Data Structures and Algorithm Analysis - Computer Science at ...

398 Chap. 11 GraphsInitialA B C D E FStep 1 A BProcess edge (C, D)1CDEFStep 2AProcess edge (E, F)B1CDE1FStep 3AProcess edge (C, F)BC1 2DFE 1Figure 11.24 Illustration of the first three steps of Kruskal’s MST algorithm asapplied to the graph of Figure 11.20.The only tricky part to this algorithm is determining if two vertices belong tothe same equivalence class. Fortunately, the ideal algorithm is available for thepurpose — the UNION/FIND algorithm based on the parent pointer representationfor trees described in Section 6.2. Figure 11.25 shows an implementation for thealgorithm. Class KruskalElem is used to store the edges on the min-heap.Kruskal’s algorithm is dominated by the time required to process the edges.The differ and UNION functions are nearly constant in time if path compressionand weighted union is used. Thus, the total cost of the algorithm is Θ(|E| log |E|)in the worst case, when nearly all edges must be processed before all the edges ofthe spanning tree are found and the algorithm can stop. More often the edges of thespanning tree are the shorter ones,and only about |V| edges must be processed. Ifso, the cost is often close to Θ(|V| log |E|) in the average case.