1. Binhai Zhu Computer Science Department, Montana State University
Graph Algorithms, 3
Binhai Zhu
Computer Science Department, Montana State University
Frequently, presenters must deliver material of a technical nature to an audience unfamiliar with the topic or vocabulary. The material may be complex or heavy with detail. To present technical material effectively, use the following guidelines from Dale Carnegie Training®.
Consider the amount of time available and prepare to organize your material. Narrow your topic. Divide your presentation into clear segments. Follow a logical progression. Maintain your focus throughout. Close the presentation with a summary, repetition of the key steps, or a logical conclusion.
Keep your audience in mind at all times. For example, be sure data is clear and information is relevant. Keep the level of detail and vocabulary appropriate for the audience. Use visuals to support key points or steps. Keep alert to the needs of your listeners, and you will have a more receptive audience.
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2. Minimum Spanning Trees
Given a weighted, undirected graph G=(V,E) of n vertices, we want to find an acyclic subset T of E which connects all the vertices in V and whose weight
w(T) = Σ(u,v) in T w(u,v)
is minimized.
This is also known as the Minimum Spanning Tree problem.
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3. Minimum Spanning Trees
Given a weighted, undirected graph G=(V,E) of n vertices, we want to find an acyclic subset T of E which connects all the vertices in V and whose weight
w(T) = Σ(u,v) in T w(u,v)
Is minimized.
This is also known as the Minimum Spanning Tree problem.
It has a lot of application in practice, say in computer networks.
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4. MST example
We can’t use a brute-force method, as it will take exponential time. v4
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2.9 1 v1
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5. MST example
We can’t use a brute-force method, as it will take exponential time. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
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6. MST example
We can’t use a brute-force method, as it will take exponential time. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
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7. MST example
We can’t use a brute-force method, as it will take exponential time. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
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8. MST example
We can’t use a brute-force method, as it will take exponential time. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
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9. MST property
We can’t use a brute-force method, as it will take exponential time.
To achieve this, we must have some
useful property. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
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10. MST property
We can’t use a brute-force method, as it will take exponential time.
To achieve this, we must have some
useful property. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
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11. MST property
We can’t use a brute-force method, as it will take exponential time.
To achieve this, we must have some
useful property. v4
1.5
2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
V-U U
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12. MST property
We can’t use a brute-force method, as it will take exponential time.
To achieve this, we must have some
useful property.
MST-Property:
The lightest edge e between
U and V-U must be in some minimum
spanning tree of G.
How do we prove this? v4
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2.9 1 v1
1.7 v5
1.6 2
1.3 v3 v2
V-U U
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13. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
2.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v} 1 v1
1.7 v5
1.6 2
1.3 v3 v2
V-U U
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14. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v} 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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15. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v} 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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16. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v} 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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17. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v} 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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18. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v} 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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19. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v}
Running time? 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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20. Prim’s algorithm Let G=(V,E)=({1,2,…,n},E)
1.5
0.9
Let G=(V,E)=({1,2,…,n},E)
Prim(G,T) //T will be the output
T ← Ø
U ← {1}
While U ≠ V
let (u,v) be the lightest edge with u ε U and v ε V-U
T ← T \union {(u,v)}
U ← U \union {v}
Running time? O(|E||V|) using no advanced data structure; with priority queue, O((|V|+|E|)∙log |V|). 3 v1
1.3 v5
1.6 2
1.7 v3 v2
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21. Detailed Prim’s algorithm
Prim(G,w,r) //r is the root of the tree
//key[v] is the minimum weight of an edge connecting v to a vertex in the current tree
For each u in V[G]
do key[u] ← ∞, π[u] ← NIL
key[r] ← 0
Q ← V[G] //Q is the priority queue, keyed on key[-]
While Q ≠ Ø
do u ← EXTRACT-MIN(Q)
for each v in Adj[u]
if v is in Q and w(u,v) < key[v]
then π[v] ← u
key[v] ← w(u,v)
Running time? O(|E||V|) using no advanced data structure; with priority queue, O((|V|+|E|)∙log |V|).
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22. Kruskal’s algorithm
Kruskal(G,w) //which involves Union-Find operations
A ←Ø //A is a forest
For each vertex v in V[G]
do MAKE-SET(v)
Sort edges of E in nondecreasing order by weight w
For each edge (u,v) in E (in nondecreasing order by weight)
do if FIND-SET(u) ≠ FIND-SET(v)
then A ← A U {(u,v)}
UNION(u,v)
Return A
Running time? O(|E| x log |V|) --- if UNION,
FIND-SET can be done in O(log |V|) time. v4
1.6
0.9 3 v1
1.3 v5
1.5 2
1.7 v3 v2
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23. Kruskal’s algorithm
Kruskal(G,w) //which involves Union-Find operations
A ←Ø //A is a forest
For each vertex v in V[G]
do MAKE-SET(v)
Sort edges of E in nondecreasing order by weight w
For each edge (u,v) in E (in nondecreasing order by weight)
do if FIND-SET(u) ≠ FIND-SET(v)
then A ← A U {(u,v)}
UNION(u,v)
Return A
Running time? O(|E| x log |V|). v4
1.6
0.9 3 v1
1.3 v5
1.5 2
1.7 v3 v2
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24. Kruskal’s algorithm
Kruskal(G,w) //which involves Union-Find operations
A ←Ø //A is a forest
For each vertex v in V[G]
do MAKE-SET(v)
Sort edges of E in nondecreasing order by weight w
For each edge (u,v) in E (in nondecreasing order by weight)
do if FIND-SET(u) ≠ FIND-SET(v)
then A ← A U {(u,v)}
UNION(u,v)
Return A
Running time? O(|E| x log |V|). v4
1.6
0.9 3 v1
1.3 v5
1.5 2
1.7 v3 v2
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25. Kruskal’s algorithm
Kruskal(G,w) //which involves Union-Find operations
A ←Ø //A is a forest
For each vertex v in V[G]
do MAKE-SET(v)
Sort edges of E in nondecreasing order by weight w
For each edge (u,v) in E (in nondecreasing order by weight)
do if FIND-SET(u) ≠ FIND-SET(v)
then A ← A U {(u,v)}
UNION(u,v)
Return A
Running time? O(|E| x log |V|). v4
1.6
0.9 3 v1
1.3 v5
1.5 2
1.7 v3 v2
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26. Kruskal’s algorithm
Kruskal(G,w) //which involves Union-Find operations
A ←Ø //A is a forest
For each vertex v in V[G]
do MAKE-SET(v)
Sort edges of E in nondecreasing order by weight w
For each edge (u,v) in E (in nondecreasing order by weight)
do if FIND-SET(u) ≠ FIND-SET(v)
then A ← A U {(u,v)}
UNION(u,v)
Return A
Running time? O(|E| x log |V|). v4
1.6
0.9 3 v1
1.3 v5
1.5 2
1.7 v3 v2
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27. Union-Find ADT Motivation: We have disjoint sets of elements. s4 s1 s6
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28. Union-Find ADT Motivation: We have disjoint sets of elements.
Problem: Find the set containing a given element.
Compute the union of two sets.
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29. Union-Find ADT
How do we do that?
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30. Union-Find ADT How do we do that? We will use an array!
Example: { {5,9,1,3}, {8,4,11,6,10},{2,7},{12}} 5 11 7 12 9 8 4 2 1 3 6
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31. Union-Find ADT How do we do that? We will use an array!
Example: { {5,9,1,3}, {8,4,11,6,10},{2,7},{12}} i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
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32. Union-Find ADT How do we do Find or Find-Set, say Find1(6)? i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
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33. Union-Find ADT How do we do Find or Find-Set, say Find1(6)?
It should return 11 (the root of the tree containing 6)! i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
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34. Union-Find ADT How do we do Union, say Union(11,7)?
We should union the two trees into one! i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
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35. Union-Find ADT How do we do Union, say Union(11,7)?
We should union the two trees into one! But how? i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
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36. Union-Find ADT How do we do Union, say Union(11,7)?
We should union the two trees into one! But how? i
parent[i] 5 11 7 12 9 8 4 2
Why? 1 3 6
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37. Union-Find ADT How do we do Union, say Union(11,7)?
We should union the two trees into one! But how? i
parent[i] 5 11 7 12 9 8 4 2
Why? You want to keep the height of the tree as small as possible, as that has sth to do with the cost of Find1. 1 3 6
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38. Union-Find ADT How do we do Union, say Union(11,7)?
We should union the two trees into one! i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
But the size of the new tree has changed. How to deal with this?
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39. Union-Find ADT How do we do Union, say Union(11,7)?
We should union the two trees into one! i
parent[i] 5 11 7 12 9 8 4 2
But the size of the new tree has changed. How to deal with this? Root stores `–size’. 1 3 6
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40. Union-Find ADT Can we improve Find1 to a new version Find2?
Say Find2(10)? i
parent[i] 5 11 7 12 8 4 9 2 1 3 6
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41. Union-Find ADT Can we improve Find1 to a new version Find2?
Say Find2(10)? i
parent[i] 5 11 7 12 9 8 4 2
While returning 11, all the nodes from 10 to the root (11) become the children of the root. 1 3 6
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42. Union-Find ADT Can we improve Find1 to a new version Find2?
Say Find2(10)? i
parent[i] 5 11 7 12 9 8 4 2
While returning 11, all the nodes from 10 to the root (11) become the children of the root. 1 3 6
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