Showing papers on "Prim's algorithm published in 1999"

Finding the k most vital edges with respect to minimum spanning tree

Abstract: For a connected, undirected and weighted graph G = (V,E), the problem of finding the k most vital edges of G with respect to minimum spanning tree is to find k edges in G whose removal will cause greatest weight increase in the minimum spanning tree of the remaining graph. This problem is known to be NP-hard for arbitraryk. In this paper, we first describe a simple exact algorithm for this problem, based on t he approach of edge replacement in the minimum spanning tree of G. Next we present polynomial-time randomized algorithms that produce optimal and approximate solutions to this problem. For $|V|=n$ and $|E|=m$ , our algorithm producing optimal solution has a time complexity of O(mn) with probability of success at least $e^{-\frac{\sqrt{2k}}{k-2}}$ , which is 0.90 for $k\geq 200$ and asymptotically 1 when k goes to infinity. The algorithm producing approximate solution runs in $O(mn+nk^2\log k)$ time with probability of success at least $1-\frac{1}{4}(\frac{2}{n})^{k/2-2}$ , which is 0.998 for $k\geq 10$ , and produces solution within factor 2 to the optimal one. Finally we show that both of our randomized algorithms can be easily parallelized. On a CREW PRAM, the first algorithm runs in O(n) time using $n^2$ processors, and the second algorithm runs in $O(\log^2n)$ time using mn/logn processors and hence is RNC.

Spanning tree based state encoding for low power dissipation

Abstract: In this paper we address the problem of state encoding for synchronous finite state machines The primary goal is the reduction of switching activity in the state register At the beginning the state transition graph is transformed into an undirected graph where the edges are labeled with the state transition probabilities Next a maximum spanning tree of the undirected graph is constructed and we formulate the state encoding problem as an embedding of the spanning tree into a Boolean hypercube of unknown dimension At this point a modification of Prim's maximum spanning tree algorithm is presented to limit the dimension of the hypercube for area constraints Then we propose a polynomial time embedding heuristic, which removes the restriction of previous works, where the number of state bits used for encoding of a k-state FSM was generally limited to [log/sub 2/ k] Next a more sophisticated embedding algorithm is presented, which takes into account the state transition probabilities not covered by the spanning tree The resulting encodings of both algorithms often exhibit a lower switching activity and power dissipation in comparison with a known heuristic for low power state encoding

A New Genetic Algorithm for the Optimal Communication Spanning Tree Problem

Abstract: This paper proposes a new genetic algorithm to solve the Optimal Communication Spanning Tree problem. The proposed algorithm works on a tree chromosome without intermediate encoding and decoding, and uses crossovers and mutations which manipulate directly trees, while a traditional genetic algorithm generally works on linear chromosomes. Usually, an initial population is constructed by the standard uniform sampling procedure. But, our algorithm employs a simple heuristic based on Prim’s algorithm to randomly generate an initial population. Experimental results on known data sets show that our genetic algorithm is simple and efficient to get an optimal or near-optimal solution to the OCST problem.

An approximation of the minimum vertex cover in a graph

Abstract: For a given undirected graphG withn vertices andm edges, we present an approximation algorithm for the minimum vertex cover problem. Our algorithm finds a vertex cover within $$2 - \frac{{8m}}{{13n^2 + 8m}}$$ of the optimal size inO(nm) time.

Greedy algorithms in disordered systems

Abstract: We discuss search, minimal path and minimal spanning tree algorithms and their applications to disordered systems. Greedy algorithms solve these problems exactly, and are related to extremal dynamics in physics. Minimal cost path (Dijkstra) and minimal cost spanning tree (Prim) algorithms provide extremal dynamics for a polymer in a random medium (the KPZ universality class) and invasion percolation (without trapping) respectively.

On the complexity of bicriteria spanning tree problems for a set of points in the plane

Abstract: A spanning tree is a very simple and common communication network model. The minimum (cost) spanning tree (MST) of a connected graph G = (V, E) may not be unique if two or more edges have the same cost. The transmission delay between two nodes in a spanning tree network is measured in terms of its path length. When the minimum cost spanning tree is not unique, the one that minimizes transmission delay is desired. The bicriteria (total cost and transmission delay) optimization problem, the so called minimum radius minimum cost spanning tree or the minimum diameter minimum cost spanning tree problems is known to be NP-hard. Its geometrical version was open. In this dissertation, it is shown that for given point set in the plane, the problem of determining the existence of a spanning tree with total weight and radius (or diameter) upper-bounded, by two given parameters C and R (or D), respectively is NP-complete. An O(m log n) time algorithm is presented for computing the union graph and the intersection graph of all MST's, where m = |E| and n = |V|. Also presented are some heuristic algorithms for finding a suboptimal minimum radius (or diameter) minimum spanning tree. It is shown that the heuristic algorithm based on a locally optimal connection strategy can have an O(n1/3) performance ratio in some cases, where n is the number of points. Experimental results indicate that the heuristic based on a modified Prim's MST algorithm, MST_H1 outperforms Prim's MST (minimum spanning tree algorithm) overall and the output of the heuristic can be used as a new oracle MST for other problems. This may contribute to improving performance of many heuristic algorithms for other problems where an initial (arbitrary) MST is needed.

Distributed Computation of a Spanning Tree

Abstract: A typical architecture that one encounters is that of a number of computers located in the nodes of a sparsely connected, but otherwise arbitrary network. One of the difficulties with such an architecture is that the flow of control and the information exchange between the various computers are not always easily orchestrated, simply because the network is so arbitrary. This brings about the so-called routing problem. There are, however, at least two general ways to reduce these difficulties.

A real-time generation algorithm for progressive meshes in dynamic environments

Abstract: This paper presents an efficient method for real-time generation of progressive mesh. In our algorithm, only the current mesh and local information of each vertex, such as vertex position, edge length, neighboring vertices, adjacent faces, and face normals are considered at each simplification step. All edge collapse costs are calculated and sorted into a binary tree using heap sort algorithm at initial stage. The whole complexity of the algorithm is O(n*lg(n)). Other properties such as vertex color and face texture are processed in the same way as geometry. Test shows that the algorithm is viable in real-time simplification for medium scale virtual models on PC platforms.

A simulated annealing approach for the capacitated minimum spanning tree problem

Abstract: Presents a novel approach to solve an NP-complete problem that is very important from the theoretical and practical point of view, namely the capacitated minimum spanning tree (CMST) problem. This approach has the following features: (a) it is based on the simulated annealing (SA) algorithm; (b) it represents a tree with N edges using N-2 integers; and (c) it defines a consistent ordering between feasible and infeasible trees. Our SA implementation was tested against the most referenced algorithms for the CMST: the Essau-Williams (1966) algorithm, the Prim (1957) algorithm and the Kruskal (1993) algorithm. The results indicate that our novel approach is very promising for solving CMST problem instances, because it consistently obtains the best results (for the tested cases) but it takes more time.

Drawing a Tree on Parallel Lines

Abstract: We consider a problem of drawing a tree on parallel lines. In this problem we given a tree and an infinite number of parallel lines in the plane. The object is to draw the tree so that 1. (i) each vertex is placed on one of the given parallel lines, 2. (ii) no two edges intersect, and 3. (iii) the ‘height’ of each vertex is nondecreasing, while minimizing the total number of lines used. We show that this problem is solvable in time linear on the size of the tree, by presenting an algorithm which solves it recursively.

Searches on a Binary Tree with Random Edge-Weights

Abstract: Each edge of the standard rooted binary tree is equipped with a random weight; weights are independent and identically distibuted. The value of a vertex is the sum of the weights on the path from the root to the vertex. We wish to search the tree to find a vertex of large weight. A very natural conjecture of Aldous states that, in the sense of stochastic domination, an obvious greedy algorithm is best possible. We show that this conjecture is false. We prove, however, that in a weaker sense there is no significantly better algorithm.

A parallel algorithm for generating multiple ordering spanning trees in undirected weighted graphs

Abstract: In this paper, we propose an efficient parallel algorithm for generatingk (k≥1) spanning trees of a connected, weighted and undirected graphG(V,E,W) in the order of increasing weight. It runs inO(T mat(n)+k logn) time withO(n2/logn) processors on a CREW PRAM, wheren=|V|, m=|E| andT mat (n),O(logn)≤T mat (n)≤O(log2 n) is the run time of the fastest parallel allel algorithm for finding a minimum spanning tree (MST) of G on a CREW PRAM. SinceT mat (n)=O(log2 n) for the time being, our algorithm is of the same time bound withT mat (n) whenk≤O(logn).