poscat seminar 10 : graph 3 · spanning tree definition a tree which “spans” whole vertices...
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POSCAT Seminar 10 :Graph 3
yougatup @ POSCAT
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Topic
Topic today
− Minimum Spanning Tree
• Cut Property
• Prim Algorithm
• Kruskal Algorithm
POSCAT Seminar 1-216 July 2014
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Spanning Tree
Definition
A tree which “spans” whole vertices
i.e. a tree with n vertices
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Spanning Tree
Definition
A tree which “spans” whole vertices
i.e. a tree with n vertices
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Minimum Spanning Tree
Problem
Find a spanning tree with minimum cost
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Minimum Spanning Tree
Problem
Find a spanning tree with minimum cost
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Is it MST ?
Minimum Spanning Tree
Problem
Find a spanning tree with minimum cost
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Is it MST ?
Cut Property
Theorem
Suppose edge X are part of a MST of 𝐺 = (𝑉, 𝐸). Pick any subset of
nodes S for which X does not cross between S and 𝑉\S, and let 𝑒
be the lightest edge across this partition. Then 𝑋 ∪ {𝑒} is part of
some MST
POSCAT Seminar 1-816 July 2014
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Proof ?
Cut Property
Theorem
Suppose X is a set of edges which are part of a MST of 𝐺 = (𝑉, 𝐸).
Pick any subset of nodes S for which X does not cross between S
and 𝑉\S, and let 𝑒 be the lightest edge across this partition. Then
𝑋 ∪ {𝑒} is part of some MST
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Assume 𝑒 ∉ 𝑇. Then we can constructa different MST 𝑇′ containing 𝑋 ∪ 𝑒by altering T slightly.
Compare the cost(𝑇′) and cost (𝑇)
Cut Property
Theorem
Suppose X is a set of edges which are part of a MST of 𝐺 = (𝑉, 𝐸).
Pick any subset of nodes S for which X does not cross between S
and 𝑉\S, and let 𝑒 be the lightest edge across this partition. Then
𝑋 ∪ {𝑒} is part of some MST
POSCAT Seminar 1-1016 July 2014
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by cut property, we can derivebeautiful greedy algorithm !
Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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?
Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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?
It make a cycle !∴ we can’t select it
Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-2016 July 2014
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-2116 July 2014
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Done ! we have no remaining edges
Kruskal Algorithm
Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
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Done ! we have no remaining edges
Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ?
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
If a edge connects two vertices with different group, it will never
make a cycle.
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
If not, it will make a cycle !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
POSCAT Seminar 1-2916 July 2014
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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No !!
Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
POSCAT Seminar 1-4416 July 2014
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Question
How can we determine whether adding a edge makes a cycle or
not ? by using Disjoint Set !
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Kruskal Algorithm
Implementation
− We have to sort the set of edges via their costs with STL
− Union & Find
Quite simple !
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Kruskal Algorithm
Implementation
− We have to sort the set of edges via their costs with STL
− Union & Find
Quite simple !
Analysis ?
POSCAT Seminar 1-4816 July 2014
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Kruskal Algorithm
Implementation
− We have to sort the set of edges via their costs with STL
− Union & Find
Quite simple !
Analysis ? Sorting takes O(𝐸 log𝐸)
Each iteration takes almost O(1)
because we use Union & Find
POSCAT Seminar 1-4916 July 2014
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Kruskal Algorithm
Implementation
− We have to sort the set of edges via their costs with STL
− Union & Find
Quite simple !
Analysis ? Sorting takes O(𝐸 log𝐸)
Each iteration takes almost O(1)
because we use Union & Find O(𝑬 𝐥𝐨𝐠𝑬 )
POSCAT Seminar 1-5016 July 2014
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