int. j. high performance computing and networking, vol. 11...

Int. J. High Performance Computing and Networking, Vol. 11, No. 3, 2018 191

Copyright © 2018 Inderscience Enterprises Ltd.

Quick convergence algorithm of ACO based on convergence grads expectation

Zhongming Yang* College of Computer Engineering Technical, Guangdong Institute of Science and Technology, Zhuhai, Guangdong, China Email: [email protected] *Corresponding author

Yong Qin College of Computer Science, Dongguan University of Technology, Dongguan, Guangdong, China Email: [email protected]

Yunfu Jia Department of Information Technology, Zhuhai Technician College, Zhuhai, Guangdong, China Email: [email protected]

Abstract: While the ant colony optimisation (ACO) can find the optimal path through a network, there are too many iterations and the convergence speed is also very slow. This paper proposes the Q-ACO QoSR based on convergence expectation to meet the requirement of OoS routing for a real-time and highly efficient network. This algorithm defines index expectation function of link, and proposes convergence expectation and convergence grads. As for the multi-constraint QoS routing model, the algorithm controls the iteration and searches the optimal path that meets the QoS restriction condition under the condition of the faster convergence. This algorithm can find the optimal path by comparing the convergence grads in a faster and larger probability. This algorithm improves the ability of routing and convergence speed.

Keywords: ant colony optimisation; ACO; QoS; QoSR; CG expectation; Q-ACO; convergence speed; convergence expectation; convergence grads; multi-constrain QoS.

Reference to this paper should be made as follows: Yang, Z., Qin, Y. and Jia, Y. (2018) ‘Quick convergence algorithm of ACO based on convergence grads expectation’, Int. J. High Performance Computing and Networking, Vol. 11, No. 3, pp.191–198.

Biographical notes: Zhongming Yang is currently an Associate Professor in the College of Computer Engineering Technical, Guangdong Institute of Science and Technology. He obtained his Information Engineering degree from Guangdong University of Technology in 1999, and received his Master’s in Software Engineering from Huazhong University of Science Technology in 2008. His research interests include computer network, network security and intelligent algorithm.

Yong Qin is a Professor in the College of Computer Science, Dongguan University of Technology, China. His research interests include computer network and intelligent algorithm.

Yunfu Jia is a University Instructor in the Department of Information Technology, Zhuhai Technician College. He received his Master’s degree from Taiyuan University of Technology in 2009. His research interests include computer network.

This paper is a revised and expanded version of a paper entitled ‘Quick convergence algorithm of ACO based on convergence grads expectation’ presented at The 7th International Symposium on Intelligence Computation and Applications (ISICA 2015), Guangzhou, China, 21–22 November 2015.

192 Z. Yang et al.

1 Introduction

ACO is a heuristic searching algorithm method based on the simulation of foraging behaviour of ants. It is frequently used in the solution of combinatorial optimisation problems. ACO was firstly proposed by Italian scholar Colorni et al. (1991) in the early 1990s. Since this algorithm does not hinge on mathematical description of specific problems, it was successfully used to solve travelling salesman problem (TSP) (Dorigo et al., 1996), quadratic assignment problem (QAP) (Gambardella et al., 1999), graph colouring problem (GCP) (Costa and Hertz, 1997), job-shop scheduling problem (FSP) (Colorni et al., 1994), and other NP-complete problems.

As the traditional QoS routing algorithm mainly includes distance vector algorithm and routing state algorithm (Cui and Lin, 2004a), there are still some weaknesses or limitations. First, these algorithms only consider either the biggest bandwidth or smallest that is obtainable at the current service flows, whereas whether such a QoS service flow path leads to the barrages (Cui and Lin, 2004b). Secondly due to micro-level congestion (such as link and router), it is hard to detect and control the congestion from the internet (Ahmed and Saito, 2007). The other problem is that these algorithms cannot efficiently enhance the speed of convergence and calculation. These limitations need to be addressed in applying QoSR to solving problems in the future. If we set QoS parameter as the optimisation condition, multi-constrained-based QoS routing decision-making problem is a NP-complete problem (Cui et al., 2002). In consideration of positive feedback, self organisation, and distributive computation of the algorithm, many scholars apply the ACO to solving routing optimisation and load balance. Some progress has been made in both the research and the application of routing optimisation. In terms of load balance, Yuan Li’s work has yielded a better result compared with the former algorithms, solving the problem of loss rate, average delay, and load balance. It has also effectively solve the multi-constrain QoS routing optimisation problem. To address the limitations of network barrages and low network efficiency, Li and Ma (2005) proposed MS-ACO algorithm, which mainly solves the problem of pheromone delay. Regarding the routing optimisation, Wang and Zhang (2005) applied the updating rule of partial pheromone and whole pheromone to multi-path routing. The result shows that it could quickly find the routing and improve the convergence speed as well as the network efficiency. Following the line of thought in parallel ACO algorithm, Lian et al. (2008) expanded the multi-path routing protocol in the ad hoc network, which largely enhanced the transmission speed of data packets; moreover, the parallel ACO increased the routing forecast probability and convergence speed. Koyama et al. (2008) proposed a new QoS multicast routing protocol, which has better genetic operations than the conventional algorithm, resulting in a better performance. Some scholars (e.g., Qi et al., 2012; Wang and Li, 2014) obtained the optimal convergence speed and optimal result

through adjusting the expectation of the partial optimisation of pheromone density, pheromone evaporation factors, the pheromone updating rule and the state shifting rule. Sustersic and Hurson (2006) outlined an experimental framework to verify the potential of the proposed scheme as a viable coherence solution for QoS applications.

Since convergence analysis theory only informs us that it is possible (Dorigo and Blum, 2005) to find the final optimal result in the ACO algorithm, it is difficult to apply in the comparison and evaluation of the performance of actual algorithms. Only by analysing the convergence speed of ACO algorithm can we know the time spent in generating the optimal result by using ACO algorithm. However, there has been little research on this aspect. Dorigo regarded the problem of research on convergence speed of ACO as the first public problem in 2005, and proposed that scholars should try to analyse the convergence speed problem in some simple ACO algorithms to fill this research gap. Hao and Huang’s (2006) paper addressed this public problem for the first time. Nevertheless, the conclusion is limited to the solution based on linear function of binary system single ant ACO algorithm.

Huang and Hao (2007) proposed the qualitative analysis of the ACO algorithm convergence based on the mathematical modelling of the Markov process, which is at absorption state in the ant colony algorithm. This paper defined the convergence expectation and convergence grads, proposed the thought of Q-ACO (quick searching ant colony algorithm) and discussed the QoSR computation. As for the multi-constrain QoS routing model, the algorithm controls the iteration and searches the optimal path that meets the QoS restriction condition under the condition of the higher convergence speed. This quick searching algorithm can be applied to routing algorithm, as well as the ant colony algorithm optimisation in other fields, such as the solution of task scheduling and load balance problems (Zhang, 2013; Duan and Fu, 2014) in cloud computing. We can also form the expectation function, with the function decided by restriction conditions such as the execution speed, scheduling time and cost calculation in the process of task scheduling. We can generate the optimal result of quick convergence through controlling the iteration.

2 Explanation of symbols

G(V, E) a network

V a set of optional nodes in the network

E a set of edges in the network

P a set of nodes in the network

S network transmission source node

D network transmission aim node

i, j network transmission node

x network bandwidth

y network delay

Quick convergence algorithm of ACO based on convergence grads expectation 193

z network loss ratio

Bw(i, j) network transmission path bandwidth

D(i, j) network transmission path delay

Pl(i, j) network transmission path loss ratio

Cost(i, j) path cost from node i and node j

constants

fij the expectation function of path from node i to node j

dij path length between node i and node j

t iteration times

tijT the amount of pheromones that ants release at the

t time of iteration on the path from node i to node j

Tij(t) the amount of pheromones at the t time of iteration on the path from node i to node j

1

mk

ij

k

T the amount of newly added pheromones at the t

time of iteration on the path from node i to node j

evaporation coefficient of pheromones

k the ant on the path

( )kijP t the ratio of the ant k on the node i select the

node j

A a set of optional nodes which is jointed with node i

pheromones value of ACO

heuristic value of ACO

a convergence grads

Fi convergence grads function.

3 Description of ACO-based QoS routing problem

Definition 1 (quality of service): QoS is the collection of service flow’s need for network service in the network’s transmitting service flow. Also service flow is the specified QoS-related packet flow from the source to the aim.

Definition 2 (quality of service routing): QoS routing allows the network to determine a path that supports the QoS needs of one or more flows in the network. Otherwise, QoSR can be regarded as a dynamic routing protocol of which the selection of path may include optional bandwidth, delay, jump times, loss ratio, shake, and other QoS parameters. QoSR seeks a feasible path for QoS traffic with two objectives:

a providing the QoS guarantee for QoS traffic

b maximising the utilisation of the entire network.

3.1 QoS routing constraint mode

Based on the characteristics of the path and the constraint conditions of the request, the network is regarded as a weighted undirected connected graph. We suppose G<V, E> as network, V as a set of nodes in the network, E as a set of edges in the network, and E = {(i, j)|i, j P} with i and j referring to node and P referring to a set of nodes.

Here we suppose that the path with a length of n from source node S to aim node D as Path(i1, …., in). The general expressions of QoS’s index which is correspondent with path Path are as follows. x, y and z mean bandwidth, delay and loss ratio, respectively. Bw(i, j), D(i, j), Pl(i, j) mean path bandwidth, path delay and path loss ratio, respectively. Particularly, i, j mean nodes, k means constant.

Definition 3 (path bandwidth): It is the amount of data that can be transmitted in a fixed amount of time from node i to node j on the path of network, or the capability that the data can be transmitted in the transmission pipelines.

2,3,...,( , ) min ( , )

j nBw i j Bw i j (1)

Here the formula of Bw(i, j) means the bandwidth index of path(i, j).

Definition 4 (path delay): It means the iterative time of that from data packet’s first byte’ inputting into routing to the last byte’ outputting of routing in the path transmission from node i to node j.

2 1

( , ) ( , ) ( )n n

kj k

D i j D i j D i (2)

Here the formula of D(i, j) and D(i) means the delay indexes of path(i, j) and node i respectively.

Definition 5 (path loss ratio): It means the ratio of the lost data packet to the data packet sending in the path test from node i to node j.

1

( , ) 1 (1 ( , ))n

j

Pl i j Pl i j (3)

Here PL(i) means the loss ratio index of node i. QoS routing is to search for the path that meets the constraint conditions as follows.

( ), ( ), ( ).x Bw i y D i z PL i

As for the multiple constraint routing problem, the value of indexes will show a great difference accounting for the inconsistent index dimension. Given that different service have different requests for service quality and the importance of individual index differs from one another, we should make an optimisation of all the indexes. The mode of multiple QoS index constraint routing is presented below.

194 Z. Yang et al.

min( , ) or max( )( )

s.t. ( )( )

y z xBw i xD i yPL i z

(4)

3.2 The ACO-based QoS routing optimisation algorithm

3.2.1 Path cost

Definition 6: Path cost is the relation function of bandwidth, delay and loss ratio that data are transmitted for a length of i. Cost(i, j) is the path cost from node i to node j.

The path cost can be expressed as follows.

( , )( , )

( , ) * ( , )Bw i j

Cost i jD i j Pl i j

(5)

is a constant.

3.2.2 Pheromone update

Definition 7: Pheromone update is the variable relationship of the presence of increase and the disappearance between the amount of path pheromone and ants over time. The shorter the path, the more ants that going by. The quicker the pheromone trace increases, the faster the evaporation and disappearance go as the time elapses.

4 CG mode

4.1 Expectation function

Definition 8: The function which is decided by the delay, bandwidth and other iterative conditions is the expectation function. fij is the expectation function on the path from node i to node j. is a constant. dij is the length of path that is from node i to node j.

* os ( , )ij

ij

C t i jf

d (6)

4.2 The probability of selecting path

Pheromone on the path is one of determinative factors that the ant selects the path. Pheromone is decided by the path cost. So the more the iterative times are, the stronger the pheromone in the path with low costs becomes. Tij(t) is the pheromone at t time iterative from node i to node j. t

ijT is the pheromone that the ant releases at t time iterative from node i to node j. Then the pheromone update formula is shown below:

( ) (1 / )ij ij ij ijT t T f T (7)

1

mk

ij ij

k

T T (8)

( ) ( )

( ) * ( )( )

0 others

ij ij

is is

s A

kij

T t f t

T t f tP t

(9)

1

mk

ij

k

T is the increased pheromone on the path which is

from node i to node j by t time iterative. is pheromone evaporation parameter. k is the ant on the path. ( )k

ijP t is probability that the ant k on the node i select the node j. The formula is presented above.

A is a set of optional nodes which is jointed with node i. Ant k(k = 1, 2, …, m) decides its shift direction in the movement according to the pheromone on the path. From formula (9), ( ) * ( )is is

s A

T t f t is the sum of all optional

pheromone on the path of ant on the node i select node j. Each path is equivalent. We can know that divert probability ( )k

ijP t increases along with the increase of

( ) ( ).ij ijT t f t and are two weight parameters that determine the relative importance of pheromone and heuristic pheromone the ants accumulated in the movement on the decision of selecting path.

Theorem 1: At the beginning of iteration, we suppose that the original pheromone and path cost is equivalent. When

0, 0, then the bigger ( ),kijP t the shorter the path

length dij.

Proof: Suppose d1, d2, …, dm as the optional path length from node i to node j and d1 < d2 < … < dm. Because

1 2 3* os ( , )

..., .ij mij

C t i jf f f f f

d And

( ) (1 / ) .ij ij ij ijT t T f T

There we suppose that every ant releases the same quantity of pheromone, because

1.1 2.1 3.1 .1 1.1 2.1 3.1 .1... ..., .m mT T T T P P P P

Theorem 2: If 0, 0 and d1 < d2 < … < dm (di is optional path length), d1 is the path length. The more the iterative times are, the bigger the probability of that the path with the shortest length is selected.


Proof: We suppose . 11.

.

1

*

*

i kk m

i k j

j

T fP

T f

as the probability of

the number i path by k time iteration as follows:

2. 2 3. 3 .

1. 1 1. 1 1. 1

2 2. 3 3. . 1. 1. 1

1 1. 1 1. 1 1.

2 3

1 1

1

2. 2. 1 3. 3. 1

1. 1. 1 1. 1. 1

.

1.

1* * *

1 ...* * *

1 1 1

1 * * ... *

* *

... *

k k m k m

k k k

k k m m k k k

k k k

m

k k k k

k k k k

m k

T f T f T fT f T f T f

f T f T f T P Pf T f T f T

f ff f

ff

T T T TT T T T

TT

. 1

1. 1

m k

k k

TT

So put Tik, T1k into

1

1 1 1

1

1 1 1 1 1

(1 / )(1 / )

.

ik ik

k k

ik i ik ik

k k k

T TT T

T f T TT f T T

So

1 1

1 1 1 1

1 1 1 1 1

[ (1 / ) ]

[ (1 / ) ][ (1 / ) ]

.

ik i ik k

k k ik

k k k

T f T T

T f T TT f T T

There we suppose every ant releases the same values of pheromone as the original values of pheromone on each path.

1 1

1 1 1 1

1 1 1 1 1

1

[ (1 / ) ]

[ (1 / ) ][ (1 / ) ]

ik i ik k

k k ik

k k k

i

T f T T

T f T TT f T T

f f

Because f1 > fi (fi is the expectation function of selected path on node i),

1,

if f

Then

1

1 1 1

1 1 1

1 10, 0,

1 1 1.

ik ik

k k

k k

T TT T P k P kP P

The more the iteration times, the bigger the probability of the path with the longest length is selected.

Theorem 3: If fi > 0 and 0, 0, the probability of the path of which the expectation function is the biggest will approach 1, when the iteration times approach ∞. .i kP is the probability of the path i selected at the path k iteration time when the fi is the biggest expectation function.

Proof: We can get the relation from Theorem 2 as follows: because and d1 < d2 < … < dm, P1k P1k–1. When fi is the biggest function, then

1. 11.

1.

lim 0, lim 1.kk

k kk

PP

P

Only by analysing the convergence speed of ACO algorithm can we know the time spent working out the optimal result of optimisation algorithm. However, the research on the convergence speed is scarce. This paper discusses the convergence speed problem based on the definition of convergence grad function. The convergence speed increases through changing judgment conditions of convergence.

5 CG convergence

Definition 9 (convergence grads function): It refers to the relation function that the pheromone on the path becomes bigger along with the iterative times increasing. When F is the convergence grads function, the formula is as follows:

[ (1 / ) ]ti i iFi T f T t (10)

We suppose that convergence grads are the ratio between differential coefficient of convergence grads function and iteration times.

Fa t (11)

Theorem 4: The bigger the expectation function fi is, the bigger the convergence grads is.

Proof: Because (1 / ), .ti i i i

Fa a T f Tt From

the Theorem 1 and Theorem 3, because fi is the biggest expectation function, Pi is the biggest probability, so ai is the biggest convergence grads.

From the theorem above, the convergence grads is the biggest on the path of the biggest expectation function. We can stop the iteration by comparing the convergence speed of every path when we find that convergence grads are bigger than the other convergence grads. We can select the best path of which the convergence grads is the biggest and greatly reduce the time of selecting path and thus improve the network’s efficiency.

196 Z. Yang et al.

Definition 10 (convergence grads expectation): The function that reflects the relation between the change of convergence grads and the increase of iteration times

1

(1 / )t

t i iy m T f (12)

m is the number of ants, and Ti is the pheromone on the ith path.

Theorem 5: The convergence grads expectation is along with iteration time.

Proof: From Theorem 2 and Theorem 3, we know that 1

(1 / )t

t i iy m T f is increased along with the iteration times’ increasing. ai will be close to a specified value when iteration time k approach N.

1

lim (1 / )

(1 / )

t

ti i ik N

i i

a m T f

m T f (13)

Theorem 6: If the iteration is close to ∞, then the speed of convergence grads expectation is close to 0: 0. Here, is the speed of convergence grads expectation.

Proof: From Theorem 5,

1

10

11

lim 0

1

t t

t tt

it t

t t i

y um mt t

T fm m

Because (1 )ii

T f is invariable value, 0.

From Theorem 5 and Theorem 6, when iteration t , the convergence grads expectation reaches an invariable value. The speed of convergence is optimised and the speed of convergence grads expectation approaches 0. Then ants always select the best path at this time.

6 The algorithm description of quick QoS convergence grads expectation based on ACO

6.1 Algorithm description

It is supposed that k is the amount of ants, and m is the amount of edges of the graph. S refers to the set of nodes. The steps are as follows:

Step 1 For a graph with n nodes, supposing that N = {1, 2, …., n}, A = {(i, j) | i, j N)}, every edge is given an original value of pheromone. m ants is searching for the optimal path on the graph in parallel, and all ants start from node 1.

Step 2 If it meets stop conditions of algorithm, we stops the algorithm and output the result. Otherwise, ants shall start from node 1. S will note the node of ants and the original value of S is .

Step 3 Count the ant’s path from 1 to m, if the ant is on node i and S = N or there is no next jumping node on node i, then the ant’s counting will finish. Otherwise, we select the next jump according to the next jump’s ratio and put the next jumping node into S. Repeat step 2.

Step 4 Record the ant’s path and update the path’s pheromone. Compare the convergence grads on every path with each other, stop the iteration when one path convergence grads is obviously greater than others. This path is selected as the optimal path. Otherwise, repeat step 1.

Algorithm flow chart is shown in Figure 1.

Figure 1 Algorithm flow chart


The description of pseudocode is presented below:

INPUT: a optimisation graph with weight G <V, E, >

S InitialisePheromoneValues ( i) InitialiseAntNumber (K)

While (the condition not meet ) do ( 1,2 | |)m m E

For i 1 to K do Si For j 1 to do P ConstructPath (S (j), T (j)) UpdateLocalPheromoneValues ( i) Si Si {P (j)} EndFor If (P (d) Si) then UpdateGlobalPheromoneValues (Si) S S {Si} EndFor CheckCondition( ) EndWhile

SGlobalBest MaxBest{ Si | Si S } OUTPUT: the Best Solution SGlobalBest

6.2 Analysis of algorithm convergence

Definition 11: Given a natural number two functions F(n) and G(n). If there is only one positive constant K and one n0, and n n0, then F(n) KG(n). G(n) is regarded as the boundary of F(n). We express it as follows:

( ) O )(F n G n (14)

With the application of progressive notation of time complexity, we conduct an analysis from the view of analysing the amount of steps in the program’s execution and work out the time complexity of this algorithm as follows:

There are n nodes, K ants and the variable value in the condition of finishing the circulated iteration is Nc. Some key steps of time complexity are shown below:

1 original parameter: O(n2 + m)

2 calculation of amount of pheromone updating: O(n2 m)

3 judging the finish condition of iteration circulation according to Nc: O(n m).

When n is the biggest, the time complexity of ACO algorithm is expressed as follows:

2( )T n O Nc n m (15)

7 Value experiment and result analysis

We have simulated the Q-ACO algorithm in the environment of VC. We took a totally undirected graph coordinate of ten nodes as the dataset and calculated the optimal path length by different iteration time using the standard ACO algorithm and Q-ACO algorithm stated in this paper respectively. The parameter value of algorithm was as follows: m = 20, = 0.999, = 5, ρ = 1. In Table 1, IACO refers to the iteration times of ACO algorithm IQ-ACO refers to the iteration times of Q-ACO algorithm. CG is the biggest grads value of Q-ACO algorithm at current time iteration. λ refers to the current error ratio when we obtained the optimal result by using Q-ACO algorithm and standard ACO algorithm.

The experiment result shows that Q-ACO quickly reduces the iteration times x under the condition that it is closest to the optimal result. Q-ACO will get the better iteration result at the same iteration times. In addition, the error ratio of Q-ACO compared with standard ACO is around 10% at the obvious reduction of iteration times.

8 Conclusions

This paper has analysed some primary factors that constrain the QOS routing problem and stated an ACO-based optimised algorithm of multiple constraint QOS routing problem. This algorithm stops the iteration through comparing the path convergence grads in Table 1 and gets the optimal path in a faster and bigger ratio. It improves the ACO-based QoSR speed of calculation and of searching for path in the limited node network, as well as algorithm efficiency of QoS routing service of searching for path.

Table 1 Analysis on the convergence expectation value of ACO and Q-ACO

IACO ACO current optimal result IQ-ACO Q-ACO current optimal result CG λ

50 3.312916 12 3.547081 0.1049 7.07%

100 3.074772 25 3.338388 0.6867 8.57%

200 2.918033 42 3.185981 0.3725 9.18%

400 2.830737 51 3.067977 0.1570 8.38%

198 Z. Yang et al.

Acknowledgements

This study was supported by a Science and Technology Project of Special fund for High-tech development by Guangdong Provincial Department of Finance in 2013 (2013B010401036), Guangdong Province outstanding young teacher training plan (Yq2014187), Guangdong Provincial Department of Education Science and Technology Innovation Project (2013KJCX0178).

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