Flow and Matching
Flow and Matching
Flow digraph:D(V,E) source:s sink:t capacity:c:E→R+ 0/1 0/1 0/1 0/1 0/3 S 0/3 0/4 0/1 0/4 0/1
Flow digraph: D(V,E) capacity: c : E R+ 0/1 0/4 0/1 0/1 0/1 0/1 0/1 0/4 0/3 0/3 s t source: s sink: t
Flow digraph:D(V,E) source:s sink:t capacity:c:E→R+ flow:f:E→R 1/1 1/3 S 1/3 1/1 214 capacity:∀(u,w)∈E,fuv≤cuw conservation:u∈V\{s,t},∑(u,wEBfwu=∑u,)EBfw
Flow 1/1 2/4 0/1 1/1 1/1 1/1 1/1 2/4 1/3 1/3 digraph: D(V,E) capacity: c : E R+ source: s sink: t s t flow: f : E R+ ⇤(u, v) ⇥ E, fuv cuv ⇥u V \{s, t}, (w,u)E fwu = (u,v)E fuv capacity: conservation:
14 1/1 1/3 1/3 2 1/1 214 0/T capacity::(u,w)∈E,fuu≤cuu conservation:u∈V\{s,t,∑(u,)E fwu=∑(u,)EEfuv value of ∑∑fsu=∑ft flow: (s,u)∈E (w,t)∈E maximum flow
1/1 2/4 0/1 1/1 1/1 1/1 1/1 2/4 1/3 1/3 s t ⇤(u, v) ⇥ E, fuv cuv ⇥u V \{s, t}, (w,u)E fwu = (u,v)E fuv capacity: conservation: (s,u)E fsu = (v,t)E fvt value of flow: maximum flow
0/1 1/1 1/1 2/3 23 1/1 314 0/T capacity:(u,v)∈E,fuw≤cuu conservation::u∈V八{s,t,∑(u,)EBSwu=∑(u,w)EBfw value of ∑∑fsu=∑ft flow: (s,u)∈E (w,t)∈E maximum flow
1/1 3/4 0/1 1/1 1/1 0/1 1/1 3/4 2/3 2/3 s t ⇤(u, v) ⇥ E, fuv cuv ⇥u V \{s, t}, (w,u)E fwu = (u,v)E fuv capacity: conservation: (s,u)E fsu = (v,t)E fvt value of flow: maximum flow