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08 - Introduction to network flow

08 - Introduction to network flow - ̅ such that s ∈ A t...

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2/22/08 - Introduction to network fow Network Flow Def: A ± ow network is a directed graph G = (V, E) with edge capacities c e (e E), c e 0 (usually integers) vertices s (source), t (sink) Notations: A vertex v s, t is an i nternal node. In(v) = {edges (u, v) E} Out(v) = {edges (v, w) E} S V In(S) = {edges (u, v) | v S} Out(S) = {edges (u, v) | u S} Assume: In(S) = Out(t) = Ø Def: A ± ow is an assignment of numbers f:E -> R + satisfying (i) [Conservation] internal node v e In(v) f(e) = e Out(v) f(e) (ii) [Capacity] edge e f(e) ce 17-1 17
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Def: The value of a Fow f is v(f) = e Out(s) f(e) = e In(t) f(e) v e Out(v) f(e) = c E f(e) = v e In(v) f(e) e Out(s) f(e) + int. v e Out(v) f(e) p In(t) f(e) + int. v e In(v) f(e) Maximum Fow problem Given a Fow network (V, E, c ̅ > , s, t), compute a Fow of maximum value 17-2
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Def: A cut in a Fow network is a partition of V into A, A
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Unformatted text preview: ̅ such that s ∈ A, t ∈ A ̅ . The capacity of a cut is Cap(A) = ∑ c e e=(u,v) u ∈ A, v ∈ A ̅ Lemma: If G is a Fow network, f is a Fow, A, A ̅ is a cut, then v(f) = Cap(A). Proof. v(f) = ∑ f(e) = ∑ f(e) - ∑ f(e) e ∈ Out(s) e ∈ Out(A) e ∈ In(A) = ∑ (f(e) - f(e)) + ∑ f(e) - ∑ f(e) e ∈ Out(A) ∩ In(A) e ∈ Out(A)\In(A) e ∈ out(A)\In(A) ≤ ∑ f(e) e ∈ Out(A)\In(A) ≤ ∑ c e = Cap(A) e ∈ Out(A)\In(A) Cap(A) = ∑ c e e = (u,v) u ∈ A, v ∈ A ̅ 17-3 Def: Given a Fow network G and a Fow f, the residual graph G f has vertex set V, edge set E f = E ∪ {(v,u) | edge e ∈ (u,v) has Fow > 0} 17-4 Capacities are c e- f(e) if e is in G f( < e ̅ ) if < e ̅ is in G e = (u,v) < e ̅ = (v,u) 17-5...
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