Minimum Spanning Tree: Difference between revisions

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* spanning, meaning it connects all nodes
* spanning, meaning it connects all nodes


= MST Problem =
The '''MST problem''' takes a connected graph <math>G</math> and outputs an MST for that graph.


The '''MST problem''' takes a connected graph <math>G</math> and outputs
Algorithm that solves this problem includes [[Kruskal's Algorithm]] and [[Prims Algorithm]].
an MST for that graph.


== Approach: Greedy ==
= Suboptimality =


The approach is to try to add the smallest edges as long as they do not
The MST problem exhibits the optimal substructure property.
create a cycle; add an edge to the tree that is minimum across the cut
of <math>T</math> vs. <math>V - T</math>


Given the MST of <math>V_{n - 1} = v_1, v_2, \ldots, v_{n-1} </math>,
Given MST tree for the graph G
the MST of <math>V</math> should be that of <math>V_{n-1}</math> plus
the edge that connects to <math>v_n</math> that is the shortest.


<math>
<math>
OPT(n) = OPT(n-1) + min( (v_i, v_n) \in E )
OPT = \text{set of edges}
</math>
</math>
where the number of edges is <math>n - 1</math>, the weight of edges is
minimum, and the tree is spanning.
Consider <math>(x,y)</math>, where X is a leaf node.
Prove that <math>A = OPT - (x, y)</math> is a MST for the subproblem
<math>G - X</math>
By contradiction, assume that <math>A</math> is not optimal. There must
be <math>B</math> such that
<math>
w(B) < w(A)
</math>
Adding the edge (x, y) back to both graphs
<math>B + (x, y)</math> is a viable tree.
<math>
w(B + w(x,y)) < w(A + w(x,y)) = w(OPT)
</math>
Therefore, OPT is not the optimal solution.
By contradiction, <math>A</math> must be optimal.




[[Category:Algorithms]]
[[Category:Algorithms]]

Latest revision as of 17:40, 20 March 2024

A minimum spanning tree is

  • a tree, meaning it has no cycle
  • minimum, meaning it has minimum weight
  • spanning, meaning it connects all nodes

The MST problem takes a connected graph and outputs an MST for that graph.

Algorithm that solves this problem includes Kruskal's Algorithm and Prims Algorithm.

Suboptimality

The MST problem exhibits the optimal substructure property.

Given MST tree for the graph G

where the number of edges is , the weight of edges is minimum, and the tree is spanning.

Consider , where X is a leaf node.

Prove that is a MST for the subproblem

By contradiction, assume that is not optimal. There must be such that

Adding the edge (x, y) back to both graphs

is a viable tree.

Therefore, OPT is not the optimal solution.

By contradiction, must be optimal.

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