Prim algorithm
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Prim's algorithm finds a minimum spanning tree of a connected weighted graph. It starts with a minimum weight edge and adds the minimum weight edge incident to the growing tree at each step, as long as it does not form a cycle. The algorithm is demonstrated on a sample weighted graph, finding a minimum spanning tree of weight 10 using Prim's algorithm and alphabetical order to break ties.
Prim algorithm
- 1. Prim’s Algorithm We consider a weighted connected graph G with n vertices. Prim’s algorithm finds a minimum spanning tree of G. procedure Prim(G: weighted connected graph with n vertices) T := a minimum-weight edge for i = 1 to n − 2 begin e := an edge of minimum weight incident to a vertex in T and not forming a circuit in T if added to T T := T with e added end return(T) Example: Use Prim’s algorithm to find a minimum spanning tree in the following weighted graph. Use alphabetical order to break ties. r. .................................................................................................................................................................................................... r . .................................................................................................................................................................................................... r. ............................................................................................................................................................................................................................................................ r . ............................................................................................................................................................................................................................................................ r. .................................................................................................................................................................................................... r . ..................................................................................................................................................................................................... ........................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................................................... . ........................................................................................................................................................................ a z c e b d 3 4 3 2 2 2 5 16 Solution: Prim’s algorithm will proceed as follows. First we add edge {d, e} of weight 1. Next, we add edge {c, e} of weight 2. Next, we add edge {d, z} of weight 2. Next, we add edge {b, e} of weight 3. And finally, we add edge {a, b} of weight 2. This produces a minimum spanning tree of weight 10. A minimum spanning tree is the following. r. .................................................................................................................................................................................................... r r. ............................................................................................................................................................................................................................................................ r r. .................................................................................................................................................................................................... r . ........................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................................................... a z c e b d 3 2 2 2 1 Gilles Cazelais. Typeset with LATEX on December 6, 2006.