Minimum Spanning Tree
Undirected Weighted Graph
- A undirected graph whose edges have weights.
Definition
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- Example:
- \(V=\{1,2,3,4\}\)
- \(E=\{\)
\((1,3), (1,2),\)
\((2,3), (4,2),\)
\((3,4)\)
\(\}\) - \(W\left(\left(1,2\right)\right)=13\)
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$$G=(V,E)$$
- \(V\) is a set of vertices of \(G\) and \(E\) is a set of edges of \(G\).
- \(\forall e \in E\),
we define the weight function \(W(e)\) is the weight of \(e\).
Spanning Tree
- Give graph \(G=(V,E)\)
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Spanning Tree is a connected tree \(T=(V,E')\) where
$$E' \subseteq E$$
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Minimum Spanning Tree(MST)
- Give graph \(G=(V,E)\)
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Minimum Spanning Tree is a Spanning Tree which
the sum of edge weights is minimum.
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Greedy Algorithms to find MST
- Kruskal's algorithm
- Prim's algorithm
- Borůvka's algorithm
- ...
Kruskal's algorithm
- Input \(G=(V,E)\)
- Let \(E_s = E, E' = \{\}\)
- while ( \(E_s\) is not empty)
- Remove an edge \(e\) with minimum weight from \(E_s\)
- If add \(e\) into \(E'\) will not create a circle
- Add \(e\) into \(E'\)
- If \(T=(V,E')\) is a tree
- output \(T\) is a minimum spanning tree
Kruskal's algorithm
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Kruskal's algorithm
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Kruskal's algorithm
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Kruskal's algorithm
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Kruskal's algorithm
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Kruskal's algorithm
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Kruskal's algorithm
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Kruskal's algorithm
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Complexity
- Sort the edges by weight
- Using Union Find Set to maintain trees
O\left(E \log E + E \alpha\left(V\right)\right) = O(E \log E)
How
to
Proof?
Edge inclusion lemma
$$G=(V,E)$$
- Let \(S\) be a subset of \(V\), and suppose \(e = (u, v)\) is the minimum cost edge of \(E\), with \(u\) in \(S\) and \(v\) in \(V-S\).
- Then \(e\) is in a certain minimum spanning tree of \(G\).
\(u\)
\(v\)
\(V-S\)
\(S\)
\(e\)
Proof
- Suppose \(T\) is a minimum spanning tree that does not contain \(e\).
\(v\)
\(u\)
\(S\)
\(V-S\)
\(u'\)
\(v'\)
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\(e\)
\(e'\)
Proof
- Add \(e\) to \(T\), this creates a cycle.
- The cycle must have some edge \(e' = (u', v')\) with \(u'\) in \(S\) and \(v'\) in \(V-S\).
\(v\)
\(u\)
\(S\)
\(V-S\)
\(u'\)
\(v'\)
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\(e\)
\(e'\)
Proof
- By definition, we have \(W(e) \le W(e')\)
- \(W(e) = W(e')\) because \(T\) is a minimum spanning tree.
\(v\)
\(u\)
\(S\)
\(V-S\)
\(u'\)
\(v'\)
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\(e\)
\(e'\)
Proof
- By definition, we have \(W(e) \le W(e')\)
- \(W(e) = W(e')\) because \(T\) is a minimum spanning tree.
- We can remove \(e'\) and add \(e\) to create a new tree \(T'\)
- \(T'\) must be a minimum spanning tree!
\(v\)
\(u\)
\(S\)
\(V-S\)
\(u'\)
\(v'\)
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\(e\)
\(e'\)
Prim's algorithm
- \(G=(V,E)\)
- Let \(S=\{v_0\}\) where \(v_0 \in V\)
- Let \(E'=\{\}\)
- while(\(V-S\) not empty)
- Let \(e = (u, v)\) is the minimum cost edge of \(E\), with \(u\) in \(S\) and \(v\) in \(V-S\)
- Add \(e\) into \(E'\)
- add \(v\) into \(S\)
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If \(T=(V,E')\) is a tree
- output \(T\) is a minimum spanning tree
Prim's algorithm
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\(S = \{a\}\)
Prim's algorithm
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\(S = \{a,d\}\)
Prim's algorithm
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\(S = \{a,d,f\}\)
Prim's algorithm
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\(S = \{a,d,f,b\}\)
Prim's algorithm
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\(S = \{a,d,f,b,e\}\)
Prim's algorithm
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\(S = \{a,d,f,b,e,c\}\)
Prim's algorithm
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\(S = \{a,d,f,b,e,c,g\}\)
Complexity
- Using Binary heap maintain edges
- Using adjacency list
O\left(E\log V\right)
Complexity
- Using Fibonacci heap maintain edges
- Using adjacency list
O\left(E + V\log V\right)
UVa 908
Tioj 1211
Minimum Bottleneck Path
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\(G=(V,E)\)
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For any path \(p\), we call a \(Bottleneck(p)\) is an edge \(e \in p\)
which \(W(e)\) is maximum -
A Minimum Bottleneck Path from \(u\) to \(v\)
is a path \(p\) from \(u\) to \(v\) which \(W(Bottleneck(p))\) is minimum
Minimum Bottleneck Path
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Minimum Bottleneck Path
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Minimum Bottleneck Lemma
- \(G=(V,E)\)
- Let \(T=(V,E')\) is a Minimum Spanning Tree of \(G\)
- \(\forall u,v \in V\):
- \(path(u,v)\) in \(T\) is a Minimum Bottleneck Path in \(G\)
Proof
We ignore proof because it is similar to
Edge Inclusion lemma
UVa 10369
UVa 1395
Minimum Spanning Tree
By jacky860226
Minimum Spanning Tree
- 436
