A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 11.6 Further Reading 417Example 11.4 Figure 11.23 shows the first three steps of Kruskal’s Algorithmfor the graph of Figure 11.19. Edge (C, D) has the least cost, andbecause C and D are currently in separate MSTs, they are combined. Wenext select edge (E, F) to process, and combine these vertices into a singleMST. The third edge we process is (C, F), which causes the MST containingVertices C and D to merge with MST containing Vertices E and F. Thenext edge to process is (D, F). But because Vertices D and F are currentlyin the same MST, this edge is rejected. The algorithm will continue on toaccept edges (B, C) and (A, C) into the MST.The edges can be processed in order of weight by using a min-heap. This isgenerally faster than sorting the edges first, because in practice we need only visita small fraction of the edges before completing the MST. This is an example offinding only a few smallest elements in a list, as discussed in Section 7.6.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 11.24 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.11.6 Further ReadingMany interesting properties of graphs can be investigated by playing with the programsin the Stanford Graphbase. This is a collection of benchmark databases andgraph processing programs. The Stanford Graphbase is documented in [Knu94].11.7 Exercises11.1 Prove by induction that a graph with n vertices has at most n(n−1)/2 edges.11.2 Prove the following implications regarding free trees.(a) IF an undirected graph is connected and has no simple cycles, THENthe graph has |V| − 1 edges.