11_Max_Flows_4 - 15.082 and 6.855J Max Flows 4 1 Overview...

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1 15.082 and 6.855J March 13, 2003 Max Flows 4
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2 Overview of today’s lecture Very quick review of Preflow Push Algorithm The Excess Scaling Algorithm O(n 2 log U) non-saturating pushes O(nm + n 2 log U) running time. A proof that Highest Preflow Push uses O(n 2 m 1/2 ) non-saturating pushes.
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3 A Feasible Preflow The excess e(j) at each node j s, t is the flow in minus the flow out. s 3 4 2 5 t 3 3 3 2 2 2 2 1 2 1 0 0 Note: total excess = flow out of s minus flow into t.
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4 Active nodes Nodes with positive excess are called active . s 3 4 2 5 t 3 3 3 2 2 2 2 1 2 1 0 0 The preflow push algorithm will try to push flow from active nodes towards the sink, relying on d( ).
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5 Review of Distance Labels Distance labels d( ) are valid for G(x) if i. d(t) = 0 ii. d(i) d(j) + 1 for each (i,j) G(x) Defn. An arc (i,j) is admissible if r ij > 0 and d(i) = d(j) + 1. Lemma. Let d( ) be a valid distance label. Then d(i) is a lower bound on the distance from i to t in the residual network.
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6 Goldberg-Tarjan Preflow Push Algorithm Procedure Preprocess begin x :=0; compute the exact distance labels d(i) for each node; x sj := u sj for each arc (s,j) A(s); d(s) := n; end Algorithm PREFLOW-PUSH; begin preprocess; while there is an active node i do begin select an active node i; push/relabel(i); end; end;
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Preprocess Step 4 1 1 4 2 2 2 3 3 1 s 2 4 5 3 t Saturate arcs out of node s. 0 5 4 3 2 1 t 4 5 3 s 2 Move excess to the adjacent arcs 0 2 1 1 1 3 2 3 6 s Relabel node s after all incident arcs have been saturated.
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Select, then relabel/push 4 1 1 4 2 2 2 3 3 1 s 2 4 5 3 t Select an active node. 0 5 4 3 2 1 t 4 5 3 s 2 Push/Relabel 0 2 1 1 1 3 2 3 6 s Update excess after a push 1 Note : Each arc gets saturated O(n) times. 8
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Select, then relabel/push 4 1 1 4 2 2 2 3 3 1 s 2 4 5 3 t Select an active node. 0 5 4 3 2 1 t 4 5 3 s 2 0 2 1 1 1 3 2 6 s No arc incident to the selected node is admissible. So relabel. 1 2 3 Note. Each node gets relabeled O(n) times. 9
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Select, then relabel/push 4 1 1 4 2 2 2 3 3 1 s 2 4 5 t Select an active node 0 5 4 3 2 1 t 4 5 s 2 0 2 1 1 3 2 6 s Push/Relabel 1 2 3 3 1 1 Bounding non-saturating pushes is more complex. Note that the active node became inactive. 10
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11 Excess Scaling Approach Let be a “scaling” parameter. In the -scaling phase e(j) for all j. At the end of the scaling phase e(j) < /2, for all j, at which point the /2-scaling phase begins. We start with > U. The last scaling phase is the 1- scaling phase. e(j) 1 for all j. At the end of the 1-scaling phase, we have a flow, and the flow is optimal. Note: the number of phases is O(log U+1)
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12 Algorithm excess scaling begin preprocess := 2 log U ; while 1 do begin while the network contains a node j with e(j) /2 do begin among nodes j with e(j) /2 (called large excess nodes ), choose i with minimum distance label perform push/relabel(i) while ensuring that no node excess exceeds ; end := /2 end
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13 Pushing in the 64-scaling phase 0 5 4 3 2 1 t 4 5 s 2 s 7 9 8 7 6 9 6 3 8 5 j 32 e(j) 64 “large excess” i e(i) < 32 4 7 5 62 0 25 2 40
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