Showing papers on "Prim's algorithm published in 1995"

A randomized linear-time algorithm to find minimum spanning trees

Abstract: We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree. Our computational model is a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.

Computing the 2‐median on tree networks in O(n lg n) time

Abstract: Location of facilities on tree networks is an important problem in transportation and telecommunication systems. For tree networks, the best-known algorithm to find 2-medians has a time complexity of O(n 2 ). By exploiting the properties relating the 1-median and the 2-medians in tree networks, and the properties inherent in tree structure, an improved algorithm is developed for computing the 2-median. The time complexity of this algorithm is O(n lg n). The details of the algorithm are described along with an illustrative example.

Optimal distributed algorithm for minimum spanning trees revisited

Abstract: In an earlier paper, Awerbuch presented an innovative distributed algorithm for solving minimum spanning tree problems that achieved optimal time and message complexity through the introduction of several advanced features. In this paper, we show that there are some cases where his algorithm can create cycles or fail to achieve optimal time complexity. We then show how to modify the algorithm to avoid these problems, and demonstrate both the correctness and optimality of the revised algorithm.

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.

Computing Hierarchies of Clusters from the Euclidean Minimum Spanning Tree in Linear Time

Abstract: A new hierarchical clustering method for point sets is presented, called diameter clustering, whose clusters belong to most other natural clusterings. For each cluster it holds that its diameter is small compared to the distance to a nearest point outside the cluster. Given a Euclidean minimum spanning tree of the input point set, it is shown that the diameter clustering can be computed in linear time. In addition we derive a nice property of this hierarchy which makes it particularly useful as a building block. It is shown in this paper that it can be employed to obtain a good approximation for the known single linkage clustering in roughly linear time. Other examples of its usefulness include computing the greedy triangulation, the complete linkage hierarchy, and a data structure for faster range queries.

Finding subsets maximizing minimum structures

Abstract: We consider the problem of nding a set of k vertices in a graph that are in some sense remote. Stated more formally, given a graph G and an integer k, nd a set P of k vertices for which the total weight of a minimum structure on P is maximized. In particular, we are interested in three problems of this type, where the structure to be minimized is a spanning tree (Remote-MST), Steiner tree, or traveling salesperson tour. We study a natural greedy algorithm that simultaneously approximates all three problems on metric graphs. For instance, its performance ratio for Remote-MST is exactly 4, while this problem is NP-hard to approximate within a factor of less than 2. We also give a better approximation for graphs induced by Euclidean points in the plane, present an exact algorithm for graphs whose distances correspond to shortest-path distances in a tree, and prove hardness and approximability results for general graphs.

Fast data structures for shortest path routing: a comparative evaluation

Abstract: The execution time of Dijkstra's (1959) algorithm for shortest path routing strongly depends upon the underlying priority queue data structure used to store intermediate path costs for the graph during the algorithm's iterations. In particular, the efficiency is determined by how quickly the data structure can execute the operations of EXTRACT-MIN and DECREASE-KEY. Using Dijkstra's algorithm on graphs of various sizes and topologies, this paper compares the actual performance of the standard implicit binary heap with that of several proposed data structures: the binomial heap, relaxed heap, run relaxed heap, Fibonacci heap and splay tree. In particular, we compare the data structures' performance of the key operations, EXTRACT-MIN and DECREASE-KEY.

Phase unwrapping based on maximum cross-amplitude spanning tree algorithm: a comparative study

Abstract: A comparative study is made of the phase unwrapping algorithms that adopt either phase gradients or fringe amplitudes as a reliability measure in the selection of an unwrapping path. To compare with the minimum phase-gradient spanning tree algorithm, a maximum cross- amplitude spanning tree algorithm is proposed, which searches a spanning tree that maximizes overall edge weights given by the cross-amplitudes, i.e., the products of the fringe amplitudes of neighboring pixels. Noise-immunity of the cross-amplitude spanning tree algorithm is demonstrated by experiment and computer simulation.

Linear-Time Algorithms for Parametric Minimum Spanning Tree Problems on Planar Graphs

Abstract: We give a linear-time algorithm for the minimum-ratio spanning tree problem on planar graphs. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators.

A new algorithm for minimum spanning tree using depth-first-search in an undirected graph

Abstract: A new algorithm is presented for the minimum spanning tree problem in any undirected graph. The algorithm does not use any sorting algorithm or priority queue or binary heaps. The algorithm is based on Depth-First-Search (DFS) and the heaviest edge of the seen cycle in DFS is deleted. It takes in worst case a time of 0(Ke) for a graph with v vertices and e edges and K = e — (v — 1).

Recognizing depth-first-search trees in parallel

Abstract: Consider that T. is a given spanning tree of an undirected graph G which contains a vertices and m (/spl ges/n-1) edges. In this paper, we propose an O(m/p+log m)-time parallel algorithm using p processors on the EREW PRAM model to determine whether T is a depth-first-search tree of G. Our algorithm is optimal in time complexity and speed-up. >

Distributed Algorithm for Finding a Core of a Tree Network

Abstract: A core of a graph G = (V, E) is a path P in G which minimizes d(P)=∑ v∈V d(v, P) where d(v,P) is the distance of vertex v from P. Finding a core of a network is essential in locating the best sites to set up service facilities. Here we present the first distributed algorithm which finds a core of a tree network in O(D) time using O(n) messages, where D and n are the diameter and the number of vertices of G respectively.

On the Design and Verification of a Simple Distributed Spanning Tree Algorithm 1

Abstract: The design of a distributed algorithm for computing a minimal distance spanning tree is carried out as a case study for the systematic derivation of a distributed algorithm in a functional setting. A distributed algorithm is derived and proved correct.