Push Re Label –Maximum Flow
Preflow Push Algorithm 1

Preflow Push Algorithm 2
Preflow Push Algorithm for Maxflow problem
 Relaxation algorithm:
 Performs local updates repeatedly until global constraint is
satisfied
 Similar to Stencil computations
 Fluid flows from a higher point to a lower point
 In the beginning
 “Source” is the highest point
 “Sink” and all other nodes are at the lowest point
 Source sends maximum flow on its outgoing edges
 Sending flow to out-neighbors is called “Push” operation
 The height of Source’s neighbors is then increased so
that fluid can flow out
 Increasing height of a node is called “Relabel” operation

Preflow Push Algorithm 3
Preflow Push Algorithm for Maxflow problem
 A node is allowed (temporarily) to have
more flow coming into it than flow going
out
i.e. a node can have “excess flow”
But the edges must respect the capacity
condition
Source can have arbitrary amount of excess
flow
 A node is said to be “active” if it has
excess flow in it

Preflow Push Algorithm 4
Preflow Push Algorithm for Maxflow problem
 We increase the height of the “active” node with
a “Relabel” operation
 If h is the minimum height among neighbors that can
accept flow
 Then height is relabeled to (h+1)
 Then we “push” the excess flow to the neighbors
 That are lower than the active node and can admit
flow
 Thus make them active
 Algorithm terminates when there are no “active”
nodes left

Preflow Push Algorithm 5
Example
 A simple graph:
 Nodes have two attributes:
 Height ‘h’
 Excess flow ‘e’
 Edges have pairs:
 First value is edge capacity
 Second value is flow
 Initialize:
 s has h=6 i.e. number of
nodes
 Push 10 along (s,a)
 e(a) = 10
 Push 12 along (s,c)
 e(c) = 12
s
c
a b
t
d
h=6
h=0
e=0
h=0
e=0
h=0
e=0
h=0
e=0
h=0
e=0
10,0
12,0
5,0
8,0
15,0
6,0
17,0
3,0

Preflow Push Algorithm 6
Example
 Relabel c with h=1
 Push 6 along (c,d)
 e(d) = 6
 Relabel c with h=2
 Push 5 along (c,a)
 e(a) = 5
 Relabel c with h=7
 Push -1 along (s,c)
s
c
a b
t
d
h=6
h=0
e=10
h=0
e=10
h=0
e=0
h=1
e=12
h=0
e=0
10,10
12,12
5,0
8,0
15,0
6,0
17,0
3,0

Preflow Push Algorithm 7
Example
 Relabel a with h=1
 Push 15 along
(a,b)
 e(b) = 15
s
c
a b
t
d
h=6
h=1
e=15
h=0
e=0
h=0
e=0
h=7
e=0
h=0
e=5
10,10
12,11
5,5
8,0
15,0
6,6
17,0
3,0

Preflow Push Algorithm 8
Example
 Relabel b with h=1
 Push 3 along (b,t)
 e(t) = 3
 Push 8 along (b,d)
 e(d) = 14
 Relabel b with h=2
 Push -4 along (a,b)
 e(a) = 4
s
c
a b
t
d
h=6
h=1
e=0
h=1
e=15
h=0
e=3
h=7
e=0
h=0
e=6
10,10
12,11
5,5
8,0
15,15
6,6
17,0
3,0

Preflow Push Algorithm 9
Example
 Relabel d with h=1
 Push 14 along (d,t)
 e(t) = 17 s
c
a b
t
d
h=6
h=1
e=4
h=2
e=0
h=0
e=3
h=7
e=0
h=1
e=14
10,10
12,11
5,5
8,8
15,11
6,6
17,0
3,3

Preflow Push Algorithm 10
Example
 Relabel a with h=3
 Push 4 along (a,b)
 e(b) = 4 s
c
a b
t
d
h=6
h=3
e=4
h=2
e=0
h=0
e=17
h=7
e=0
h=1
e=0
10,10
12,11
5,5
8,8
15,11
6,6
17,14
3,3

Preflow Push Algorithm 11
Example
 Relabel b with h=4
 Push -4 along (a,b)
 e(a) = 4 s
c
a b
t
d
h=6
h=3
e=0
h=4
e=4
h=0
e=17
h=7
e=0
h=1
e=0
10,10
12,11
5,5
8,8
15,15
6,6
17,14
3,3

Preflow Push Algorithm 12
Example
 Relabel a with h=5
 Push 4 along (a,b)
 e(b) = 4 s
c
a b
t
d
h=6
h=5
e=4
h=4
e=0
h=0
e=17
h=7
e=0
h=1
e=0
10,10
12,11
5,5
8,8
15,11
6,6
17,14
3,3

Preflow Push Algorithm 13
Example
 Relabel b with h=6
 Push -4 along (a,b)
 e(a) = 4 s
c
a b
t
d
h=6
h=5
e=0
h=6
e=4
h=0
e=17
h=7
e=0
h=1
e=0
10,10
12,11
5,5
8,8
15,15
6,6
17,14
3,3

Preflow Push Algorithm 14
Example
 Relabel a with h=7
 Push -4 along (s,a)
 e(a) = 0 s
c
a b
t
d
h=6
h=7
e=4
h=6
e=0
h=0
e=17
h=7
e=0
h=1
e=0
10,10
12,11
5,5
8,8
15,11
6,6
17,14
3,3

Preflow Push Algorithm 15
Example
 No active nodes
left
 The algorithm
terminates
s
c
a b
t
d
h=6
h=7
e=0
h=6
e=0
h=0
e=17
h=7
e=0
h=1
e=0
10,6
12,11
5,5
8,8
15,11
6,6
17,14
3,3

Preflow Push Algorithm 16
Pseudo code
1. Worklist wl = initializePreflowPush();
2. while (!wl.isEmpty()) {
3. Node n = wl.remove();
4. n.relabel();
5. for (Node w in n.neighbors()) {
6. if (n can push flow to w) {
7. pushflow(n, w);
8. wl.add(w);
9. }
10. }
11. if (n has excess flow) {
12. wl.add(n);
13. }
14. }

Preflow Push Algorithm 17
Conclusion
 Preflow Push is an algorithm to find the
max-flow in a graph
 It exhibits amorphous data parallelism
Parallelism is dependent on the structure of
the graph