The.Algorithm.Design.Manual.Springer-Verlag.1998

Kruskal's Algorithm
a tree of weight at most . Therefore, we could not have made a mistake in selecting (x,y), and the correctness follows.
What is the time complexity of Kruskal's algorithm? Inserting and retrieving m edges from a priority queue such as a heap
takes time. The while loop makes at most m iterations, each testing the connectivity of two trees plus an edge. In
the most simple-minded approach, this can be implemented by a breadth-first or depth-first search in a graph with at most n
edges and n vertices, thus yielding an O(mn) algorithm.
However, a faster implementation would result if we could implement the component test in faster than O(n) time. In fact, the
union-find data structure, discussed in Section , can support such queries in time. With this data structure,
Kruskal's algorithm runs in time, which is faster than Prim's for sparse graphs. Observe again the impact that the
right data structure can have in implementing a straightforward algorithm.
Next: Shortest Paths Up: Minimum Spanning Trees Previous: Prim's Algorithm
Algorithms
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