Minimum Spanning Tree: Difference between revisions
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= Suboptimality = | |||
The MST problem exhibits the optimal substructure property. | |||
Given MST tree for the graph G | |||
<math> | |||
OPT = \text{set of edges} | |||
</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]] | ||
Revision as of 01:27, 8 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
MST Problem
The MST problem takes a connected graph Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and outputs an MST for that graph.
Approach: Greedy
The approach is to try to add the smallest edges as long as they do not create a cycle; add an edge to the tree that is minimum across the cut of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} vs.
Given the MST of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{n - m} = v_1, v_2, \ldots, v_{ n - m} } , the MST of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} should be that of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{n-1}} plus the edge that connects to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_n} that is the shortest.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle OPT(n) = OPT(n-1) + min( (v_i, v_n) \in E ) }
Suboptimality
The MST problem exhibits the optimal substructure property.
Given MST tree for the graph G
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle OPT = \text{set of edges} }
where the number of edges is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n - 1} , the weight of edges is minimum, and the tree is spanning.
Consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)} , where X is a leaf node.
Prove that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = OPT - (x, y)} is a MST for the subproblem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G - X}
By contradiction, assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is not optimal. There must be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} such that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(B) < w(A) }
Adding the edge (x, y) back to both graphs
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B + (x, y)} is a viable tree.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(B + w(x,y)) < w(A + w(x,y)) = w(OPT) }
Therefore, OPT is not the optimal solution.
By contradiction, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} must be optimal.
