networks

# networks - EE236A(Fall 2007-08 Lecture 8 Network...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 8 Network optimization • network flows • minimum cost network flow problem • extreme flows • shortest path problem • bipartite matching 8–1 Networks network (directed graph): m nodes connected by n directed arcs • arcs are ordered pairs ( i, j ) • we assume there is at most one arc from node i to node j • we assume there are no self-loops (arcs ( i, i ) ) arc-node incidence matrix A ∈ R m × n : A ij = 1 arc j starts at node i- 1 arc j ends at node i otherwise column sums of A are zero: 1 T A = 0 reduced arc-node incidence matrix ˜ A ∈ R ( m- 1) × n : the matrix formed by the first m- 1 rows of A Network optimization 8–2 example ( m = 6 , n = 8 ) 1 2 3 4 5 6 1 2 3 4 5 6 7 8 A = 1 1- 1- 1 1 1- 1- 1- 1 1 1 1- 1- 1 1- 1 Network optimization 8–3 Network flow flow vector x ∈ R n • x j : flow (of material, traffic, charge, information, . . . ) through arc j • positive if in direction of arc; negative otherwise total flow leaving node i : n summationdisplay j =1 A ij x j = ( Ax ) i i x j A ij =- 1 x k A ik = 1 Network optimization 8–4 External supply supply vector b ∈ R m • b i : external supply at node i • negative b i represents external demand from the network • must satisfy 1 T b = 0 (total supply = total demand) i x j A ij =- 1 x k A ik = 1 b i balance equations: Ax = b reduced balance equations: ˜ Ax = ( b 1 , . . . , b m- 1 ) Network optimization 8–5 Minimum cost network flow problem...
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networks - EE236A(Fall 2007-08 Lecture 8 Network...

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