Mastering Algorithms with C

Description of Minimum Spanning Trees

Picture a number of pegs on a board connected by pieces of string. Assuming that every peg is reachable from any other by traveling along one or more strings, imagine a game in which the object is to remove some of the strings until all of the pegs remain connected using the least amount of string. This is the idea behind a minimum spanning tree. Formally stated, given an undirected, weighted graph G = (V, E ), a minimum spanning tree is the set T of edges in E that connect all vertices in V at a minimum cost. The edges in T form a tree because each vertex ends up with exactly one parent that precedes it in the span, with the exception of the first vertex, which is the root of the tree.

Prim’s Algorithm

One approach to computing a minimum spanning tree is Prim’s algorithm. Prim’s algorithm grows a minimum spanning tree by adding edges one at a time based on which looks best at the moment. The fact that Prim’s algorithm adds edges using this approach makes it greedy (see Chapter 1). Although greedy algorithms often yield approximations rather than optimal solutions, Prim’s algorithm actually provides an optimal result.

Fundamentally, the algorithm works by repeatedly selecting a vertex and exploring the edges incident on it to determine if there is a more effective way to span the vertices explored thus far. The algorithm resembles breadth-first search because it explores all edges incident on a vertex before moving deeper in the graph. To determine ...