Lecture 4: Minimum Spanning Trees
Minimum Spanning Tree (MST):
Given a connected graph G=(V,E), a spanning tree is a connected subgraph on V. A minimal spanning tree is a spanning tree with the minimal number of edges.
“A spanning tree is a tree.”
To not be a tree, it must (a) be not connected or (b) have a cycle. By definition, (a) is false, because an MST is defined as a connected subgraph. For (b), if a cycle exists we can erase any edge in the cycle and still maintain connectedness.
Each edge has a cost. The minimum spanning tree is a least total cost tree. For example:
MST Algorithm:
- Choose all edges, dropping expensive edges until you have a complete tree
- Choose least cost edges that do not make a cycle (Kruskal’s)
- Build a single tree, always adding least cost edges (Prim’s)
Kruskal’s Algorithm:
Sort edges by cost
T = {}
for e in edges do
if e does not create a cycle then
add e to T
endif
endfor
Prim’s Algorithm:
Choose a start vertex s
T = {s}
While there is a vertex not in T do
choose least cost edge from T to some vertex not in T
add this edge to T
endwhile
“Kruskal’s algorithm produces an MST”
Let k1, … , kn-1 be edges added by Kruskal’s algorithm in the order added. Assume k1, … , kn-1 is not an MST. Then let ki be the first edge added that makes it impossible to extend k1, … , ki to an MST. Consider a true MST that uses k1, … , ki-1. Let k1, … , ki-1,mi, …, mn-1 be the true MST. If we add ki to k1, … , ki-1,mi, …, mn-1 (true MST) then we get a cycle. This cycle contains some edge mj where j >= i. The cycle cannot be all k’s because Kruskal’s algorithm would not have added k.
Consider the graph from adding ki and removing mj in k1, … , ki-1,mi, …, mn-1. We still have a spanning tree, with three cases:
- |ki| < |mj| => Contradiction, because the new tree has less cost than the true MST
- |ki| > |mj| => Contradiction, because mj would have been chosen by Kruskal's algorithm before ki.
- |ki| = |mj| => Contradiction, because now both trees are MST and have the same cost, but ki was chosen to not be in an MST with k1, … , ki.
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