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Stoer-Wagner Algorithm
Group 4
Stoer-Wagner Algorithm Group 4 Introduction Theorem 1 G: G(s, - - PowerPoint PPT Presentation
Stoer-Wagner Algorithm Group 4 Introduction Theorem 1 G: G(s, - - PowerPoint PPT Presentation
Stoer-Wagner Algorithm Group 4 Introduction Theorem 1 G: G(s, t): MinimumCutPhase(G, w, a) MinimumCut(G, w, a) {1} {2, 3, 4, 5, 6, 7, 8} {8} {1, 2, 3, 4, 5, 6, 7} W = 5 W = 5 {7, 8} {1, 2, 3, 4, 5, 6} {4, 7, 8} {1, 2, 3, 5, 6} W
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Theorem 1
G: G(s, t):
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- MinimumCutPhase(G, w, a)
- MinimumCut(G, w, a)
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{1} {2, 3, 4, 5, 6, 7, 8} {8} {1, 2, 3, 4, 5, 6, 7} W = 5 W = 5
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{7, 8} {1, 2, 3, 4, 5, 6} {4, 7, 8} {1, 2, 3, 5, 6} W = 7 W = 7
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{3,4,7,8}{1,2,5,6} {1,5}{2,3,4,6,7,8} {2}{1,3,4,5,6,7,8} W = 4 W = 7 W = 9
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Lemma
2 3 4 5 7 8 6 1
t s
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…… t s u v
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……
u v
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w(Au, u)
……
u v
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w(Cu)
……
u v
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Induction
1) w(Au, u) = w(Cu) ≤ w(Cu)
u
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Induction
……
u v 2)
≤
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Induction
……
u v 2)
≤
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Induction
……
u v 2)
≤
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Induction
……
u v 2)
≤
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Induction
……
u v 2)
≤
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Running Time
- MinimumCut :
𝑊 − 1 MinimumCutPhase
- MinimumCutPhase : using Fibonacci heap
- ExtractMax :
𝑃 log 𝑊 𝑊 times
- IncreaseKey:
𝑃 1 𝐹 times
- Every MinimumCutPhase:
𝑃 𝐹 + 𝑊 log 𝑊
- Sum:
𝑃 𝑊 𝐹 + 𝑊 2 log 𝑊
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