Maximum Flow in Planar Graph
Planar Graph and i Dual
Dualiy i defined for direced planar graph a well
Minimum - cu in undireced planar graph
An - cu (undireced graph)
An - cu
The dual o he cu
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)
The face conaining
and he face conaining
Le P be he hore pah beween hem
Any cycle ep. and croe P
The hore cycle will cro P once
The hore cycle will cro i once We are afer he hore cycle ep. and and croe P once.
Finding uch hore cycle
Finding uch hore cycle Claify edge inciden o he pah a lef or righ
Finding uch hore cycle Cu he pah open
Finding uch hore cycle Direc he edge inciden o he pah
Finding uch hore cycle v 1 v 2 Find hore pah beween every pair v 1, v 2
Finding uch hore cycle v 1 v 2 Find hore pah beween every pair v 1, v 2
Finding uch hore cycle v 1 v 2 Find hore pah beween every pair v 1, v 2
Finding uch hore cycle v 1 v 2 Take he hore among hee hore pah
Speeding up by divide and conquer v 1 v 2
Speeding up by divide and conquer Shore cycle do no cro
Speeding up by divide and conquer Shore cycle do no cro
Take v 1 and v 2 o be he middle pair v 1 v 2
Take v 1 and v 2 o be he middle pair v
Spli he problem v
Spli he problem
Add new ource/ink v
In fac Thi ae i ymmeric o our aring poiion v
Analyi Ob1: Pah are horer by a facor of 2 deph of recurion log n v
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
Summary Toal ime O(nlog 2 n) uing Dijkra or O(nlog(n)) uing he O(n) SSSP algorihm for planar graph v
Circulaion and price
Circulaion and price β β β β β β
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
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
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.
2 applicaion for hi connecion
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)
Max - flow when and are on he ame face Find max flow from o when and are on he ame face
Max - flow when and are on he ame face Add an edge from o wih capaciy
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
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
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)
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 )