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 = 7 W = 7

{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

Lemma s 5 1 3 4 8 6 2 7 t

u s …… t v

v …… u

w(A u , u) …… v u

w(C u ) …… v u

Induction 1) u w(A u , u) = w(C u ) ≤ w(C u )

Induction 2) …… ≤ v u

Induction 2) …… ≤ v u

Induction 2) …… ≤ v u

Induction 2) …… ≤ v u

Induction 2) …… ≤ v u

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 𝑃 𝑊 𝐹 + 𝑊 2 log 𝑊 • Sum:

Thanks