Abstract

Cayley's Formula tells us that a complete undirected graph on n vertices has n n-2 distinct spanning trees. Prfer's proof of this result establishes a one-to-one correspondence between the spanning trees on n vertices and the strings of length (n - 2) over an alphabet of n symbols. The proof describes two algorithms. One identifies the edges in the spanning tree corresponding to a string, and the other builds the string corresponding to a list of edges in a spanning tree. Naive implementations of these algorithms have times that are quadratic in the number n of vertices. Elaborating the data structures they use reduces both times to O(n log n). This paper describes these faster implementations. The data structures they use are familiar to any student who has completed CS2. They include priority queues implemented with heaps and adjacency lists implemented with trees. 1. Introduction A spanning tree of a connected, undirected graph G is a subgraph of G that connects all of G 's ve...

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