SEQUENTIAL AND PARALLEL ALGORITHM TO FIND MAXIMUM FLOW ON EXTENDED MIXED NETWORKS BY REVISED POSTFLOW-PULL METHODS

Jan Zizka et al. (Eds) : ICAITA, SAI, CDKP, Signal, NCO - 2015
pp. 29–41, 2015. © CS & IT-CSCP 2015 DOI : 10.5121/csit.2015.51504
SEQUENTIAL AND PARALLEL
ALGORITHM TO FIND MAXIMUM FLOW
ON EXTENDED MIXED NETWORKS BY
REVISED POSTFLOW-PULL METHODS
Lau Nguyen Dinh and Tran Quoc Chien
University of Da Nang, Danang City, Vietnam
launhi@gmail.com
dhsp@dng.vnn.vn
ABSTRACT
The problem of finding maximum flow in network graph is extremely interesting and
practically applicable in many fields in our daily life, especially in transportation. Therefore, a
lot of researchers have been studying this problem in various methods. Especially in 2013, we
has developed a new algorithm namely, postflow-pull algorithm to find the maximum flow on
traditional networks. In this paper, we revised postflow-push methods to solve this
problem of finding maximum flow on extended mixed network. In addition, to take more
advantage of multi-core architecture of the parallel computing system, we build this
parallel algorithm. This is a completely new method not being announced in the world. The
results of this paper are basically systematized and proven. The idea of this algorithm is using
multi processors to work in parallel by postflow_push algorithm. Among these processors,
there is one main processor managing data, sending data to the sub processors, receiving data
from the sub-processors. The sub-processors simultaneously execute their work and send their
data to the main processor until the job is finished, the main processor will show the results of
the problem.
KEYWORDS
Processor, alogrithm, maximum flow, extended mixed network, parallel.
1. INTRODUCTION
The maximum flow problem on the network is one of the optimization problems on graphs
that is widely applicable in practice as well as in combinatorial theory. The problem was
proposed and solved by two American mathematicians Ford and Fulkerson in the early
1950 [2] and more and more scientists are interested in research. Edmonds and Karp gave
method with complexity O(|V|.|E|2
) [3]. In 1986, A. Goldberg and R.E. Tarjan [4] have
developed pre-flow push method with complexity O(|V|2
.|E|) and a lot of paper
concerning parallel algorithm are written by many interested researchers [6], [7], [8], [9].
Especially in 2013 we has developed a new algorithm namely, postflow-pull algorithm to

30 Computer Science & Information Technology (CS & IT)
find the maximum flow on traditional networks [10]. The work of Naveen Garg and Jochen
Konemann in 2007 and the above works just concentrate on traditional traffic networks
without any specific steps and correct proof. In fact, it is necessary to build extended
mixed networks. 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. Therefore, a lot of researchers have been studying this problem in
various methods. In an ordinary graph the weights of edges and vertexes are considered
independently where the length of a path is the sum of weights of the edges and the
vertexes on this path. However, in many practical problems, weights at a vertex are not
the same for all paths passing this vertex, but depend on coming and leaving edges.
The paper develops a model of extended mixed network that can be applied to
modelling many practical problems more exactly and effectively. Currently, parallel
processing method is a promising and effective solution for the deadlock problems that
sequential method encounters such as: program execution time, processing speed, the
ability of memory storage, the advantage of multi-core architecture, large-scale data
processing. The main contribution of this paper is the revised postflow-push [10] algorithm
finding maximal flow on extended mixed network and we build parallel algorithms on multi
processors. This is a completely new approach aiming to take advantage of multi-core
architecture, to reduce computation time and to solve the problem with large-scale data
[10].
2. EXTENDED MIXED NETWORK
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. [12], [16].
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.
3. FLOW EDGE ON EXTENDED MIXED NETWORK
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
(ii) For any vertex k is not a source or sink
( )∑∈Ekv
kvf
),(
, = ( )∑∈Evk
vkf
),(
,
(iii) For any vertex k is not a source or sink

Computer Science & Information Technology (CS & IT) 31
( )∑∈Ekv
kvf
),(
, ≤cv(k)
•••• Theorem 3.1 Given f = {f(x,y) | (x,y)∈E} is flow edge on extended mixed network G, where s
is source vertex, t is sink vertex, that is
( )∑∈Evs
vsf
),(
, − ( )∑∈Esv
svf
),(
, = ( )∑∈Etv
tvf
),(
, − ( )∑∈Evt
vtf
),(
,
Namly total flow go from source vertex equal to total flow going to sink vertex
Proof. ∀ x,y∈V|∄ (x,y) ∈E, then assign f(x,y)= 0, where
( ) ( )∑∑∑∑ ∈ ∈∈ ∈
=
Vv VuVu Vv
uvfvuf ,, ⇔ ( ) ( )∑ ∑ ∑∈ ∈ ∈






−
Vv Vu Vu
uvfvuf ,, = 0
⇔ ( ) ( )
{ }
∑ ∑ ∑∈ ∈ ∈






−
tsVv Vu Vu
uvfvuf
,
,, + ( ( )∑∈Esu
suf
),(
, − ( )∑∈Eus
usf
),(
, ) + ( ( )∑∈Etu
tuf
),(
, −
( )∑∈Eut
utf
),(
, ) = 0
From (ii) in section 3, the first term equal to zero, so
( ( )∑∈Esu
suf
),(
, − ( )∑∈Eus
usf
),(
, ) +( ( )∑∈Etu
tuf
),(
, − ( )∑∈Eut
utf
),(
, ) = 0
⇔ ( )∑∈Eus
usf
),(
, − ( )∑∈Esu
suf
),(
, = ( )∑∈Etu
tuf
),(
, − ( )∑∈Eut
utf
),(
,
The value of flow:
val(f) = ( )∑∈Eus
usf
),(
, − ( )∑∈Esu
suf
),(
, is called value of flow f.
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.
•••• Theorem 3.2. Given an extended mixed network G(V, E, ce, cv, be, bv), where s is source
vertex, t is sink vertex , then exist is the maximal flow.

4. MAXIMUM FLOW AND THE MINIMUM CUT
Given an extended mixed network G(V, E, ce, cv, be, bv), where s is source vertex, t is sink
vertex. For any set S, T ⊂V, symbol (S, T) is a set of all edges reached and an unreached going
from S input T, (S,T) = {(x, y) ∈ E |x∈ S &y∈ T}.
If S, T ⊂ V| S∪T = V & S∩T = ∅ and s∈ S, t∈T, then (S, T) is called cut (source-sink) of G.
Given f = {f(x,y) | (x,y)∈E} is flow edge on extended mixed network G. Symbols
f(S,T) = ( )∑∈ ),(),(
,
TSyx
yxf
vertex, t is sink vertex.
Given f = {f(x,y) | (x,y)∈E} is flow edge on extended mixed network G and (S, T) is cut of G.
Where, val(f) = f(S,T)−f(T,S)
Proof. ∀ x,y∈V|∄ (x,y) ∈E, then assign f(x,y)= 0, we have
val(f) = ( )∑∈Eus
usf
),(
, − ( )∑∈Esu
suf
),(
, = ( )∑∈Vu
usf , − ( )∑∈Vu
suf , = ( )∑∑∈ ∈Sv Vu
uvf , − ( )∑∑∈ ∈Sv Vu
vuf ,
(as ∀v∈S{s}, ( )∑∈Vu
uvf , − ( )∑∈Vu
vuf , =0)
= ( )∑∑∈ ∈Sv Su
uvf , + ( )∑∑∈ ∈Sv Tu
uvf , − ( ( )∑∑∈ ∈Sv Su
vuf , + ( )∑∑∈ ∈Sv Tu
vuf , )
= ( )∑∑∈ ∈Sv Su
uvf , − ( )∑∑∈ ∈Sv Su
vuf , + ( )∑∑∈ ∈Sv Tu
uvf , − ( )∑∑∈ ∈Sv Tu
vuf ,
= ( )∑∑∈ ∈Sv Tu
uvf , − ( )∑∑∈ ∈Sv Tu
vuf , = f(S,T)−f(T,S).
Given (S,T) is cut. Symbol S(T) = {u∈S| ∃v∈T, (u,v)∈(S,T)}
vertex, t is sink vertex.
Given f = {f(x,y) | (x,y)∈E} is flow edge on extended mixed network G and (S, T) is cut of G.
Where, ∀S’⊂S(T) we have
f(S,T) ≤ ( )∑∈ 'Sv
V vc + ( )∑∈ )',(),(),(
,
TSTSyx
E yxc
Proof. we have
f(S,T) = ( )∑∈ ),(),(
,
TSyx
yxf = ( )∑∈ )',(),(
,
TSyx
yxf + ( )∑∈ )',(),(),(
,
TSTSyx
yxf = ( )∑ ∑∈ ∈' )},({),(
,
Sx Txyx
yxf + ( )∑∈ )',(),(),(
,
TSTSyx
yxf
≤ ( )∑∈ 'Sv
V vc + ( )∑∈ )',(),(),(
,
TSTSyx
E yxc

The capacity of slice cut
Given (S, T) is slice cut of G. Symbol cap(S, T) is capacity of (S, T) slice cut. We have
cap(S,T) = min{ ( )∑∈ 'Sv
vcv + ( )∑∈ )',(),(),(
,
TSTSyx
yxce |S’⊂S(T)}
From Theorem 4.1 and Theorem 4.2 infered that Theorem 4.3
•••• Theorem 4.3. Given f = {f(x,y) | (x,y)∈E} is flow edge on extended mixed network G and (S,
T) is cut of G. Where val(f) ≤cap(S,T).
5. POSTFLOW-PULL METHODS
5.1. Some basic concept
5.1.1. Residual extended mixed 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 mixed 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)
- 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
),(
, .
5.1.2. 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
( )∑∈Ekv
kvf
),(
, ≥ ( )∑∈Evk
vkf
),(
,
(iii) for any vertex k is not a source or sink
( )∑∈Ekv
kvf
),(
, ≤ cv(k)
5.1.3. Postflow
For extended mixed network G = (V, E, ce, cv, be, bv). Postflow 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, outflow is not smaller than inflow, that is
( )∑∈Ekv
kvf
),(
, ≤ ( )∑∈Evk
vkf
),(
,

(iii) for any vertex k is not a source or sink
( )∑∈Ekv
kvf
),(
, ≤ cv(k)
Each vertex whose outflow is larger than its inflow is called the unbalanced vertex. The
difference between a vertex’s inflow and outflow is called excess. The concept of residual
extended mixed 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 depth 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}.
5.1.4. Depth function
Depth function of the Postflow in the extended mixed network G = (V, E, ce, cv, be, bv), is a set
of non-negative vertex weights d(0), ..., d(|V| −1) such that d(s) = 0(s is source vertex) and d(u)+1
≥ d(v) for every edge (u,v) in the residual extended mixed network for the flow. An eligible
edge is an edge (u,v) in the residual extended mixed network with d(u)+1=d(v).
A trivial depth function is d(0) = d(1) = ... = d(|V| − 1) = 0. Then if we set d(u)= 1, any positive
edge to u is the priority edge.
We define a more interesting depth 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 s. This depth
function is valid because d(s)= 0, and for any pair of vertices u and v connected by an edge (u,v)
in residual mixed network Gf, then d(u)+1≥ d(v), because the path from a to v with edge (u,v)
(d(u)+1 must be not shorter than the shortest path from s to v i.e d(v)).
Property 5.1. For any flow f in extended mixed network G and associated depth function d. a
vertex’s depth d(v) is not larger than the length of the shortest path from vertex s to vertex v in
residual extended mixed network Gf .
Proof: For any given vertex v, assume l be the shortest-path length from s to v in the residual
extended mixed network Gf. And let (s=v1, v2, ..., vl=v) from s to v. then
d(v) = d(v1) ≤ d(vl-1) + 1
≤ d(vl-2) + 2
:
≤ d(v1) + l = d(s) + l = l (because d(s)=0)
The intuition behind depth function is the following: when an unbalanced node’s depth is less
then the depth of the sink, it is possible that there is some way to push flow from that node down
to the source; else, if an unbalanced node’s depth exceeds the depth of the sink, we know that
node’s flow needs to be pushed back to the sink.
Corollary: if a vertex’s depth is greater then |V|, then there is no path from the source to that
vertex in the residual extended mixed network Gf.

5.2. General Postflow-pull methods
General Postflow-push methods is briefly described as follows:
Step1:
Initialize: the only Postflow is in the edges leaving for the sink vertices is the following:
f(v,t)=min{ce(v,t), cv(v)}
The other flows are 0.
Select any available depth function d in the extended mixed network G.
Step 2:
Condition to terminate : If there are no available unbalanced vertices, then postflow f
becomes max flow.
Step 3: (pull flow)
Choose unbalanced vertex v.
If exists priority edge (u, v) ∈Ef then
If f(v,u)>0, then pull along the edge (u,v) a flow with value min{-delta,cef(u,v)}(where
delta<0 is the excess of the vertex v).
If (u,v)∈E and cvf(u)>0, then pull along the edge (u,v) a flow with value min{-delta, cef(u,v),
cvf(u)}(where delta<0 is the excess of the vertex v).
If it does not exists the priority edge from v, then increased the depth of the vertex v as
follows:
d(v): = 1 + min {d(u) | (u, v) ∈ Ef}
Back to step 2.
◊ Note. In the general Postflow-pull methods, we do not give the detailed steps how to select the
initial depth function, how to choose the unbalanced vertices as well as how to choose the priority
edges. Performing these detailed steps for many algorithms belongs to the general Postflow-pull
methods.
Property 5.2. Postflow-pull methods always preserve the validity of the depth function.
Proof:
(i) Where it exists priority edge (u,v) ∈ Ef: We have d(u)+1 = d (v). After pulling along edge
(u,v) a flow, we still have d(v) +1= d(u) +2≥d(u).
(ii) if it does not exist priority edges to v: we have∀u: (u,v) ∈ Ef⇒ d(u)+1 >d(v). After
incrementing d(v):d(v):=1+min{d(u)|(u,v)∈Ef} then d(v) still satisfied ∀u, (u,v) ∈Ef : d(u)+1 ≥
d(v).
Property 5.3. While Postflow-pull algorithm is in execution, there always exists a directed path
from sink vertex to the unbalanced vertex in the residual extended mixed network, and there are
no directed paths from source vertex to sink vertex in the residual extended mixed network.

Proof. (by induction)
Initially, the only Postflow is in the edges leaving for the sink vertices is the following:
f(v,t)=min{ce(v,t),cv(v)} and other flows are 0. Then the first vertices of those edges directed to
the sink are unbalanced. With any unbalanced vertex u, we have (t, u) ∈Ef and (u,t) ∉Ef, inferred
there exists paths from t to u, and there are no directed paths from source vertex a to sink vertex
in the residual extended mixed network Gf. So the properity is true with the initial flow.
Next, the new unbalanced vertex u only appears when a flow is pushed to the old unbalanced
vertex v on the priority edge (u,v). Then the residual extended mixed network will have more
edge (v,u). Due to exist of the path from residual extended mixed network from t to v based on
inductive hypothesis, there exists a path from t to u in the residual extended network.
To prove that there are no paths from source vertex s to sink t in the residual extended mixed
network. It can be argued as follows.
First, vertices u adcajent to sink vertices t, (u, t) ∈ E, since the initial flow on the edge (u,t) is
f(u,t)=min{ce(u,t),cv(u)}, if (u,t) ∈Gf, then the flow pushed back along t to u. Where (t, u) is the
priority edge, d(t)+1 = d(u) > t(t). Thus each vertex a can reach to t in the residual extended
mixed network must have the depth which is greater than the depth of t.
For any u to t in the residual extended mixed network. There exists paths from u to t in the
residual extended mixed network: (u→u1→u2→ ... uk→t). Similarly argued as above we have
d(u) > d(u1) > ... > d(uk-1) > d(uk) > d(t)
Thus each vertex to t must have a depth which is greater than t. Besides, the depth of the source
vertex is 0, so it's impossible to reach to t. So there are no directed paths from source vertex to
sink vertex in the residual extended network.
• Corollary. Vertex’s depth is always less than 2.|V|.
Proof. We need to consider only unbalanced vertices, the depth 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 as in the proof of Property 5.1, the path from s source vertex to a given unbalanced
vertex in the residual extended mixed network Gf implies that unbalanced vertex’s depth is not
greater than the sink vertex’s depth plus |V| -2 (the source vertex can not be on the path). Since
the depth of the sink never changes, and it is initially not greater than |V|, the given unbalanced
vertex’s depth is not greater than 2.|V| - 2, and no vertex has depth 2|V| or greater.
• Theorem 5.4 General Postflow-pull methods is true.
Proof. First we prove the general Postflow-pull method that terminates after perforing some
finite steps. We confirm that after implementing these finite steps there is not any unbalanced
vertex. Proof by contradiction method is used. Assume that the set of vertices are infinite, there
will exist vertex u that appears infinite times in that set. Since the number of vertices in the
network is finite so there exists vertex v≠ u so that the flow is pulled on along (u,v) and (v,u) in
infinite times. Since edge (u,v) and edge(v,u) are the priority ones in infinite residual network
and d(u)+1= d(v) and d(v)+1=d(u), then the depth of u and v will increment indefinitely, and this
conflicts with the above corollary.
When this method terminates, we receive the flow. Based on property 5.3, it does not exist a path
from the source to the sink in the residual network. According to augmenting-path algorithm, it is
max flow.

The complexity of the following method is O(|V|2
|E|) [10].
5.3. Postflow-pull algorithm
This is a particular algorithm in Postflow-pull method. Here the unbalanced vertices are pushed
into the queue. With each vertex from the queue, we will pull the flow in the priority edge until
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 depth and push it into the queue.
Now we can describe the Postflow-pull algorithm as follows:
Inputs: Extended mixed network G with source s, sink t,
Output: Maximum flow
Step 1: Initialized:
Initialize: the only postflow is in the edges for the source vertices is the following:
f(v,t)=min{ce(v,t), cv(v)}
The other flows are 0.
Choose depth function d(v) which is the length of the shortest path from source s to
vertex v.
Push all unbalanced vertices into the queue Q.
Step 2: Condition to terminate: If Q = ∅, then postflow f becomes maximum flow, end.
Step 3:
Get unbalanced vertex v from the queue Q.
Browsing the priority edge (u, v) ∈Ef
- If f(v,u)>0, then pull along the edge (u,v) a flow with value min{-delta,cef(u,v)}(where
- If (u,v)∈E and cvf(u)>0, then pull along the edge (u,v) a flow with value min{-
delta,cef(u,v),cvf(u)} (where delta<0 is the excess of the vertex v).
- If vertex u is the new unbalanced vertex, then push this vertex u into queue Q.
- If vertex v is still unbalanced, then increased the depth of the vertex v as follows:
d(v): = 1 + min {d(u) | (u, v) ∈ Ef}
Back to step 2.
6. POSTFLOW-PULL PARALLEL ALGORITHM TO FIND THE MAXIMUM
FLOW
6.1. The idea of the algorithm
Based on the parallel algorithm [10], we build parallel algorithms on m processors. In m
processors, there will be a main processor to manage data, divide the set of vertex V of the graph
into m-1 sub-processors, and send data to the sub-processors as well as receive data from the sub-
processors sending to [6],[7],[8],9], [10].
( ) ( ) EfF ij ∈= ji,,

Sub-processors receive the values from the main processor, then proceed to pull and replace label
(pull_relabel) and transfer the results to the main processor.
The main processor after receiving the results from the sub-processors will perform replacement
label (Relabel) until finding the maximum flow
6.2. 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, d, 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 e, cf to sub-processors
Step 2: The Condition to terminate: If Q = ∅, then postflow f becomes maximum flow, end.
Else, go to step 3.
Step 3: The main processor sends d to sub-processors
Step 4: m-1 sub-processors (P1, P2, …,Pm-1) implement
(4.1) Receive e, cf, d and the set of vertices from the main processor
(4.2) Handling unbalanced vertice v (pull and replace label). Get unbalanced vertexs v
from Q and v∈Vi (i= 1,2, ..., m-1). Browsing the priority edge (u, v) ∈Ef
- If f(v,u)>0, then pull along the edge (u,v) a flow with value min{-delta,cef(u,v)}(where
- If (u,v)∈E and cvf(u)>0, then pull along the edge (u,v) a flow with value min{-delta,
cef(u,v), cvf(u)}. (where delta<0 is the excess of the vertex v).
If vertex v is still unbalanced, then increased the depth of the vertex v as follows:
d(v): = 1 + min {d (u) | (u, v) ∈ Ef}
( 4.3) Send e, cf, d to the main processor
Step 5: The main processor implements
(5.1) Receive e, cf, d from step 4.3
( ) ( ) EfF ij ∈= ji,,
{ } )andj,iif( i VVVV iji =∪≠=∩ φ

(5.2) This step is distinctive from the sequential algorithms to synchronize our data, after
receiving the data in (5.1), the main processor checks if all the edges ( ) Evu ∈, that have
d(v)> d(u)+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<0 is the excess of the vertex
v).
- If (u,v)∈E and cvf(u)>0, then f(u,v):= min{-delta, cef(u,v), cvf(u)} (where delta<0 is the
excess of the vertex v). Put the new unbalanced vertex into set Q
(5.3) If Vu ∈∀ e(u)=0, eliminate u from active set Q. Back to step 2.
Theorem 3.1. Postflow-pull parallel algorithm is true and has complexity O(|V|2
|E|).
Proof: Similar to [10]. postflow-pull 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 depth function. A valid d
guarantees that there does not exist any paths from s to t throughout the execution of the
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 d, as it bounds the number
of pull and relabel operations to O(|V |2
|E|). 
Parallel algorithm for finding maximum flow in the extended mixed network is built on m
processors. The program written in Java with database administration system MySQL We
experimentally sampled nodes as follows: The extended mixed graph corresponds to 18000 nodes
and 25000 edge. The simulation result is shown in figure 1. This result demonstrates that the
runtime of parallel algorithms is better than sequential algorithm.
Figure 1. Chart performs the speedup of extended Mixed graph having 18000 nodes and 25000 edge

7. CONCLUSION
The detail result of this paper is building sequential and parallel algorithm by postflow-pull
methods to find maximum flow in extended mixed network. In addition, to take more advantage
of multi-core architecture of the parallel computing system and reduce the computing time of this
algorithm, we build this algorithm on multiple processors. This is a completely new method not
being announced in Vietnam and in the world. The results of this paper are basically systematized
and proven.
REFERENCES
[1] Chien Tran Quoc, 2010 “Postflow-pull methods to find maximal flow”, Journal of science and
technology - University of DaNang, 5(40), pp 31-38.
[2] L. R. Ford and D. R. Fulkerson, 1962 Flows in Networks Princeton University Press.
[3] J. Edmonds and R. M. Karp, 1972 “Theoretical improvements in algorithmic efficiency for network
flow problems,” J. ACM, vol. 19, no. 2, pp. 248–264.
[4] A. V. Goldberg and R. E. Tarjan, 1986 "A new approach to the maximum ﬂow problem," in
STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computing.
New York, NY, USA: ACM, pp. 136-146.
[5] Robert Sedgewick, 2011 Algorithms in C, Part 5: Graph Algorithms. Addison-Wesley.
[6] R. J. Anderson and a. C. S. Jo, 1992 “On the parallel implementation of goldberg’s maximum flow
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and architectures. New York, NY, USA: ACM, pp. 168–177.
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network flow algorithm and experimental evaluation of the gap relabeling heuristic” in PDCS ’05:
Proceedings of the 18th ISCA International Conference on Parallel and Distributed Computing
Systems.
[8] B. Hong, 2008 “A lock-free multi-threaded algorithm for the maximum flow problem” in IEEE
International Parallel and Distributed Processing Symposium, Aprail.
[9] Zhengyu He, Bo Hong, 2010 “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.
[10] Chien Tran Quoc, Lau Nguyen Dinh, Trinh Nguyen Thi Tu, 2013 “Sequential and Parallel Algorithm
by Postflow-Pull Methods to Find Maximum Flow”, Proceedings 2013 13th International Conference
on Computational Science and Its Applications, ISBN:978-0-7695-5045-9/13 $26.00 © 2013 IEEE,
DOI 10.1109/ICCSA.2013.36, published by IEEE- CPS pp 178-181.
[11] Lau Nguyen Dinh, Thanh Le Manh, Chien Tran Quoc, 2013 “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)
ISSN: 0866 708X, pp 109-125.
[12] Lau Nguyen Dinh, Chien Tran Quoc and Manh Le Thanh, 2014 “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.
[13] Naveen Garg, Jochen Könemann, 2007 “Faster and Simpler Algorithms for Multicommodity Flow
and Other Fractional Packing Problems”, SIAM J. Comput, Canada, 37(2), pp. 630-652.
[14] Lau Nguyen Dinh, Chien Tran Quoc, Thanh Le Manh, 2014 “Improved Computing Performance for
Algorithm Finding the Shortest Path in Extended Graph”, proceedings of the 2014 international
conference on foundations of computer science (FCS’14), July 21-24, 2014 Las Vegas Nevada, USA,
Copyright © 2014 CSREA Press, ISBN: 1-60132-270-4, Printed in the United States of America, pp
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[15] Chien Tran Quoc, Thanh Le Manh, Lau Nguyen Dinh, 2013 “Sequential and parallel algorithm by
combined the push and pull methods to find maximum flow”, Proceeding of national Conference on

Fundamental and Applied Infromation Technology Research (FAIR), Hue, Vietnam, 20-
21/6/2013.ISBN: 978-604-913-165-3, pp 538-549.
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flowcostlimits on extended traffic network”, Proceeding national Conference XVI "Some selected
issues of Information Technology and Communications” Danang 14-15/11/2013, ISBN: 978-604-67-
0251-1, pp 314-321.
[17] Lau Nguyen Dinh, Tran Ngoc Viet, 2012 “Parallelizing algorithm finding the shortest paths of all
vertices on computer cluster system”, Proceedings national Conference XVth "Some selected issues
of Ìnormation Technology and Communications” Ha Noi, 03-04-2012, pp 403-409.
[18] Lau Nguyen Dinh, Tran Ngoc Viet, 2012 “A parallel algorithm finding the shortest paths of multiple
pairs of source and destination vertices in a graph”, Journal of science and technology - University of
DaNang 9 (58),2012, pp 30-34.
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from a vertex to all vertices”, Journal of science, University of Hue, 74B, 5, pp 81-92.
AUTHORS
1.Dr. LAU NGUYEN DINH
Born in 1978 in Dien Ban, Quang Nam, Vietnam. He graduated from Maths_IT
faculty of Hue university of science in 2000. He got master of science (IT) at Danang
university of technology and hold Ph.D Degree in 2015 at Danang university of
technology. His main major: Applicable mathematics in transport, parallel and
distributed process, discrete mathemetics, graph theory, grid Computing and
distributed programming.
2.Ass. Prof. DrSc. CHIEN TRAN QUOC
Born in 1953 in Dien Ban, Quang Nam, Vietnam. He graduated from Maths_IT
faculty. He got Ph.D Degree of maths in 1985 in Charles university of Prague, Czech
Republic and hold Doctor of Science in Charles university of Prague, Czech Republic
in 1991. He received the tittle of Ass. Pro in 1992. He work for university of Danang,
Vietnam. His main major: Maths and computing, applicable mathematics in transport,
maximum flow, parallel and distributed process, discrete mathemetics, graph theory,
grid Computing, distributed programming.

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SEQUENTIAL AND PARALLEL ALGORITHM TO FIND MAXIMUM FLOW ON EXTENDED MIXED NETWORKS BY REVISED POSTFLOW-PULL METHODS