# On finding and updating spanning trees and shortest paths

If each edge has a distinct weight then there will be only one, unique minimum spanning tree.

A minimum spanning tree would be one with the lowest total cost, thus would represent the least expensive path for laying the cable.

If there are There may be several minimum spanning trees of the same weight; in particular, if all the edge weights of a given graph are the same, then every spanning tree of that graph is minimum.

The first algorithm for finding a minimum spanning tree was developed by Czech scientist Otakar Borůvka in 1926 (see Borůvka's algorithm).

Its purpose was an efficient electrical coverage of Moravia. In each stage, called Boruvka step, it identifies a forest F consisting of the minimum-weight edge incident to each vertex in the graph G, then forms the graph denotes the graph derived from G by contracting edges in F (by the Cut property, these edges belong to the MST). Since the number of vertices is reduced by at least half in each step, Boruvka's algorithm takes O(m log n) time.

One example would be a telecommunications company which is trying to lay out cables in new neighborhood.

If it is constrained to bury the cable only along certain paths (e.g.

If the minimum cost edge e of a graph is unique, then this edge is included in any MST.

Proof: if e was not included in the MST, removing any of the (larger cost) edges in the cycle formed after adding e to the MST, would yield a spanning tree of smaller weight.

The runtime complexity of a DT is the largest number of queries required to find the MST, which is just the depth of the DT.

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