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Digraphs+Theory,+Algorithms+and+Applications_Part8

Digraphs+Theory,+Algorithms+and+Applications_Part8 - 3.7...

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3.7 Unit Capacity Networks and Simple Networks 123 Lemma 3.7.2 Let L = ( V = V 0 V 1 . . . V k , A, l 0 , u 1) be a layered unit capacity network with V 0 = { s } and V k = { t } . One can find a blocking ( s, t ) -flow in L in time O ( m ) . Proof: It suffices to see that the capacity of each augmenting path is 1 and no two augmenting paths of the same length can use the same arc. Hence it follows that Dinic’s algorithm will find a blocking flow in time O ( m ). ut Lemma 3.7.3 Let N = ( V, A, l 0 , u 1) be a unit capacity network and let x * be a maximum ( s, t ) -flow in N . Then dist N ( s, t ) 2 n/ p | x * | (3.15) Proof: Let ω = dist N ( s, t ) and let V 0 = { s } , V 1 , V 2 , . . . , V ω be the first ω distance classes from s . Since N contains no multiple arcs, the number of arcs from V i to V i +1 is at most | V i || V i +1 | for i = 0 , 1 , . . . , ω - 1. Since the arcs in ( V i , V i +1 ) correspond to the arcs across an ( s, t )-cut in N , we have | x * | ≤ | V i || V i +1 | for i = 0 , 1 , . . . , ω - 1. Thus max {| V i | , | V i +1 |} ≥ p | x * | for i = 0 , 1 , . . . , ω - 1. Now we easily see that n = | V | ≥ ω X i =0 | V i | ≥ p | x * |b ω + 1 2 c (3.16) implying that ω 2 n/ p | x * | . ut Theorem 3.7.4 [232] For unit capacity networks the complexity of Dinic’s algorithm is O ( n 2 3 m ) . Proof: Let N be a unit capacity network with source s and sink t . We assume for simplicity that N has no 2-cycles. The case when N does have a 2-cycle can be handled similarly (Exercise 3.41). Let q be the number of phases performed by Dinic’s algorithm before a maximum ( s, t )-flow is found in N . Let 0 x (0) , x (1) , . . . , x ( q ) denote the ( s, t )-flows in N which have been calculated after the successive phases of the algorithm. Thus x (0) is the starting flow which is the zero flow and x ( i ) denotes the flow after phase i of the algorithm. Let τ = d n 2 3 e and let K = | x ( q ) | denote the value of a maximum ( s, t )-flow in N . By Lemmas 3.7.1 and 3.7.2 it suffices to prove that the total number of phases, q , is O ( n 2 3 ). This is clear in the case when K τ , since we augment the flow by at least one unit after each phase. So suppose that K > τ . Choose j such that | x ( j ) | < K - τ and | x ( j +1) | ≥ K - τ . By Theorem 3.4.2 and Theorem 3.4.3 the value of a maximum flow in N ( x ( j ) ) is K -| x ( j ) | > τ . Applying Lemmas 3.7.1 and 3.7.3 to N ( x ( j ) ), we see that dist N ( x ( j ) ) ( s, t ) 2 n 2 3 . Using Lemma 3.6.2 and the fact that each phase of Dinic’s algorithm results in a blocking flow, we see that j 2 n 2 3 . Thus, since at most τ phases remain after phase j we conclude that the total number of phases q is O ( n 2 3 ). ut
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124 3. Flows in Networks 3.7.2 Simple Networks A simple network is a network N = ( V, A, l 0 , u ) with special vertices s, t in which every vertex in V - { s, t } has precisely one arc entering or precisely one arc leaving. For an example see Figure 3.13. s t Figure 3.13 A simple network. Capacities are not shown. Below we assume that the simple network in question does not have any 2-cycles. It is easy to see that this is not a serious restriction (Exercise 3.42).
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