Preflow Push Algorithms for Maximum Flow

0.1 Maximum Flow Problem:[CLRS] INPUT: A directed simple graph G = [V ;E] with positive edge capacities c(e); e ∈ E and two distinguished nodes s, t ∈ V . Definition 1 A feasible flow is a function f : E → R+ that satisfies: ∑ v∈V (u,v)∈E f(u, v)− ∑ v∈V (v,u)∈E f(v, u) =    F 0 −F u = s u = s, t u = t 0 ≤ f(u, v) ≤ c(u, v) ∀(u, v) ∈ E We want a feasible flow that maximizes F . Definition 2 A preflow is a function f : E → R+ that satisfies: e (u) = ∑ v∈V (v,u)∈E f(v, u)− ∑ v∈V (u,v)∈E f(u, v) ≥ 0 ∀u = s 0 ≤ f(u, v) ≤ c(u, v) ∀(u, v) ∈ E where e(u) is called the excess at u with respect to preflow f . This algorithm maintains a preflow at all times and ends with a feasible flow that maximizes F . Definition 3 A node u for which e(u) > 0 is said to be overflowing with respect to f .