and $m$ is the number of edges.

Optimal Cost-Sensitive Distributed Minimum Spanning Tree Algorithm

Abstract: In a network of asynchronous processors, the cost to send a message can differ significantly from one communication link to another. Assume that associated with each link is a positive weight representing the cost of sending one message along the link and the cost of an algorithm executed on a weighted network is the sum of the costs of all messages sent during its execution. We present a distributed Minimum Cost Spanning tree algorithm that is optimal with respect to this cost measure.

Yet another efficient algorithm for replacing the edges of a minimum spanning tree

Abstract: Given T, a minimum spanning tree of the graph G = (V, E), e' E E Tis a replacement fore E TiffT e + e' is a minimum spanning tree ofG' = (V, Ee). In this paper, we propose a solution that computes a replacement for every edge of the minimum spanning tree in O(max{ Cmst. V log V} ), where Cmst is the cost of computing a minimum spanning tree ofG' = (~r, ET). The problem of computing the minimum spanning tree of a graph has been studied extensively by researchers [4, 5, 8, 9]. The problem of finding a replacement for every edge of a minimum spanning tree was first considered by Chin and Houck [2] who give an O(V ) algorithm. Tarjan [10] presented an O(Eo:(E, V)) algorithm, where o:(E, V) is the functional inverse of Ackermann's function. Also, Hsu et. al., [7] presented an algorithm for the replacement problem that requires O(E log E) time. Ballet. al., [II] and Corley et. al., [3] studied a similar problem in networks. They are concerned with the failure of a link in a network and its impact on the links that carry the maximum flow or form the shortest path between two extreme nodes. To the best of our knowledge, when V log V E O(E), our solution is the best so far and comparatively it is quite easy to implement. The running time of any algorithm that attempts to compute the replacement problem is a function of the size of the edge set, S, from which the replacement edges are selected. However, we will show that there is a subset of the remaining edges (the set of graph edges minus the edges of the minimum spanning tree) that can be computed efficiently and that it contains a set of replacement edges as a subset. In § 1, we present an algorithm that computes the set of the replacement edges in terms of an unspecified edge set, S. The largest choice for S is the set of graph edges minus the edges of the minimum spanning tree. In §2. we show how to prune the set of the remaining edges, to form a smaller set S for the algorithm that we present in § 1. This results in an algorithm that runs in O(max(Cmst• V log V)), where Cmin = O(E + V log V) is the cost of computing a second minimum spanning tree of the input