Prim's algorithm

About: Prim's algorithm is a research topic. Over the lifetime, 775 publications have been published within this topic receiving 17971 citations. The topic is also known as: DJP algorithm & Jarník algorithm.

An ant-based algorithm for finding degree-constrained minimum spanning tree

Abstract: A spanning tree of a graph such that each vertex in the tree has degree at most d is called a degree-constrained spanning tree. The problem of finding the degree-constrained spanning tree of minimum cost in an edge weighted graphis well known to be NP-hard. In this paper we give an Ant-Based algorithm for finding low cost degree-constrained spanning trees. Ants are used to identify a set of candidate edges from which a degree-constrained spanning tree can be constructed. Extensive experimental results show that the algorithm performs very well against other algorithms on a set of 572 problem instances.

Fault-Tolerant Geometric Spanners

Abstract: We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces.We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal.Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.

On the Implementation of MST-Based Heuristics for the Steiner Problem in Graphs

Abstract: Some of the most widely used constructive heuristics for the Steiner Problem in Graphs are based on algorithms for the Minimum Spanning Tree problem. In this paper, we examine efficient implementations of heuristics based on the classic algorithms by Prim, Kruskal, and Bor?vka. An extensive experimental study indicates that the theoretical worst-case complexity of the algorithms give little information about their behavior in practice. Careful implementation improves average computation times not only significantly, but asymptotically. Running times for our implementations are within a small constant factor from that of Prim's algorithm for the Minimum Spanning Tree problem, suggesting that there is little room for improvement.

Estimating the weight of metric minimum spanning trees in sublinear-time

Abstract: In this paper we present a sublinear time (1 + e)-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an n-point metric space. The running time of the algorithm is U(n/eO(1)). Since the full description of an n-point metric space is of size Θ(n2), the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. Furthermore, it has been previously shown that no o(n2) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum.

Computing a Diameter-Constrained Minimum Spanning Tree in Parallel

Abstract: A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. The Diameter-Constrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NPcomplete, for all values of k; 4 ≤ k ≤ (n - 2). Therefore, one has to depend on heuristics and live with approximate solutions. In this paper, we explore two heuristics for the DCMST problem: First, we present a one-time-treeconstruction algorithm that constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting edges to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. It is particularly suited when the specified values for k are small--independent of n. The second algorithm starts with an unconstrained MST and iteratively refines it by replacing edges, one by one, in long paths until there is no path left with more than k edges. This heuristic was found to be better suited for larger values of k. We discuss convergence, relative merits, and parallel implementation of these heuristics on the MasPar MP-1 -- a massively parallel SIMD machine with 8192 processors. Our extensive empirical study shows that the two heuristics produce good solutions for a wide variety of inputs.