Ford-Fulkerson

[[Category:Max-Flow]

General Information

Algorithmic problem:Max-Flow Problems Prerequisites: Type of algorithm: loop

Auxiliary data: A pointer p of type "pointer to a B-tree node".

Abstract View

Invariant: Before and after each iteration:

1. p points to some node N of the B-tree and
2. the searched key is in the range of N.

Variant: p is redirected from the current node N to some child of the current node.

Break condition:

1. p points to a leaf of the B-tree or (that is, inclusive-or)
2. the searched key is in the node to which p points.

Induction Basis

Abstract view: We start with some feasible flow, for example, the zero flow.

Implementation: [math]\displaystyle{ \forall a \in A:f(a):=0. }[/math]

Proof: Obvious.

Induction Step

Abstract view: Find a flow-augmenting path and increase [math]\displaystyle{ f }[/math] along this path by the maximal value such that the flow value of each [math]\displaystyle{ a\in A }[/math] remains in the interval [math]\displaystyle{ [0...c(a)] }[/math].
Implementation:

1. Apply a graph traversal algorithm from [math]\displaystyle{ s }[/math] as follows:
   1. if [math]\displaystyle{ f(v,w)\lt c(v,w) }[/math], [math]\displaystyle{ (v,w)\in A }[/math] can be used for going forward in the direction [math]\displaystyle{ v \rightarrow w }[/math];
   2. if [math]\displaystyle{ f(v,w)\gt 0 }[/math], [math]\displaystyle{ (v,w)\in A }[/math] can be used for going forward in the direction [math]\displaystyle{ w\rightarrow v }[/math]; .
2. Terminate this graph traversal once either is seen or all reachable nodes were seen (whatever occurs first).
3. In the latter case, the break condition of the loop applies.
4. Otherwise,
   1. let [math]\displaystyle{ p }[/math] denote the current path of the traversal (which is an [math]\displaystyle{ (s,t) }[/math]-path in this case);
   2. let [math]\displaystyle{ x }[/math] denote the minimum of the values [math]\displaystyle{ c(a)-f(a) }[/math] on all forward arcs of [math]\displaystyle{ p }[/math];
   3. let [math]\displaystyle{ y }[/math] denote the minimum of the values [math]\displaystyle{ f(a) }[/math] on all backward arcs of [math]\displaystyle{ p }[/math].
   4. For each arc [math]\displaystyle{ a \in A }[/math] on [math]\displaystyle{ p }[/math], increase the flow value by [math]\displaystyle{ min \{x,y \} }[/math] if [math]\displaystyle{ a }[/math] is a forward arc on [math]\displaystyle{ p }[/math], otherwise, decrease the flow value by [math]\displaystyle{ min \{x,y \} }[/math].

Correctness: If the graph traversal does not hit [math]\displaystyle{ t }[/math], the break condition is fulfilled, so nothing is to show. So consider the case that the graph traversal does hit [math]\displaystyle{ t }[/math]. Then [math]\displaystyle{ p }[/math] is an [math]\displaystyle{ (s,t) }[/math]-path. By definition of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], the capacity constraints are preserved. To see that the flow conservation conditions are preserved as well, only the internal nodes [math]\displaystyle{ p }[/math] of are relevant. Let [math]\displaystyle{ v }[/math] be such an internal node, and let [math]\displaystyle{ u }[/math] and [math]\displaystyle{ w }[/math] denote the immediate predecessor and successor of [math]\displaystyle{ v }[/math] on [math]\displaystyle{ p }[/math], respectively. Basically, there are four cases:

• Either [math]\displaystyle{ (u,v) }[/math] is on [math]\displaystyle{ p }[/math] as a forward arc or [math]\displaystyle{ (v,u) }[/math] is on [math]\displaystyle{ p }[/math] as a backward arc.
• Either [math]\displaystyle{ (v,w) }[/math] is on [math]\displaystyle{ p }[/math] as a forward arc or [math]\displaystyle{ (w,v) }[/math] is on [math]\displaystyle{ p }[/math] as a backward arc.

It is easy to check preservation of the flow conservation conditions for each of these four cases.

Pseudocode

B-TREE-FIND(x,k)
1 i = 1
2 while i ≤ x.n and k > x.keyi 
3        i = i + 1
4 if i ≤ x.n and k == x.keyi 
5        return (x.i)
6 elseif  x.leaf
7        return NIL
8 else DISK-READ(x.ci) 
9        return B-TREE-FIND(x.ci,k)

Complexity

Statement: The asymptotic complexity is in [math]\displaystyle{ \Theta(\log n) }[/math] in the worst case.

Proof: Follows immediately from the fact that the height of B-tree with n nodes is in [math]\displaystyle{ \Theta(\log n) }[/math] (cf. the remark clause of the B-Trees page).