Preflow Push Algorithm
M. Amber Hassaan
Max Flow Problem



Given a graph with “Source” and “Sink” nodes we want to compute:
 The maximum rate at which fluid can flow from Source to Sink
 The rate of flow through each edge of the graph
Given a graph:
 Edges have flow capacities
 First value is flow capacity
 Second value is current flow rate
 Edges behave like pipes
 Nodes are junctions of pipes
History:
 Ford and Fulkerson 1956 (Augmenting Paths)
 Dinic 1970
 Edmonds and Karp 1972
a
 Malhotra, Kumar and Maheshwari 1978
 Goldberg and Tarjan 1986 (Preflow Push)
5, 2
 Boykov and Kolmogorov 2006
4, 4
6, 2
s
t
8, 7
Preflow Push Algorithm
Min-cut
5, 5
b
2
Max Flow Problem


Flow f(u,v) is a real-valued function defined for
every edge (u,v) in the graph
The flow needs to satisfy the following 3
properties:
≤ c(u,v) i.e. capacity of edge (u,v)
= - f(v,u)
 Flow coming into node v = Flow leaving v
 f(u,v)
 f(u,v)
a
5, 2
4, 4
6, 2
s
t
8, 7
Preflow Push Algorithm
5, 5
b
3
Max Flow Problem

Residual Graph:
 A Graph
that contains edges that can admit more flow
 Define a Graph G(V,E’) for G(V,E)
 We define residual flow r(u,v) = c(u,v) – f(u,v)
 If r(u,v) > 0 then (u,v) is in E’
Original Graph
a
5, 2
Residual Graph
a
2
4
4, 4
3
6, 2
s
t
4
s
t
2
1
8, 7
5, 5
b
5
7
Preflow Push Algorithm
b
4
Preflow Push Algorithm for Maxflow problem

Relaxation algorithm:




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


Performs local updates repeatedly until global constraint is
satisfied
Similar to Stencil computations
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
5
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
6
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
7
Example

a
A simple graph:



Height ‘h’
Excess flow ‘e’

3,0
s
First value is edge capacity
Second value is flow
Initialize:


e(a) = 10
t
8,0
h=6
17,0
12,0
c

s has h=6 i.e. number of
nodes
 Push 10 along (s,a)
b
5,0
Edges have pairs:

15,0
h=0
e=0
10,0
Nodes have two attributes:


h=0
e=0
h=0
e=0
6,0
h=0
e=0
d
h=0
e=0
Push 12 along (s,c)

e(c) = 12
Preflow Push Algorithm
8
Example
h=0
e=10
a

Relabel c with h=1
 Push



s
5 along (c,a)
Relabel c with h=7
t
8,0
h=6
17,0
12,12
c
e(a) = 5
 Push
3,0
5,0
Relabel c with h=2

b
10,10
6 along (c,d)
e(d) = 6
 Push
15,0
h=0
e=10
h=1
e=12
6,0
h=0
e=0
d
h=0
e=0
-1 along (s,c)
Preflow Push Algorithm
9
Example
h=1
e=15
a

Relabel a with h=1
 Push

e(b) = 15
b
10,10
3,0
15 along
(a,b)
15,0
h=0
e=0
5,5
s
t
8,0
h=6
17,0
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
h=0
e=0
d
h=0
e=5
10
Example
h=1
e=0
a

Relabel b with h=1
 Push



s
t
8,0
h=6
8 along (b,d)
17,0
12,11
Relabel b with h=2

3,0
5,5
e(d) = 14
 Push
b
10,10
3 along (b,t)
e(t) = 3
 Push
15,15
h=1
e=15
-4 along (a,b)
c
h=7
e=0
6,6
h=0
e=3
d
h=0
e=6
e(a) = 4
Preflow Push Algorithm
11
Example
h=1
e=4
a

Relabel d with h=1
 Push

b
10,10
3,3
14 along (d,t)
e(t) = 17
15,11
h=2
e=0
5,5
s
t
8,8
h=6
17,0
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
h=0
e=3
d
h=1
e=14
12
Example
h=3
e=4
a

Relabel a with h=3
 Push

b
10,10
3,3
4 along (a,b)
e(b) = 4
15,11
h=2
e=0
5,5
s
8,8
h=6
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
t
h=0
17,14 e=17
d
h=1
e=0
13
Example
h=3
e=0
a

Relabel b with h=4
 Push

b
10,10
3,3
-4 along (a,b)
e(a) = 4
15,15
h=4
e=4
5,5
s
8,8
h=6
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
t
h=0
17,14 e=17
d
h=1
e=0
14
Example
h=5
e=4
a

Relabel a with h=5
 Push

b
10,10
3,3
4 along (a,b)
e(b) = 4
15,11
h=4
e=0
5,5
s
8,8
h=6
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
t
h=0
17,14 e=17
d
h=1
e=0
15
Example
h=5
e=0
a

Relabel b with h=6
 Push

b
10,10
3,3
-4 along (a,b)
e(a) = 4
15,15
h=6
e=4
5,5
s
8,8
h=6
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
t
h=0
17,14 e=17
d
h=1
e=0
16
Example
h=7
e=4
a

Relabel a with h=7
 Push

b
10,10
3,3
-4 along (s,a)
e(a) = 0
15,11
h=6
e=0
5,5
s
8,8
h=6
12,11
c
h=7
e=0
Preflow Push Algorithm
6,6
t
h=0
17,14 e=17
d
h=1
e=0
17
Example
h=7
e=0
15,11
a


No active nodes
left
The algorithm
terminates
h=6
e=0
b
10,6
3,3
5,5
s
8,8
h=6
12,11
c
6,6
h=7
e=0
t
h=0
17,14 e=17
d
h=1
e=0
Min-cut
Preflow Push Algorithm
18
Pseudo code
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Worklist wl = initializePreflowPush();
while (!wl.isEmpty()) {
Node n = wl.remove();
n.relabel();
for (Node w in n.neighbors()) {
if (n can push flow to w) {
pushflow(n, w);
wl.add(w);
}
}
if (n has excess flow) {
wl.add(n);
}
}
Preflow Push Algorithm
19
Amorphous Data Parallelism in Preflow Push






Topology: graph
Operator: local computation (data driven)
Active nodes: nodes with excess flow
Neighborhoods: active nodes and their
neighbors
Ordering: unordered
Parallelism: Shoots in the beginning and then
drops down
 Dependent
on number of active nodes
Preflow Push Algorithm
20
Parallelism profile for Preflow Push
Input Graph: 512x512 Grid
Preflow Push Algorithm
21
Preflow Push: Global Relabeling

The height of the nodes directs the flow towards the sink:





It affects the flow of fluid globally
But is updated locally
Which causes the fluid to flow in arbitrary directions
Increases the total number of push/relabel operations
Global Relabeling:


After every N push/relabel operations we compute the height of
every node from the sink
The height is computed by a breadth-first scan on the Residual
Graph




starting at the sink
The height is incremented at each next level of BFS
It reduces the number of push/relabel operations significantly
N is determined heuristically
Preflow Push Algorithm
22
Preflow Push: Global Relabeling
h=2
e=0
h=1
e=0
4
a
4
b
11
3
6
5
s
t
8
h=6
14
11
1
h=0
e=17
3
c
h=2
e=0
6
d
h=1
e=0
Preflow Push Algorithm
23
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
Preflow Push Algorithm
24