On finding and updating spanning trees and shortest paths

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More generally, any undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.There are quite a few use cases for minimum spanning trees.If the weights are positive, then a minimum spanning tree is in fact a minimum-cost subgraph connecting all vertices, since subgraphs containing cycles necessarily have more total weight. Then deleting e will break T1 into two subtrees with the two ends of e in different subtrees. If S = {A, B, D, E}, thus V-S = {C, F}, then there are 3 possibilities of the edge across the cut(S, V-S), they are edges BC, EC, EF of the original graph.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.A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight.That is, it is a spanning tree whose sum of edge weights is as small as possible.

A minimum spanning tree would be one with the lowest total cost, thus would represent the least expensive path for laying the cable.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.along roads), then there would be a graph representing which points are connected by those paths.Some of those paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights.

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If each edge has a distinct weight then there will be only one, unique minimum spanning tree.This is true in many realistic situations, such as the telecommunications company example above, where it's unlikely any two paths have exactly the same cost. If the edge weights are not unique, only the (multi-)set of weights in minimum spanning trees is unique, that is the same for all minimum spanning trees.

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