Maximum Flow Algorithms

1. Maximum Flow Algorithms Emanuele Altieri CSC252, Fall 2000

2. What is the maximum amountof material that can be transportedfrom s (source) to t (sink) ? Town C 8 Town A 9 5 7 Port (t) Mine (s) 2 3 18 10 Town B Town D 12

3. Maximum Flow Algorithms Given a graph G = (V, E) Ford-Fulkerson method Preflow-Push algorithm Lift-To-Front algorithm T(V, E) = O(E |f*|) T(V, E) = O(V2E) T(V, E) = O(V3) Edmons-Karp algorithm T(V, E) = O(V E2)

4. Ford-Fulkerson and Edmons-Karp algorithms Flow of a Graph A flow of a graph G = (V, E) is a function f : V V R that satisfies the following properties: Skew Simmetry Flow Conservation Capacity Constraints 4:10 10 u v u v u c (u, v) = 10 c (v, u) = 0 f (u, v) = 4 f (v, u) = –f (u, v) = – 4 In general  u, v  V  f (u, v) = – f (v, u) c (u, v) = 10 f (u, v)  c (u, v) In general  u, v  V  f (u, v)  c (u, v)  u  V – {s, t}  (v  V)f (u, v) = 0 also called net flow into u: f (V, u) = 0 Residual Capacity Given u, vV, the residual capacity cf of (u, v) shows how many flow units can still flow from u to v. cf (u, v) = c (u, v) – f (u, v) 6 u v 6 cf (u, v) = 10 – 4 = 6 4:10 u v u v 4 cf (v, u) = 0 – (– 4) = 4 4 u v Residual Graph 3 5:8 The residual network of a graph G = (V, E) is defined as: s s 5 2:3 5:5 1 G Gf = (V, Ef) Ef = { (u, v)  V  V such that cf (u, v) > 0 } 2 Gf 5 t t 2 2:2

5. Ford-Fulkerson and Edmons-Karp algorithms Ford-Fulkerson Method for( each edge (u, v)  E [G] ) { f [u, v] = 0 f [v, u] = 0 } while(  p (s  t)  Gf) { cf (p) = min { cf (u, v) where (u, v)  Ef [p] } for( each edge (u, v)  Ef){ f [u, v] = f [u, v] + cf (p) f [v, u] = – f [u, v] } } O(E) s t Time to find p using any traversal: O(E) Max Number of iterations: O(| f * | ) Any path s t is augmented at each iteration T(V, E) = O(E |f*|) Edmons-Karp Algorithm for( each edge (u, v)  E [G] ) { f [u, v] = 0 f [v, u] = 0 } while(  p (s  t)  Gf such that p =  (s, t) ) { cf (p) = min { cf (u, v) where (u, v)  Ef [p] } for( each edge (u, v)  Ef){ f [u, v] = f [u, v] + cf (p) f [v, u] = – f [u, v] } } O(E) s t Time to find  using BFS: O(E) Max Number of iterations: O(VE) The shortes path s t (in length) is augmented at each iteration T(V, E) = O(V E2)

6. Preflow-Push Algorithm Preflow, instead of flow A preflow of a graph G = (V, E) is a function f : V V R that satisfies the following properties: Capacity Constraints NO Flow Conservation c (u, v) u v u f (u, v)  c (u, v) 15 10 u reservoir Skew Simmetry  u  V – {s} excess flow e(u) = f (V, u)  0  u  V – {s, t} if e(u) > 0 then u is overflowing 5 f (u, v) e(u) = f (V, u) = 5 u v f (v, u) = –f (u, v) Residual Capacity Given u, vV, the residual capacity cf of (u, v) shows how many flow units can still flow from u to v. cf (u, v) = c (u, v) – f (u, v) Height Function Let G = (V, E) be a flow network with source s and sink t, and let f be a preflow in G. A function h : VN is a height function if h(s) = |V| h(t) = 0 h(u)  h(v) + 1,  (u, v)  Ef 5 4 3 2 1 0 s Residual Graph Gf = (V, Ef) Ef = { (u, v)  V  V such that cf (u, v) > 0 and h(u)  h(v) + 1 } t

7. Preflow-Push Algorithm INITIALIZE-PREFLOW (G, s) LIFT (u) PUSH (u, v) • u  V – { s, t } • Applies when • e(u) > 0 •  v  V (u, v)  Ef h[u]  h[v] • u, v  V • Applies when • e(u) > 0 • cf (u, v) > 0 • h(u) = h(v) + 1 G = (V, E) s  V // Source of the graph for(each vertex u  V [G]) { h[u] = 0 e[u] = 0 } for (each egde (u, v)  E[G]) { f [u, v] = 0 f [v, u] = 0 } h[s] = |V [G] | for(each vertex u  Adj[G]) { f [s, u] = c(s, u) f [s, u] = – c(s, u) e [u] = c (s, u) } h[u] = 1 + min { h[v] : (u, v)  Ef } f = min {e[u], cf (u, v)} f [u, v] = f [u, v] + f f [v, u] = – f [u, v] e[u] = e[u] – f e[v] = e[v] + f GENERIC-PREFLOW-PUSH (G) a a a 0:10 0:10 0:10 15:15 15:15 15:15 INITIALIZE-PREFLOW (G, s) while ( an applicable push or lift) do select an applicable push or lift operation and perform it LIFT(a) PUSH(a, b) 0:8 0:8 8:8 s t s t s t f = 8 10:10 0:20 10:10 0:20 10:10 0:20 b b b 5 4 3 2 1 0 e(u) 5 4 3 2 1 0 e(u) 5 4 3 2 1 0 e(u) s s s 10 10 10 10 15 15 15 10 10 a a 8 20 20 20 8 8 t t t a b b b 15 10 15 10 7 18

8. Lift-To-Front Algorithm DISCHARGE (u) Vertices List current [u] = head [ N [u] ] while( e[u] > 0 ) v = current [u] if (v == NULL) LIFT (u) current [u] = head [ N [u] ] elseif(cf (u, v) > 0 and h[u] = h[v] + 1) PUSH (u, v) else current [u] = next_neighbor [v] end if end while L = V – { s, t } Neighbor Lists u  L, N[u] = {v  V such that (u, v)  E or (v, u)  E } 0:8 y z 0:10 0:5 14:14 0:7 6 5 4 3 2 1 0 s x t 0:16 12:12 s LIFT-TO-FRONT (G, s) INITIALIZE-PREFLOW (G, s) L = V[G] – { s, t } u = head[L] while( u  NULL) old_height = h[u] DISCHARGE (u) if (h[u] > old_height) move u to the front of list L u = next[u] end while 14 16 12 7 5 x y 8 z 10 t INITIALIZE-PREFLOW (G, s) LIFT (u) PUSH (u, v) • u  V – { s, t } • Applies when • e(u) > 0 • v  V (u, v)  Ef h[u]  h[v] • u, v  V • Applies when • e(u) > 0 • cf (u, v) > 0 • h(u) = h(v) + 1 u, v  V f [u, v] = c(u, v) if u == s – c(v, u) if v==s 0 otherwise h [u] = |V| if u == s 0 otherwise L N N[x] N[y] N[z]

9. References • Introduction to Algorithms, Thomas Cormen, ch.27, 27.4-5 (Preflow-Flush and Lift-To-Front algorithms. Ford-Fulkerson method and Edmons-Karp algorithm) • Algorithms and Data Structures, Jeffrey Kingston, 12.3 (Ford-Fulkerson method and Edmons-Karp algorithm)