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.

Extending greedy multicast routing to delay sensitive applications

Abstract: Given a weighted undirected graph G(V,E) and a subset R of V, a Steiner tree is a subtree of G that contains each vertex in R We present an online algorithm for finding a Steiner tree that simultaneously approximates the shortest path tree and the minimum weight Steiner tree, when the vertices in the set R are revealed in an online fashion This problem arises naturally while trying to construct source-based multicast trees of low cost and good delay The cost of the tree we construct is within an O(log |R|) factor of the optimal cost, and the path length from the root to any terminal is at most O(1) times the shortest path length The algorithm needs to perform at most one reroute for each node in the tree Our algorithm extends the results of Khuller etal and Awerbuch etal, who looked at the offline problem We conduct extensive simulations to compare the performance of our algorithm (in terms of cost and delay) with that of two popular multicast routing strategies: shortest path trees and the online greedy Steiner tree algorithm

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.

Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm

Abstract: We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.

Relation-Algebraic Verification of Prim’s Minimum Spanning Tree Algorithm

Abstract: We formally prove the correctness of Prim’s algorithm for computing minimum spanning trees. We introduce new generalisations of relation algebras and Kleene algebras, in which most of the proof can be carried out. Only a small part needs additional operations, for which we introduce a new algebraic structure. We instantiate these algebras by matrices over extended reals, which model the weighted graphs used in the algorithm. Many existing results from relation algebras and Kleene algebras generalise from the relation model to the weighted-graph model with no or small changes. The overall structure of the proof uses Hoare logic. All results are formally verified in Isabelle/HOL heavily using its integrated automated theorem provers.