comp360_hw3sol

comp360_hw3sol - COMP-360 Homework 3 Solutions 1....

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Unformatted text preview: COMP-360 Homework 3 Solutions 1. Edmonds-Karp (10 points) (a) See the graphs shown on the next page. The left column shows the original flow network, and the flows after each iteration. The maximum flow is shown at the bottom of the left column. The right column shows the residual networks corresponding to the flow at each iteration. The augmenting path in the residual network that is chosen at each iteration is highlighted with thick, blue arrows. (b) The size of the maximum flow is 13. The cut S = { s, a, b, c, e } , T = { d, t } has 13 capacity. 1 2 2. Flow out of s equals flow into t (10 points) From the skew-symmetry property of flows, we know that f ( u, v ) = f ( v, u ) for any u, v . Equiv- alently, f ( u, v ) + f ( v, u ) = 0 . Suppose the vertices are named as V = { v 1 , v 2 , . . ., v N } . Then we have that N 1 summationdisplay i =1 N summationdisplay j = i +1 f ( v i , v j ) + f ( v j , v i ) = 0 Observe that for any u, v V with u negationslash = v , the term f...
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comp360_hw3sol - COMP-360 Homework 3 Solutions 1....

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