Maximum Flow in Planar Graphs
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1 Maximum Flow in Planar Graph
2 Planar Graph and i Dual
3 Dualiy i defined for direced planar graph a well
4 Minimum - cu in undireced planar graph
5 An - cu (undireced graph)
6 An - cu
7 The dual o he cu
8 Cu/Cycle A cu ha eparae he graph ino wo conneced componen one conaining and one conaining (we can aume he min-cu i like hi) A imple cycle wih inide and ouide Look for a hore uch cycle in he dual (lengh in he dual are capaciie in he primal)
9 The face conaining
10 and he face conaining
11 Le P be he hore pah beween hem
12 Any cycle ep. and croe P
13 The hore cycle will cro P once
14 The hore cycle will cro i once We are afer he hore cycle ep. and and croe P once.
15 Finding uch hore cycle
16 Finding uch hore cycle Claify edge inciden o he pah a lef or righ
17 Finding uch hore cycle Cu he pah open
18 Finding uch hore cycle Direc he edge inciden o he pah
19 Finding uch hore cycle v 1 v 2 Find hore pah beween every pair v 1, v 2
20 Finding uch hore cycle v 1 v 2 Find hore pah beween every pair v 1, v 2
21 Finding uch hore cycle v 1 v 2 Find hore pah beween every pair v 1, v 2
22 Finding uch hore cycle v 1 v 2 Take he hore among hee hore pah
23 Speeding up by divide and conquer v 1 v 2
24 Speeding up by divide and conquer Shore cycle do no cro
25 Speeding up by divide and conquer Shore cycle do no cro
26 Take v 1 and v 2 o be he middle pair v 1 v 2
27 Take v 1 and v 2 o be he middle pair v
28 Spli he problem v
29 Spli he problem
30 Add new ource/ink v
31 In fac Thi ae i ymmeric o our aring poiion v
32 Analyi Ob1: Pah are horer by a facor of 2 deph of recurion log n v
33 Analyi Ob2: Each red verex i in one ubproblem (+ and ) Toal ize of ubproblem a level k log(n) i O(n + 2 k ) = O(n) v
34 Summary Toal ime O(nlog 2 n) uing Dijkra or O(nlog(n)) uing he O(n) SSSP algorihm for planar graph v
35 Circulaion and price
36 Circulaion and price β β β β β β
37 Circulaion and price Decompoe he flow ino CCW cycle Sar wih poenial of 0 For each CCW cycle of value β, add β o he poenial of he face inide he cycle The flow along an edge i he difference in he poenial of i inciden face β 1 β 2 -β 1 β 2
38 Circulaion and price Any face price define a circulaion he ame way β 1 -β 5 β 1 β 5 β 2 -β 1 β 4 β 5 -β 4 β 2 β 3 β 3 -β 2 β 4 -β 3 β The flow i feaible iff β β u(e) Iff e u(e) + β β 0 (nonnegaive reduced co) β
39 Circulaion and price Flow i feaible iff e u(e) + β β β β We can ge poenial from any hore pah ree in he dual The reduced co equal he reidual capaciie of he correponding flow.
40 2 applicaion for hi connecion
41 Feaible circulaion Negaive capaciy i a lower bound on he flow on he revere arc u(e) < 0 β β A circulaion exi iff here are feaible poenial iff no negaive cycle in he dual Can decide via a hore pah algorihm ha can handle negaive weigh O(mn)
42 Max - flow when and are on he ame face Find max flow from o when and are on he ame face
43 Max - flow when and are on he ame face Add an edge from o wih capaciy
44 Max - flow when and are on he ame face f 2 f 1 Infinie face pli Compue hore pah from f 1 and define a flow according o hee poenial
45 Max - flow when and are on he ame face f 2 f 1 Infinie face pli Compue hore pah from f 1 and define a flow according o hee poenial
46 Max - flow when and are on he ame face f 2 Delee he new edge and you ge a maximum flow from o f 1 Proof. I feaible (corre. o p. in he dual), i maximum becaue i equal he minimum -cu (=hore pah from f1 o f2)
47 Max - flow when and are on he ame face f 2 f 1 Proof. I feaible (corre. o p. in he dual), i maximum becaue i equal he minimum -cu (=hore pah from f 1 o f 2 )
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