Network Flows UPCOPENCOURSEWARE number 34414

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1 Nework Flow UPCOPENCOURSEWARE number Topic : F.-Javier Heredia Thi work i licened under he Creaive Common Aribuion- NonCommercial-NoDeriv. Unpored Licene. To view a copy of hi licene, vii hp://creaivecommon.org/licene/by-nc-nd/./ or end a leer o Creaive Common, Caro Sree, Suie 9, Mounain View, California, 9, USA.

2 .- Maximum flow algorihm (MF) (Chap. 6 & 7 Ahuja,Magnani,Orlin) Noaion and hypohei. Claificaion of he MF algorihm. Cu and Flow. Peudopolynomial MF augmening pah algorihm. Generic augmening pah algorihm. Labeling algorihm (Ford-Fulkeron). Convergence. Complexiy. Polynomial MF augmening pah algorihm. Improvemen of he Ford-Fulkeron algorihm Shore Augmening Pah Algorihm. Exercice. Convergence and complexiy. MFP-

3 Noaion and Aumpion. MFP: conidering a direced nework G=(N,A) wih a nonnegaive capaciy u, he maximum flow problem find he maximum flow from he ource node o he ink node. max.a.: x { j:( i, j) A} { j:( j, i) A} x ij x ji u z = v Aumpion: i. The nework i direced. ii. ij v = v ij ( i, i N j) A i = i = {, } The capaciie u ij are nonnegaive ineger. (.a) (.b) (.c) iii. G doe no conain any direced pah - wih infinie capaciy arc. iv. If (i,j) A, hen (j,i) A (probably wih u ji =). v. G doe no conain parallel arc. MFP-

4 MFP algorihm: MFP algorihm claificaion Augmening-Pah algorihm: mainain ma balance conrain a every node of he nework oher ha he ource () and ink () node. Thi algorihm incremenally augmen flow along pah from he ource node o he ink node. Algorihmic complexiy: O(n m) he ucceive hore pah algorihm i he mo efficien augmening-pah algorihm. Wih uncapaciaed arc, imple implemenaion may no converge o an opimal oluion. Pre-flow-Puh algorihm: flood he nework o ha ome node have exce of flow. Thee algorihm incremenally relieve flow from node wih exce by ending flow o he forward oward he ink node () or backward o he ource node (). Algorihmic complexiy: O(n m) Highe-label pre-flow-puh algorihm i mo efficien maximum flow algorihm. MFP-

5 Reidual nework: Flow and Cu (/) Reidual capaciy r ij of arc (i,j) A, w.r. he flux x ij : max flow ha can be en from node i o node j hrough arc (i,j) and (j,i). u ij - x ij i x ji j r ij = ( u ij - x ij ) + x ji Reidual nework G(x) : formed by he arc wih poiive reidual capaciy w.r.. flow x: i i j ( x ij, u ij ) (,) G j r ij G(x) (p.e.:r = ( - ) + = ) MFP-

6 i j (x ij,u ij ) i j r ij Flow and Cu (/) (,) A G(x) Flow acro an - cu : - cu capaciy, u[s,s] : um of capaciie of he foward arc (i,j) (S,S) in he - cu. S={,}, S={,}, (i,j) (S,S) = {(,),(,)} u[s,s] = + = 7 Minimum cu : - cu wih he minimum capaciy among all cu -: Example, min{ 6, 7, 8, 6} = 6 Reidual capaciy of he - cu, r[s,s] : um of he reidual capaciie of he forward arc in he cu (S,S) : v = S={,}, S={,}, (i,j) (S,S) = {(,)} r[s,s] = i S x ji = (.) { j:( i, j) A} { j:( j, i) A} x ij x ij ( i, j) ( S, S ) x ij ( i, j) ( S, S ) MFP- 6

7 Flow and Cu (/) Propery.: The value of any flow x i le han or equal o he capaciy of any - cu in he nework Proof: ubuing x ij u ij in he fir expreion of (.) and x ij in he econd expreion follow ha: v ( i, j) ( S, S ) [ S S] uij = u, Reul: if we find a flow x equal o he value of any cu -, hi flow i he opimum. I ha been demonraed ha alway exi a flow of hi ype. Propery.: For any flow x of value v in he nework A, he addiional flow ha can be en from he ource o he ink node i le han or equal o he reidual capaciy of any - cu. MFP- 7

8 Example (prob. 6. A-M-O) Conidering he following nework: (,) (,) (,) (,8) (,) (,) 7 i (x ij, u ij ) j Find: 6 (,6) Four - cu, each one wih forward arc, indicaing heir capaciy, reidual capaciy and he flow acro he cu. Repreen he reidual graph G(x) and indicaing wo augmening pah - (i.e. any direced pah - over G(x)). MFP- 8

9 Generic Augmening Pah Algorihm (GAP). GAP Algorihm : given a direced graph G=(N,A) wih ineger capaciie find he maximum flow beween node and. begin x:=; find G(x); while G(x) conain a direced pah from node do end; end idenify an augmening pah P ; e δ := min{ r ij : (i,j) P }; augmen δ uni of flow along P; updae G(x); MFP- 9

10 Generic Augmening Pah : example. Solve he following MFP wih he GAP algorihm: i u ij j MFP-

11 GAP Algorihm oluion example Begin: x := G( x ) ; P ={(,),(,)} ; δ = G( x ) ; P ={(,),(,),(,)} ; δ = i r ij j G( x ) ; P ={(,),(,)} ; δ = G( x ) ; P= ; STOP MFP-

12 Ford-Fulkeron Algorihm: decripion. Ford-Fulkeron Algorihm : le he direced pah G=(N,A) wih ineger capaciie, find he maximum flow beween and. begin end node i labeled; find an augmening pah P from o uing pred( ); unlabel all node; eδ := min{ r ij : (i,j) P}; e pred( j ):= for each j N ; augmen δ uni of flow along P; label node and e LIST:={ }; updae he reidual capaciie; while LIST and i unlabeled do end; remove a node i from LIST; for each arc ( i, j ) in he reidual nework emanaing from node i do if node j i unlabeled hen e pred( j ):= i; label node j and add j o LIST; end; if i labeled hen augmen(r,pred); end; if i labeled do end if; Procedure augmen(r,pred); begin MFP-

13 Ford-Fulkeron algorihm : example. See he Apple for main idea of he algorihm: hp://www-b.i.okuhima-u.ac.jp/~ikeda/uuri/maxflow/maxflowapp.hml?demo Solve now he following MFP wih he Ford-Fulkeron algorihm: 6 i u ij j MFP-

14 Ford-Fulkeron Algorihm: convergence (I). Convergence of he algorihm: Theorem.: A erminaion of he Ford-Fulkeron algorihm, he flow x are he oluion of he MFP. Proof: Le S be he e of labeled node a he la ieraion. Thu: S and S=N-S a) r ij = (i,j) (S,S) (here i no any augmening pah beween S and S.) b) r ij = (u ij -x ij ) + x ji ; x ij u ij ; x ji a) and b) x ij =u ij (i,j) (S,S) ; x ij = (i,j) (S,S). Subiuing in (.) we ge: ij ij ij (, ij) ( SS, ) (, ij) ( SS, ) (, ij) ( SS, ) [, ] v= x x = u = uss Therefore, according o propery (.), he flow x are he maximum flow and [S,S] cu i a minimum cu. MFP-

15 Ford-Fulkeron Algorihm: convergence (II). Ford-Fulkeron Algorihm: convergence (II). Max-Flow Min-Cu Theorem: The maximum flow beween he ource node and he ink node in a capacied nework i equal o he minimum capaciy among all - cu. Augmening Pah Theorem: A flow x* i a maximun flow if and only if he reidual nework G(x*) conain no augmening pah. Inegraliy Theorem: If all arc capaciie are ineger, he maximum flow problem ha an ineger maximum flow. MFP-

16 Ford-Fulkeron Algorihm: Algorih: complexiy. Wor-cae complexiy: we compue an upper bound of he number of ieraion of Ford-Fulkeron: Calculaion of he augmening pah: each augmenaion vii each arc a mo once, and herefore he co i O(m). Number of augmening pah: If all capaciie are ineger and bounded by U, hen he capaciy of he cu (,N-{}) i nu, and he maximum flow i nu. The algorihm increae he flow a lea one uni in any augmenaion, i.e., hen he number of augmenaion i nu. Theorem.: The labeling algorihm olve he MFP in O(nmU) ime. MFP- 6

17 Improved Generic Pah Algorihm Drawback of he Ford-Fulkeron labeling algorihm:. I peudopolynomial complexiy O(nmU) i unaracive for problem wih large capaciie.. If he capaciie are irraional, he algorihm migh no converge. Improvemen: Large augmening pah algorihm: le P be a pah from o in G(x) uch ha δ* i maximum (O(m logu)) Shore augmening pah algorihm: le P be a pah from o in G(x) wih he fewe number of arc (O(m n )). MFP- 7

18 Shore augmening pah algorihm () Diance Label A diance label i a funcion d: N Z +. A diance label i aid o be valid if i aifie he following: d() =. d(i) d(j) + for each (i,j) G(x). An arc (i,j) G(x) i admiible if d(i) =d(j) +. () Adaped from: Orlin, Jame..8J Nework Opimizaion,Fall. (Maachue Iniue of Technology: MIT OpenCoureWare), hp://ocw.mi.edu (Acceed 8 Mar, ). Licene: Creaive Common BY-NC-SA MFP- 8

19 Shore augmening pah algorihm An example of valid diance label The diance label are on he node. All arc are in he reidual nework. The admiible arc are hick and red. The label would no be valid if here were an arc from o. MFP- 9

20 Shore augmening pah algorihm More on valid diance label Propery.: Le d( ) be a valid diance label. Then d(i) i a lower bound on he hore pah from i o in he reidual nework. (The diance i meaured in erm of he number of arc.) Proof. Le P be he hore pah from I o. If P conain k arc hen: d(i) k d(j) k- d(h) d(l) i j h l P ha k arc MFP-

21 Shore augmening pah algorihm On Finding Pah hore - pah Propery.: Le d( ) be a valid diance label. Then d(i) i a lower bound on he hore pah from i o in he reidual nework. (The diance i meaured in erm of he number of arc.) Propery.: If d() n, he reidual nework conain no direced pah from he ource node o he ink node. Propery.: An admiible pah i a hore augmening pah from he ource o he ink. MFP-

22 The hore augmening pah algorihm begin x:= ; Obain exac diance label d(i); i:= ; while d() < n do begin end end If i ha an admiible arc hen advance(i); if i = hen augmen and e i = ; ele rerea(i); endif Procedure advance(i) begin Le ( i, j) be an admiible arc in A( i); pred( j):= i and i:= j; end Procedure rerea(i) begin d(i) := + min { d(j) : r ij > }; if i hen i:= pred(i); end MFP-

23 Shore Augmening Pah Thi i he original nework, and he original reidual nework. MFP-

24 Iniialize Diance The node label henceforh will be he diance label. d(j) i a mo he diance of j o in G(x) MFP-

25 Repreenaion of admiible arc An arc (i,j) i admiible if d(i) = d(j) +. An - pah of admiible arc i a hore pah Admiible arc will be repreened wih hick line 6 MFP- 6

26 Look for a hore - pah Sar wih and do a deph fir earch uing admiible arc. Nex. Send flow, and updae he reidual capaciie. 7 MFP- 7

27 Updae reidual capaciie Here are he updaed reidual capaciie. We will updae diance label laer, a needed. 8 MFP- 8

28 MEIO/UPC-UB : NETWORK FLOWS MFP- 9 9 Look for a hore - pah Sar wih and do a deph fir earch uing admiible arc. Nex. Send flow, and updae he reidual capaciie.

29 Updae reidual capaciie Here are he updaed reidual capaciie. We will updae diance label laer, a needed. MFP-

30 Search for a hore - pah Sar wih and do a deph fir earch uing admiible arc. If here are no admiible arc from i, hen relabel(i) and revere along he pah leading o i. MFP-

31 Updae diance and pah Sar wih and do a deph fir earch uing admiible arc. If here are no admiible arc from i, hen relabel(i) and revere one arc along he pah leading from. MFP-

32 MEIO/UPC-UB : NETWORK FLOWS MFP- Updae diance and pah Sar wih and do a deph fir earch uing admiible arc. If here are no admiible arc from i, hen relabel(i) and revere one arc along he pah leading from.

33 Look for a hore - pah Coninue he pah from where i lef off. If he pah reache, hen end flow and updae reidual capaciie. MFP-

34 MEIO/UPC-UB : NETWORK FLOWS MFP- Updae reidual capaciie Here are he updaed reidual capaciie.

35 MEIO/UPC-UB : NETWORK FLOWS MFP- 6 6 Search for a hore - pah Search for a hore - pah aring from If here are no admiible arc from i, hen relabel(i) and revere one arc along he pah leading from.

36 MEIO/UPC-UB : NETWORK FLOWS MFP- 7 7 Search for a hore - pah Search for a hore - pah aring from If here are no admiible arc from i, hen relabel(i) and revere one arc along he pah leading from.

37 MEIO/UPC-UB : NETWORK FLOWS MFP- 8 8 Search for a hore - pah Search for a hore - pah aring from If he pah reache, hen end flow and updae reidual capaciie.

38 updae he reidual capaciie Here are he updaed reidual capaciie 9 MFP- 9

39 Search for a hore - pah Search for a hore - pah Nex: updae he reidual capaciie MFP-

40 Updae he reidual capaciie Here are he updaed reidual capaciie MFP-

41 Look for a hore - pah 6 updae diance label and pah If d() > n-, hen here i no pah from o MFP-

42 Thee are he reidual capaciie for he opimum flow 6 There i no - pah in he reidual nework A min cu ha S = {,, }. MFP-

43 Shore Augmening Pah Algorihm Convergence and Complexiy (AMO- 7.) Convergence: Lemma.: The SAPA mainain valid diance label a each ep. Moreover, each relabel operaion ricly increae he diance label of a node Theorem.: The SAPA correcly compue a maximum flow Raionale: The SAPA erminae when d()>n+ he nework conain no augmening pah (Prop. 7.). Conequenly, he flow x i a max. flow. Complexiy : We can deermine each augmenaion in O(n). The oal ime o mainain and updae all diance label i O(nm). The oal number of augmenaion i O(nm). Concluion. The oal running ime i O(n m). MFP-

44 Ford-Fulkeron Alg. Ford-Fulkeron Algorihm : exercici : exercie (II). The following exercie will be aigned o: 6.(), 6. (), 6. (), 6. (), 6.6 (), 6. (). Each one repreen an applicaion of he MFP. Tak o complee: MFP formulaion. If he aemen doe no define he MFP, formulae he problem wih a opology of he nework and i repecive co. Solve numerically wih any ofware he MFP formulaed. MFP-

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