Maximum Flows: Polynomial Algorithms

Transcription

1 Maximum Flow: Polynomial Algorihm

2 Algorihm Augmening pah Algorihm - Labeling Algorihm - Capaciy Scaling Algorihm - Shore Augmening Pah Algorihm Preflow-Puh Algorihm - FIFO Preflow-Puh Algorihm - Highe Label Preflow-Puh Algorihm - Exce Scaling Algorihm O(nmU) O( nmlogu) O( n ) O( nm n O( n m) O( n m) log U )

3 Augmening flow along a pah wih maximum reidual capaciy The pah wih maximum reidual capaciy ha capaciy a lea where v v * ( v * v) / m : he maximum flow :he curren flow value By Theorem., in Augmening pah algorihm if we augmen flow along a pah wih maximum reidual capaciy, he algorihm erminae wihin O(mlog U)

4 Augmening flow along a pah wih maximum reidual capaciy Theorem.(for Maximizaion): z k ( z : he objecive funcion value of omeoluion a hek' h ieraion k z k ) ( z * z H : The maximum poible improvmen k ) Then he algorihm erminae wihin O(log H/α). ( v k v k ) ( v * v k ) / m / m H :mu

5 Δ-reidual Nework G(x,Δ) A nework conaining arc whoe reidual capaciy i a lea Δ. (,) (,) (,) (,) (,) (,) (,) (,) (,) G(x,)

6 Capaciy Scaling Algorihm x log u while. G( x, ) conain a pah fromnode o node do..- idenify a pah p in G( x, )..- Augmen uni of flow along P and updaeg(x, )..- /..- min {r ij :(i, j) i an arcin P } O( m log U )

7 Diance label Wih repec o he reidual nework G(x), a diance funcion d from he e of node o he e of nonnegaive ineger i valid if i aifie he following wo validiy condiion: ) ( ) ( )) ( ( ), ( ) ) ( ) j d i d x G A j i b d a

8 Diance label Propery 7.. If he diance label are valid, he diance label d(i) i a lower bound on he lengh of he hore pah from node i o node in he reidual nework. Propery 7.. If d() n, he reidual nework conain no direced pah from node o node.

9 Diance label We ay a diance label i exac if for each node i, d(i ) i he lengh of he hore pah from node i o node in he reidual nework. We ay an arc (i,j) in he reidual nework i admiible if d(i)=d(j)+. A pah from node o node i admiible if all of i arc are admiible Propery 7.. An admiible pah i a hore augmening pah from ource o ink.

10 Shore Augmening pah Algorihm - x - Obain heexac diacelabel d(i) -i ; - while d() n do. if i ha an admiible arc hen...advance(i)...if i hen augmenand e i ele rerea( i)

11 Shore Augmening pah Algorihm Procedureadvance(i) - le (i, j) be an admiible arc in A(i) - (j) i, i j Procedurererea(i) - d(i) min{d(j) :(i, j) A(i), r ij } - if i, hen i (j) Procedureaugmen - Uing - he eceor indice idenify an augmening pah p frm min{r - augmen uni ij :(i, j) p} of flow along pah p. o.

12 Shore Augmening pah Algorihm

13 Shore Augmening pah Algorihm i = verex

14 Shore Augmening pah Algorihm i = verex

15 Shore Augmening pah Algorihm i = verex

16 Shore Augmening pah Algorihm i = verex

17 Shore Augmening pah Algorihm i = verex

18 Shore Augmening pah Algorihm i = verex

19 Shore Augmening pah Algorihm i = verex

20 Shore Augmening pah Algorihm i = verex

21 Shore Augmening pah Algorihm i = verex

22 Shore Augmening pah Algorihm i = verex

23 Shore Augmening pah Algorihm i = verex

24 Shore Augmening pah Algorihm i = verex

25 Shore Augmening pah Algorihm i = verex

26 Shore Augmening pah Algorihm i = verex

27 Shore Augmening pah Algorihm i = verex

28 Shore Augmening pah Algorihm i = verex

29 Shore Augmening pah Algorihm i = verex

30 Shore Augmening pah Algorihm i = verex

31 Shore Augmening pah Algorihm i = verex

32 Shore Augmening pah Algorihm i = verex

33 Shore Augmening pah Algorihm i = verex

34 Shore Augmening pah Algorihm i = verex

35 Shore Augmening pah Algorihm i = verex

36 Correcne of Shore Augmening pah Algorihm Lemma 7.. The hore pah augmening algorihm mainain valid diance label a each ep. Moreover, each relabel(rerea) operaion ricly increae he diance label of a node. Theorem 7. The hore pah augmening algorihm correcly compue a maximum flow.

37 Complexiy of Shore Augmening pah Algorihm Propery 7.7. Uing he curren arc daa rucure, if he algorihm relabel any node a mo k ime, he oal ime pen in finding admiible arc and relabeling he node i O(km). Lemma 7.8. If he algorihm relabel any node a mo k ime, he algorihm aurae arc a mo km/ ime. Lemma 7.9.(a) In he algorihm each diance label increae a mo n ime. Conequenly he oal number of relabel operaion i a mo n*n. (b) The number of augmen operaion i a mo nm/.

38 Complexiy of Shore Augmening pah Algorihm Theorem 7.. The hore augmening pah algorihm run in O( n m) ime. The number of Augmenion : O(mn/) The ime for each Augmenion :O(n) The number of Relabel operaion: O( n ) The ime pending on Advance operaion = The ime pending on finding admiible arc : O(nm)