ON THE IMPLEMENTATION OF GOLDBERG'S MAXIMUM FLOW ALGORITHM IN EXTENDED MIXED NETWORK

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 6, December 2017
DOI:10.5121/ijcsit.2017.9609 93
ON THE IMPLEMENTATION OF GOLDBERG'S
MAXIMUM FLOW ALGORITHM IN EXTENDED
MIXED NETWORK
Nguyen Dinh Lau1
, Tran Quoc Chien1
, Phan Phu Cuong2
and Le Hong Dung3
1
University of Danang, Danang, Vietnam
2
Vinaphone of Danang, Vietnam
3
College of Transport II, Ministry of Transport, Vietnam
ABSTRACT
In this paper, we solve this problem of finding maximum flow in extended mixed network by Revised
preflow-push methods of Goldberg This algorithm completely different algorithm postflow-pull in [15].
However, we share some common theory with [15].
KEYWORDS
Algorithm, maximum flow, extended mixed network, preflow, excess.
1. INTRODUCTION
In real life, we do not always have the freedom of choice that this idealized scenario suggests,
because not all pair of reduction relationships between these problems have been proved, and
because few optimal algorithms for solving any of the problem has yet been invente, and perhaps
no efficient reduction that directly relates a given pair of problems has yet been devised.
In this paper, We consider another approach to solving the maximum flow problem in network
mixed network. Using a generic method known as preflow-push method, we incrementally move
flow along the outgoing edges of vertices that have more inflow than outflow. The preflow-push
approach was developed by A. Goldberg and R.E. Tarjan in 1986 [4] on basis of various earlier
algorithms. It is widely used because of its simplicity, flexibility.
As we did in augmenting-path algorithms, we use the residual network to keep track of the edges
that we might push flow through. Every edge in the residual network represents a potential place
to push flow. If a residual network edge is in the same direction as the corresponding edge in the
flow network, we increase the flow; if it is in the opposite direction, we decrease the flow. If the
increase fills the edge or the decrease empties the edge, the corresponding edge disappears from
the residual network. For preflow-push algorithms, we use an additional mechanism to help
decide which of the edges in the residual network can help us to eliminate active vertices.
The problem of finding maximum flow in network mixed network is extremely interesting and
practically applicable in many fields in our daily life, especially in transportation. The paper
develops a model of extended mixed network that can be applied to modelling many practical
problems more exactly and effectively.

94
Given a graph network G (V, E) with a set of vertices V and a set of edges E, where edges can be
directed or undirected, with edge capacity ce:E→R*, so that ce(e) is adge capacity e ∈ E and
vertices capacity cv:V→R*, so that cv(u) is vertices capacity u ∈ V.
With edge cost be be: E→R*, be(e): cost must be return to transfer an unit transport on edge e
With each v∈V, Set Ev are set edge of vertice v.
Vertice cost bv:V×Ev×Ev→R*, bv(u,e,e’): cost must be return to transfer an unit transport from
edge e to vertice u to edge e’.
A set (V, E, ce, cv, be, bv) is called extended mixed network.
2. FLOW EDGE ON EXTENDED MIXED NETWORK
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template for papers prepared using Microsoft Word. Papers not conforming to these requirements
may not be published in the conference proceedings.
Given an extended mixed network G = (V, E, ce, cv, be, bv). where s is source vertex, t is sink
vertex. A set of flows on the edges f = {f(x,y) | (x,y)∈E} is called flow edge on extended mixed
network. So that:
(i) 0 ≤f(x,y) ≤ce(x,y) ∀(x,y)∈E (1)
(ii) For any vertex k is not a source or sink
( ) ( )∑∑ ∈∈
=
EvkEkv
vkfkvf
),(),(
,, (2)
(iii) For any vertex k is not a source or sink
( ) )(,
),(
kcvkvf
Ekv
≤∑∈
(3)
The maximum problem:
Given an extended mixed network G(V, E, ce, cv, be, bv), where s is source vertex, t is sink
vertex. The task required by the problem is finding the flow which has a maximum value. The
flow value is limited by the total amount of the circulation possibility on the roads starting from
source vertex. As a result of this, there could be a confirmation on the following theorem.
3. PREFLOW-PUSH METHODS
3.1. Some basic concept
◊ Residual extended network Gf:
For flow f on G = (V, E, ce, cv, be, bv), where s is source vertex, t is sink vertex. Residual
extended network, denoted Gf is defined as the extended network with a set of vertices V and a
set of edge Ef with the edge capacity is cef and vertices capacity is cvf as follows:
- For any edge (u, v) ∈ E, if f(u, v)> 0, then (v, u) ∈Ef with edge capacity is cef (v,u)=f(u, v)

95
- For any edge (u,v) ∈ E, if c(u,v) -f(u, v)> 0, then (u, v) ∈Ef with edge capacity is cef(u,v) =
ce(u,v) - f(u,v)
- For any vertices v∈ V then
cvf(v)= cv(v)− ( )∑∈Evx
vxf
),(
, (4)
◊ preflow:
For extended mixed network G = (V, E, ce, cv, be, bv). Preflow is a set of flows on the edges f =
{f(x, y) | (x, y)∈ G} So that
(i) 0 ≤ f(x, y) ≤ ce(x, y) ∀(x, y) ∈ E
(ii) for any vertex k is not a source or sink, inflow is not smaller than outflow, that is
( ) ( )∑∑ ∈∈
≥
EvkEkv
vkfkvf
),(),(
,, (5)
(iii) for any vertex k is not a source or sink
( ) )(,
),(
kcvkvf
Ekv
≤∑∈
(6)
Each vertex whose inflow is larger than its outflow is called the unbalanced vertex. The
difference between a vertex’s inflow and outflow is called excess. The concept of residual
extended network Gf is similarly defined as flow.
The idea of this methods is balancing inflow and outflow at the balanced vertices by pushing
along an outgoing edge and pushing against an incoming edge. Process of balancing is repeated
until no more the unbalanced vertex then we get maximum flow. We store the unbalanced
vertices on a generalized queue. A tool called a height function is used to help select the edge
available in residual network to eliminate the unbalanced vertices. Now we assume that a set of
the network is denoted as V={0,1,...,|V|-1}.
◊ height function of the pre-flow in the extended mixed network G = (V, E, ce, cv, be, bv), is a set
of non-negative vertex weights h(0), ..., h(|V| −1) such that h(z) = 0 for z and h(u) ≤ h(v) + 1 for
every edge (u,v) in the residual extended network for the flow. An eligible edge is an edge (u,v)
in the residual extended network with h(u)=h(v) + 1.
A trivial height function is h(0) = h(1) = ... = h(|V| − 1) = 0. Then if we set h(a)= 1, any positive
edge to a is the priority edge.
We define a more interesting height function by assigning to each vertex the latter’s shortest –
path distance to the sink (its distance to the root in any BFS tree of the network rooted at z. This
height function is valid because h(z)= 0, and for any pair of vertices u and v connected by an edge
(u,v) in residual network Gf, then h(u) ≤ h(v) + 1, because the path from u to z with edge (u,v)
(h(v)+1 must be not shorter than the shortest path from u to z is h(u)).
3.2. Preflow-push algorithm
This is a particular algorithm in Preflow-push method. Here the unbalanced vertices are spushed
into the queue. With each vertex from the queue, we will push the flow in the priority edge until

96
the flow becomes either balanced or does not have any priority edge. If it does not exist priority
edge but there are unbalanced vertices, then we increase the height and push it into the queue.
This algorithm inplements the genetic vertex- based preflow-push maxflow algorithm, using a
generalized queue that disallows duplicates for active nodes.
The shortest-paths function is used to initialize vertex heights in the array h, to shortest- paths
distances from the sink. The array contains each vertex’s excess flow and therefore implicitly
defines the set of active vertices. By convention, s is initially active but never goes back on the
queue and t is nener active.
The main loop choose an active vertex v, then pushes flow through each of its eligible edges
(adding vertices that receive the flow to the active list, if necessary), until either v become
inactive or all its edges have been considered. In the latter case, v’s height is incremented and it
goes back into the queue.
Initially, the only preflow is in the edges for the source vertices is the following:
f(s,v)=min{ce(s,v),cv(v)} and other flows are 0. Then the first vertices of those edges directed to
the source are unbalanced. With any unbalanced vertex v, we have (v, s) ∈Ef and (s,v) ∉Ef,
inferred there exists paths from v to a, and there are no directed paths from source vertex a to sink
vertex in the residual network Gf. So the properity is true with the initial flow.
Now we can describe the Preflow-push algorithm as follows:
--------------------------------------------------------------------
Input: Extended mixed network G(V, E, ce, cv, be, bv). with source s, sink t.
Output: Maximum flow
{
Initialize:
=0
, f(s,v)= min{ce(s,v), cv(v)}
Select any available heigth function h in the extended mixed network G.
Q u| u .
While Q do
{
Get u from Q.
If (u, v) ∈ Ef is priority edge then
{
Unbalanced vertex u:
If f(v,u)>0 then push along the edge (u,v) a flow with value min{excess(u),cef(u,v)}).
If (u,v)∈E and cvf(v)>0) then push along the edge (u,v) a flow with value
min{excess(u), cef(u,v), cvf(v)}
If excess(v) then Q=Q .
}
Else
{
h(u)= 1 + min {h (v) | (u, v) ∈ Ef}
v Q
}
}
( ) ( ) EfF ij ∈= ji,,

97
Prinf f is maximum flow
}
--------------------------------------------------------------------
There are many optionsto explore in developing preflow-push implementations. We have already
discussed three major choices:
- Edge-based versus vertex-based generic algorithm
- Generalized queue implementation
- Initial assignment of heights
If a vertex’s height is greater then |V|, then there is no path from that vertex to the sink in the
residual extended network Gf.
Theorem 3.1. The complexity of the preflow-push method is O (|V|2
| E|) [5].
Proof: We bound the number of phases using a potential function. This argument is a simple
example of a powerful technique in the analysis of algorithm and data structures.
Define the quantity Q to be 0 if there are no active vertices and to be the maximun height of the
active vertices otherwise, then consider the effect of each phase on the value of Q. Let h0(s) be the
initial height of the source. At the beginning, Q= h0(s); at the end, Q=0.
First, we note that the number of phases where the height of some vertex increases is no more
than 2|V|2
-h0(s), since each of the V vertex heights can be increased to a value of at most 2|V|, by
the corollary: “Vertex’s height is always less than 2.|V|”.
While Preflow-push algorithm is in execution, there always exists a directed path from sink
vertex to the unbalanced vertex in the residual extended network, and there are no directed paths
from source vertex to sink vertex in the residual extended network.
We need to consider only unbalanced vertices, the height of each unbalanced vertex is either the
same as or 1 greater than it was the last time that the vertex was balanced. By the same argument,
the path from a source vertex to a given unbalanced vertex in the residual extended network Gf
implies that unbalanced vertex’s height is not greater than the source vertex’s height plus |V| -2
(the sink vertex can not be on the path). Since the height of the source never changes, and it is
initially not greater than |V|, the given unbalanced vertex’s height is not greater than 2.|V| - 2, and
no vertex has height 2|V| or greater.
Since Q can increase only if the height of some vertex increases, the number of phases where Q
increases is no more than 2|V|2
- h0(s).
If, however, no vertex’s height is incremented during a phase, the Q must decrease by at least 1,
sine the effect of the phase was to push all excess flow from each active vertex to vertices that
have smaller height.
Together, these facts imply that the number of phases must be less than 4|V|2
: The value of Q is h-
0(s) at the beginning and can be incremented at most 2|V|2
- h0(s) times and therefore can be
decremented at most 2|V|2
times. The worst case for each phase is that all vertices are on the
queue and all of their edges are examined, leading to the stated bound on the total running time.
This bound is tight. The number of phases used by the preflow-push algorithm is proportional to
|V|2
.

98
Example: Finding the max flow for the following extended mixed network G (Figure 1). The
order of the vertices is 1, 2, 3, 4, 5, 6.
Table 1. Edge capacity
Table 2. Vertex capacity
Table 3. Height function
The steps of the algorithm described in the following figure:
Figure 1. Network G
Figure 2. Initialze flow
Queue Q: > 3 | 2 >. Get unbalanced vertex 2 from the queue (Balanced 2):
Push along the edge (2, 5) a flow with value 7, We have preflow as in figure 3.
Edge ce
(1,2) 10
(1,3) 9
(2,3) 5
(2,5) 7
(3,4) 7
(3,5) 6
(4,6) 10
(4,5) 5
(5,6) 9
Vertex 1 2 3 4 5 6
Cv ∞ 10 9 10 9 ∞
Edge 1 2 3 4 5 6
H 3 2 2 1 1 0

99
Table 4. Height function
Figure 3. Preflow
We increase the heigth of the vertex 2 as follows: h(2) = 1 + 3 = 4. Height function change as in
table 4
Push along the edge (2, 1) a flow with value 3, We have preflow as in figure 4.
Figure 4. Preflow
Push unbalanced vertex 5 to Queue Q. Queue Q: >5 | 3 >
- Get unbalanced vertex 3 from the queue (Balanced 3):
Push along the priority edge (3, 4) a flow with value 7, We have preflow as in figure 5.
Figure 5. Preflow
Push along the priority edge (3, 5) a flow with value 2. We have preflow as in figure 6.
Figure 6. Preflow
Edge 1 2 3 4 5 6
h 3 4 2 1 1 0

100
Figure 7. Preflow
Push unbalanced vertex 4 to Queue Q.
Queue Q: > 4 | 5>
Queue Q = ∅ (no more the unbalanced vertex), ended and the final flow is the maximum
flow with value is 16
Figure 8. Preflow
4. BUILDING THE PARALLEL ALGORITHM
Inputs: Extended mixed network G with source s, sink t m processors (P0, P1,…, Pm-1), where P0
is the main processor.
Output: Maximum flow
Step 1: The main processor P0 performs
(1.1). initialize: e, h, f, cf, Q: set of unbalanced vertices (excluding the vertices s and t) are
the vertices with positive excess.
(1.2). divide set of vertices V into sub-processors:
Let Pi be the ith
sub-processor (i = 1,2, ..., m-1)
Pi will receive the set of vertices Vi so that
(1.3). The main processor sends h, e, cf to sub-processors
( ) ( ) EfF ij ∈= ji,,
{ } )andj,iif( i VVVV iji =∪≠=∩ φ

101
Step 2: The Condition to terminate: If Q = ∅, then postflow f becomes maximum flow, end. Else;
choose unbalanced vertex u from Q. If exists priority edge (u, v) ∈ Ef then go to step 3, else
go to step 4.
Step 3: m-1 sub-processors (P1, P2, …,Pm-1) implement
(3.1) Receive e, cf, h and the set of vertices from the main processor
(3.2) Handling unbalanced vertice v (push and replace label). Get unbalanced vertexs u
(e(u)>0) from Q and u∈Vi (i= 1,2, ..., m-1).
If f(v,u)>0, then push along the edge (u,v) a flow with value min{delta,cef(u,v)}(where delta
is the excess of the vertex u).
If (u,v)∈E and cvf(v)>0, then push along the edge (u,v) a flow with value min{delta, cef(u,v),
cvf(v)}
Step 4 : Increased the height of the vertex u as follows:
h(u): = 1 + min {h (v) | (u, v) ∈ Ef}
Step 5: Send e, cf, h to the main processor
Step 6: The main processor implements
(6.1) Receive e, cf, h from step 5
(6.2) This step is distinctive from the sequential algorithms to synchronize our data, after
receiving the data in (6.1), the main processor checks if all the edges ( ) Evu ∈, that have
h(u)> h(v)+1, the main processor will relabel for vertices u, v as follows:
- e(u):= e(u)+cef(u,v), e(v):= e(v)-cef(u,v)
- If f(v,u)>0, then f(u,v):= min{delta,cef(u,v)}
(where delta is the excess of the vertex u).
- If (u,v)∈E and cvf(v)>0,
then f(u,v):= min{delta, cef(u,v), cvf(v)}
Put the new unbalanced vertex into set Q
(6.3) If Vu ∈∀ e(u)=0, eliminate u from active set Q.
Back to step 2.
Theorem 4.1. Postflow-pull parallel algorithm is true and has complexity O(|V|2
|E|).
Proof: Preflow-push parallel algorithm is build in accordance with other parallel computing
system such as: PRAM, Cluster system, CUDA, RMI, threads,… Push and replace label using
atomic, due to support of atomic ‘read-modify-write’ instructions, are executed atomically by the
architecture. Other than the two execution characteristics provided by the architecture, we do not
impose any order in which executions from multiple sub-processors can or should be interleaved,
as it will be left for the sequential consistency property of the architecture to decide.
The outcome of the execution reduces to only a few simpliﬁed scenarios. By analyzing these
scenarios, we can show that function f is maintained as a valid height function. A valid d
guarantees that there does not exist any paths from s to t throughout the execution of the

102
algorithm, and hence guarantees the optimality of the ﬁnal solution if the algorithm terminates.
The termination of the algorithm is also guaranteed by the validatity of h, as it bounds the number
of push and relabel operations to O(|V |2
|E|).
5. CONCLUSIONS
The detail result of this paper is building sequential and parallel algorithm by preflow-push
methods to find maximum flow in extended mixed network. The results of this paper are basically
systematized and proven.
REFERENCES
[1] Chien Tran Quoc. Postflow-pull methods to find maximal flow. Journal of science and technology -
University of DaNang, 5(40), 31-38, 2010.
[2] L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 1962.
[3] J. Edmonds and R. M. Karp. Theoretical improvements in algorithmic efficiency for network flow
problems. J. ACM, vol. 19, no. 2, pp. 248–264, 1972.
[4] A. V. Goldberg and R. E. Tarjan. A new approach to the maximum ﬂow problem. Proceedings of the
eighteenth annual ACM symposium on Theory of computing. New York, NY, USA: ACM, pp. 136-
146, 1986.
[5] Robert Sedgewick. Algorithms in C Part 5: Graph Algorithms. Addison-Wesley, 2001.
[6] R. J. Anderson and a. C. S. Jo. On the parallel implementation of goldberg’s maximum flow
algorithm. Proceedings of the fourth annual ACM symposium on Parallel algorithms and
architectures. New York, NY, USA: ACM, pp. 168–177, 1992.
[7] D. Bader and V. Sachdeva. A cache-aware parallel implementation of the push-relabel network flow
algorithm and experimental evaluation of the gap relabeling heuristic. Proceedings of the 18th ISCA
International Conference on Parallel and Distributed Computing Systems, 2005.
[8] B. Hong. A lock-free multi-threaded algorithm for the maximum flow problem. IEEE International
Parallel and Distributed Processing Symposium, 2008.
[9] Zhengyu He, Bo Hong. Dynamically Tuned Push-Relabel Algorithm for the Maximum Flow Problem
on CPU-GPU-Hybrid Platforms. School of Electrical and Computer Engineering-Georgia Institute of
Technology, 2010.
[10] Chien Tran Quoc, Lau Nguyen Dinh, Trinh Nguyen Thi Tu. Sequential and Parallel Algorithm by
Postflow-Pull Methods to Find Maximum Flow. Proceedings 2013 13th International Conference on
Computational Science and Its Applications, pp 178-181, 2013.
[11] Lau Nguyen Dinh, Thanh Le Manh, Chien Tran Quoc. Sequential and Parallel Algorithm by Pre-Push
Methods to Find Maximum Flow. Vietnam Academy of Science and Technology AND Posts &
Telecommunications Institute of Technology, special issue works Electic, Tel, IT; 51(4A) pp 109-
125, 2013.
[12] Lau Nguyen Dinh, Chien Tran Quoc and Manh Le Thanh. Parallel algorithm to divide optimal linear
flow on extended traffic network. Research, Development and Application on Information &
Communication Technology, Ministry of Information & Communication of Vietnam, No 3, V-1,
2014.
[13] Naveen Garg, Jochen Könemann. Faster and Simpler Algorithms for Multicommodity Flow and
Other Fractional Packing Problems. SIAM J. Comput, Canada, 37(2), pp. 630-652, 2007.
[14] Lau Nguyen Dinh, Chien Tran Quoc, Thanh Le Manh.. Improved Computing Performance for
Algorithm Finding the Shortest Path in Extended Graph. proceedings of international conference on
foundations of computer science (FCS’14), pp 14-20, 2014.
[15] Nguyen Dinh Lau, Tran Quoc Chien, Sequential and parallel Algorithm to find maximum flow on
extended mixed networks by revised postflow-pull methods, Fourth International Conference on
Advanced Information Technologies and Applications (ICAITA 2015), ISSN: 2231–5403, ISBN:
978-1-921987-43-4, DOI: 10.5121/csit.2015.51501, 2015, pp. 19-28

ON THE IMPLEMENTATION OF GOLDBERG'S MAXIMUM FLOW ALGORITHM IN EXTENDED MIXED NETWORK

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