A shortest-path network problem using an annealed ant system algorithm Shao-Han Liu, Jzau-Sheng Lin, and Zi-Sheng Lin Department of Computer Science and Information Engineering, National Chin-Yi Institute of Technology, No. 35, Lane 215, Sec. 1, Chung-Shan Rd., Taiping, Taichung, TAIWAN. TEL: 886-4-23924505 ext 7311 FAX: 886-4-23926610
[email protected]
Abstract This paper investigates a shortest-path network problem using an annealed ant system algorithm, in which an annealing strategy is embedded to calculate the probabilities to decide which path the ants will select next. The shortest-path problem is to determine the shortest route between a source and a destination in a transportation-network topology. In this approach, according the concrete problems of shortest routing, we construct two globally optimizing annealed ant algorithms that are Concentrated Model and Distributed Model. The Concentrated Model (CM) means all ants are initially concentrated in the source node while all ants randomly select a node except the destination as their starting point initially and at least one must appear in the source node for the Distributed Model (DM). The experimental results show that the proposed annealed ant algorithm with the Roulette wheel selection can obtain better performance than that generated by the traditional ant strategy with/without the Roulette wheel selection.
1. Introduction Many network services such as video conferencing and video on demand, have popularly used the multimedia communications. The attached hosts/routers are required to transmit data rapidly. In order to provide an efficient data routing, routers must support quick services. Finding shortest paths in a network is one of the fundamental problems in network services. For example, routing algorithm refers to a process of finding appropriate path so that the traffic can be relayed in some optimal ways. One important issue in routing algorithms is the speed with which they can react topology changes. Traditionally, TCP/IP uses routing information protocol (RIP) [1] in which the routing decisions are based on the number of hops between source and destination. Therefore, how to find a shortest path in a network topology is optimization problems. And, the shortest-path problem is the most common problem in the whole class of optimal path problems. Several topics are applied in shortest-path computation such as neural networks and mobile agent.
Leung [2] demonstrated neural scheduling algorithms for timing-multiplex switches. Brown [3] also presented neural network for switching problems. A neural-network model, Routron, was proposed by Lee and Chang [4] for routing of communication networks with unreliable components. The continuous Hopfield neural network was applied by Ali and Kamoun [5] to the optimal routing problem in packet-switched computer networks to minimize the network wide average time delay. Lin et al. [6] also proposed an annealed Hopfield neural network with a new cooling schedule for the shortest path computation in MOSPF protocol. Ant system algorithm, presented by Derigo and others [7-9], is an important methodology to apply on non-linear optimal problems recently. It is a stochastic combinational strategy to solve problems by artificial ants with very simple basic capability that simulates the behavior of real ants. An ant moves randomly and detects a previously laid pheromone on a path in order to find the shortest way between their nest and the food source. It must decide with a grade of priority to select a path in accordance with the amount pheromone deposited on the path. M. Dorigo and G. Di Caro [7-9] first proposed a mobile agent called AntNet to the routing problems. Next, Baran and Sosa [10] presented a new approach for AntNet routing to improve the performance of the original AntNet. In the proposed annealed ant system approach, an original ant algorithm is modified and the annealed strategy with a cooling schedule is added to calculate the probabilities that the ants will decide to select the next paths using Roulette wheel. Compare with the conventional ant techniques, the major strength of the proposed annealed ant approach is that the probability generated by the laid pheromone and the visibility can follow the annealed strategy with a cooling schedule. The Roulette wheel scheme is used to enhance the selecting ability of ants to find the path attached more gummous pheromone. Consequently, a near global-minimum result can be found in the shortest-path problem. In a simulated study, the proposed algorithm demonstrates the capability for the shortest-path problem in routing network and shows more promising results than the conventional ant algorithm.
Proceedings of the Fourth Annual ACIS International Conference on Computer and Information Science (ICIS’05) 0-7695-2296-3/05 $20.00 © 2005 IEEE Authorized licensed use limited to: CBNU. Downloaded on March 30,2010 at 10:09:11 EDT from IEEE Xplore. Restrictions apply.
The rest of the paper is organized as follows. Section 2 introduces the basic theory of ant system algorithm; Section 3 reviews the annealing techniques; in section 4, we describe the proposed annealed ant algorithms to search for the shortest path in a network. Section 5 shows the effectiveness of the proposed algorithms in simulation. Finally, the conclusion is given in section 6.
the k-th ant. When the ant has completed a tour in time interval [ t0 , t0 n ] and a cycle of n iterations is consisted, the laid trail substance is used to update the amount of substance previously laid as the following equation: (3) W ij (t n) U W ij (t ) 'W ij and m
'W ij
2. The Ant Algorithm
ants be m which is represented m simple agents at time t. The k-th ant starting from town i decides to visit town j D
D
¦ ª¬W i" (t )º¼ "
where Kij
E E
(1)
ª¬Ki" º¼
1 is the visibility of town j from town i, dij
D and E are two heuristically defined parameters. If we define the pheromone intensity on path (i, j) at time t to be W ij (t ) and to assign a random value to W ij (t ) when t = 0. Along the path from i to j, a trail substance is laid on path (i, j) and defined as: 'W ijk
Q ° ® Lk ° 0 ¯
if k th ant uses path in its tour
where U is a coefficient of persistence of the tail and 'W ijk is the quantity of trail substance laid on path (i, j)
by the k-th ant during a cycle( between time t and t + n). Therefore the ant system algorithm used in the TSP problem can be described as 1. Set t = 0; and randomly set W ij (t ) as a positive 2. 3.
value. m ants are randomly positioned on the towns as starting points. For every ant k, choose the town j with probability pijk (t ) , and move it to the town j
4.
from town i. If the tours are not completed, go to step 3, otherwise go to step 5.
5.
For every ant k, compute Lk and 'W ijk .
6.
Update the amount of substance for every path (i, j) using Eq. (3). 7. t = t +n; 8. If the stop criterion is not matched, go to step 3, otherwise computing process for the shortest path in TSP problem is completed. The ant system algorithm can also be extended to solve the shortest-path network problem to select a minimal-length way between source and destination nodes in a network topology that an ant visits each node.
3. Annealing Techniques
with the probability pijk defined as follows: ªW ij (t ) º ªKij º ¬ ¼ ¬ ¼
(4)
k 1
Ants always deposit a kind of substance called pheromone on the path between their nest and the food source so that they can find the shortest path to move food and go back efficiently in accordance with a heavy pheromone. When an ant is walking on a path, it deposits some pheromone that can be recognized by other ants. The amount of pheromone on the path decides the probability that the ants coming later choose the path. That is the more ants visit a path, the larger amount of pheromone are deposited on it. This effect can result in that the ants can find the shortest path finally. Ant system algorithm, based on behavior of real ants, is a natural approach to establish from their nest to food source. It is a parallel architecture to force ants move simultaneously, independently, and without supervisor. In the ant system algorithm, each agent (ant) is cooperative to choose a path with a heavy pheromone laid by the previous ants. The Travel Salesman Problem (TSP) can be stated as the problem to find a minimal-length closed path using the ant algorithm that an ant visits each town. Each ant chooses the next town to visit in accordance with the visibility of the town and the pheromone intensity. In the application of TSP problem, there are n towns and the Euclidean length of path between towns i and j is dij . Let the total number of
pijk (t )
¦ 'W ijk
(2)
otherwise
In Eq.(2), Q is a constant and Lk is the tour length of
The simulated annealing strategy was first proposed in 1953 by Metropolis et al. [11] to simulate molecular processes. Kirkpatrick et al. [12] used the idea as a method to resolve minimizing functions of many variables, such as NP-hard problems. Simulated annealing drives its name from an analogy between its behavior and that of a physical process of thermodynamics and metallurgy in which a metal is first melted at a very high temperature and then slowly cooled until it solidifies in a structure of minimum energy. At the beginning, an initial temperature T, used to control the probability of accepting a worsening perturbation over time, is set to a very high value; later it is multiplied by a cooling rate every iteration. Simulated annealing is a stochastic relaxation strategy used successfully to resolve optimization problems including circuit routing [12], computer network topology [13], traveling salesman [14], and
Proceedings of the Fourth Annual ACIS International Conference on Computer and Information Science (ICIS’05) 0-7695-2296-3/05 $20.00 © 2005 IEEE Authorized licensed use limited to: CBNU. Downloaded on March 30,2010 at 10:09:11 EDT from IEEE Xplore. Restrictions apply.
image processing [15]. Instead of other optimization methods such as the steepest descent approach used in the neural network, the simulated annealing technique, which allows the search to escape a local minimum, seeks the global or near global minimum of an energy function. The simulated annealing technique has nonzero probability to go from one state to another and moves temporarily toward a worse state so as to move away from local traps. The probability function depends on the temperature and the cost difference between two states. With the probabilistic hill-climbing search approach, the simulated annealing technique has a promising probability to go to a higher cost at a higher temperature. In order to converge to a near global minimum in the annealing process, a feasible cooling schedule is required. In a cooling schedule, reaching thermal equilibrium at low temperature might take a very long time. In this paper, the decrement function shown in Eq. 5, proposed by Lin [16], is used as cooling schedule. 1 ª (5) T (t ) P tanh( w)t º T (t 1) ¼ P 1 ¬ where w is a small constant and closes to unit as well as P is also a constant. Lin showed that Eq. (5) can result in a faster decrement speed than that result from the traditional decrement functions.
4. The Proposed Annealed Ant Algorithms in Shortest-path Problem In 1997, M. Dorigo and G. Di. Caro [7] firstly proposed the ant algorithm called AntNet for packet routing in communication networks. In AntNet, a group of mobile agents built paths between pair of nodes, traveling around concurrently and exchanging data to update routing tables. This work modified the probability function in the traditional ant algorithm so that the new probability function not only depends on the temperature but the cost function defined by the pheromone density and visibility. In the scheme of ant system algorithm, the total cost function for the network topology from node i to k can be defined as ¦¦ [W ik (t )]D [Kik ]E
E (t )
i
(6)
Eij (t )
pijk (t )
if
path (i, k ) exists
otherwise
Parameters Kik , W ik (t ) , D and E are same as the previous definitions. Based on the reference [18], pijk (t ) is looked upon as the probability that the k-th ant starting from node i to decide to visit node j undergo random thermal perturbations at a given temperature T conforms to a Boltzmann distribution pijk (t ) v e
'Eij (t ) / T
(7)
Then the mean field Eij (t ) can be calculated from Eq.
(8)
e
Eij (t ) / T
(9)
¦ e Ei" (t ) / T "
The normalization operation in Eq. (9) guarantees that the k-th ant will be absorbed on several nodes with certain probability degrees so that the cost function in Eq. (6) will also converge to a near-global minimum. In this paper, the proposed annealed ant algorithm is modified into two models called Concentrated Model (CM) and Distributed Model (DM). These two models are described as follows.
4.1 Concentrated-model annealed ant algorithm The Concentrated Model (CM) means all ants are initially concentrated in the source node. The detail process is depicted as 1. Set source and destination nodes as the start and stop nodes for all m ants. 2. Set t = 0; and randomly set W ij (t ) as positive 3.
values for all path (i, j). Set U as a positive value and calculate all visibilities Kij .
5.
Set w and P to be positive values in the cooling schedule and start with an initial temperature T(0). Calculate the mean field energy using Eq. (8).
6.
Compute the normalized probability
7.
using Eq. (9) that the k-th ant will move to the node j from node i in accordance with the Roulette wheel selection. Decrease T with the annealing factor T(t) shown in Eq. (5) iteratively. If the completed path from source to destination is not found, go to step 5, otherwise go to step 9.
4.
8. 9.
1/ dik ® ¯ 0
[W ij (t )]D [Kij ]E
k The probability pij (t ) that the k-th ant starting from node i and deciding to select node j can then be normalized as follows:
k
where Kik
(6) to be
pijk (t )
For every ant k, compute Lk and 'W ijk .
10. Update the amount of substance for every path (i, j) using Eq. (3). 11. t = t +n; 12. If the stop criterion is not matched, go to step 5, otherwise computing process for the shortest path in routing network is completed. In the shortest-path network problem, the stop criterion can be defined as a large number of iterations or the searched paths from source and destination changed or not near two iterations.
4.2 Distributed-model annealed ant algorithm
Proceedings of the Fourth Annual ACIS International Conference on Computer and Information Science (ICIS’05) 0-7695-2296-3/05 $20.00 © 2005 IEEE Authorized licensed use limited to: CBNU. Downloaded on March 30,2010 at 10:09:11 EDT from IEEE Xplore. Restrictions apply.
All ants randomly select a node except the destination as their starting point initially and at least one must appear in the source node for the Distributed Model (DM). The DM algorithm is depicted as 1. Randomly select a node as the start point for every ant and at least one must appear in the source node. 2. Set t = 0; and randomly set W ij (t ) as positive 3.
values for all path (i, j). Set U as a positive value and calculate all visibilities Kij .
5.
Set w and P to be positive values in the cooling schedule and start with an initial temperature T(0). Calculate the mean field energy using Eq. (8).
6.
Compute the normalized probability
7.
using Eq. (9) that the k-th ant will move to the node j from node i in accordance with the Roulette wheel selection. Decrease T with the annealing factor T(t) shown in Eq. (5) iteratively. If the completed path from source to destination is not found, go to step 5, otherwise go to step 9.
4.
8. 9.
pijk (t )
For every ant k, compute Lk and 'W ijk .
10. Update the amount of substance for every path (i, j) using Eq. (3). 11. t = t +n; 12. If the stop criterion is not matched, go to step 5, otherwise computing process for the shortest path in routing network is completed.
5. Experimental Results In order to show the computation performance for the proposed annealed ant system strategy, all simulations are executed with the interpreter language of MATLAB in a Pentium IV personal computer. In this paper, the heuristically defined parameters D and E are set to 1 and Q=1000. The network topology, shown in Fig. 1 and used in reference [10], is applied as a test example. The shortest path is computed as nodes 1-3-8-14-20 with length = 142 in Figure 1. The successful percentage for the different number of ants with distinct persistence in CM and DM algorithms for the conventional ant system algorithm with Roulette wheel selection are shown as Figures 2 (a) and (b) respectively. The successful percentage is defined that the correct times is obtained to find the shortest path in 100 experiments. From these two figures, we can find that the CM strategy can obtain better performance than the DM approach while the larger the coefficient of persistence is, the better the performance. For example, the successful percentage is 34% and 25% with 20 ants and U 0.2 in CM and CM approaches respectively. The better results can be obtained when U 1 . For 100 experiments, 99 experiments can get the shortest path
with U 1 and 180 ants in the CM algorithm as well as U 1 and 220 ants in the DM approach individually. For all experiments in the conventional ant algorithm with the Roulette wheel selection, the curves of successful percentage are situated when the numbers of ants are 180 and 220 in Figures 2 (a) and (b). These curves show that adding more ants to generate better performance does not be guaranteed. This can also prove that the CM algorithm can save more computation time to obtain the shortest path in the application of routing networks due to more ants appearing the paths near source node in the CM mode than those in the DM model. Figures 3 (a) and (b) show the experimental results obtained from the proposed annealed ant system approaches with Roulette wheel selection in CM and DM models. For these two experiments, w was set 0.98 and 0.998 as well as U was set 0.8 and 0.99 respectively. From Figure 3 , the proposed annealed ant algorithm in CM mode also obtains better performance than those in the DM mode. In Figure 3 (a), the successful percentage can reach to 100% when the number of ants is over 80. The experimental results show that the performance produced by the DM mode with U =0.99 is better than U =0.8. From all of the experiments, the proposed annealed ant system strategy in CM model can find the optimal solution in the shortest-path problem. In order to show the promising performance in the proposed approach, the proposed algorithm was also compared with conventional ant strategy in reference [9]. The successful-percentage curves with different parameters for the conventional ant algorithm in CM and DM models are shown in Figures 4 (a) and (b). In Figure 4 (a), the successful percentages are almost kept about 96% in the CM no matter what the number of ants and the constrained parameters are changed. The characteristics for the successful-percentage curves are proportional to the number of ants under 300 ants. From all experimental results, we can conclude that the proposed annealed ant system algorithm with the Roulette wheel selection can obtain more promising performance than those generated by the traditional ant strategy with/without the Roulette wheel selection.
6. Conclusion In this paper, we proposed CM and DM models with the annealed ant system strategy for the purpose of finding the shortest path in routing network. For the sake of finding the completive path efficiently, the annealed strategy and Roulette-wheel selection are also used to decide which path an ant will select next. In the experimental results, the proposed annealed ant algorithm with Roulette-wheel selection in CM model can get better performance. And, the larger the coefficient of persistence is, the better the performance. In addition, the proposed cooling schedule could satisfactorily force the ants to deposit more pheromone on a short branch in order to fined the completely
Proceedings of the Fourth Annual ACIS International Conference on Computer and Information Science (ICIS’05) 0-7695-2296-3/05 $20.00 © 2005 IEEE Authorized licensed use limited to: CBNU. Downloaded on March 30,2010 at 10:09:11 EDT from IEEE Xplore. Restrictions apply.
shortest path from source to destination node while the convergence of cost function with finite number of iterations is guaranteed.
[8]
G Di Caro, and M. Dorigo, “Ant colonies for adaptive routing in packet-switched communications networks,” Proceedings of PPSN-V, Fifth Int. Conference on Parallel Problem Solving form Nature Berlin: Springer-Verlag, 1998, pp.673-682. G Di Caro, and M. Dorigo, “AntNet: Distributed stigmergetic control for communications networks,” J. of Artificial Intelligence Researcb (JAIR), vol.9, 1998, pp. 317-365. B. Baran, and R. Sosa, “A new approach for AntNet routing,” Proceedings, Ninth International Conf. on Computer Commni. And Networks, Oct. 2000, pp. 303-308. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equations of state calculations by fast computing machines, “ J. Chem. Phys., vol. 21, 1953, pp. 1087-1092. S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, 1983, pp. 671-680. S. Pierre, M. A. Hyppolite, J. M. Bourjolly, and O. Dioume, “Topological design of computer communication networks using simulated annealing,” Engineering Applications of Artificial Intelligence, vol. 8, 1995, pp. 61-69. P. J. M. Van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Application, D. Reidel Publishing Company, Dordrecht Holland, 1987. H. L. Tan, S. B. Gelfand, and E. J. Delp, “A cost minimization approach to edge detection using simulating annealing,” Science, Vol. 220, 1983, pp. 671-680. Jzau-Sheng Lin, “Image vector quantization using an annealed Hopfield neural network,” Opt. Eng., Vol. 38, 1999, pp. 599-604.
Acknowledgement [9]
This work was partially supported by the National Science Council, TAIWAN, under the Grants NSC92-2213-E-167-003 and NSC93-2213-E-167-005.
[10]
References [1] [2] [3] [4]
[5]
[6]
[7]
C. Hedrick, “Routing information protocol,” RFC 1058, Proteon Inc. June, 1988. Y. M. Leung, “Neural scheduling algorithms for time-multiplex switches,” IEEE J. Selected Areas Commun., vol. 12, 1994, pp. 1481-1487. T. X. Brown “Neural networks for switching,” IEEE Commun. Mag., 1989, pp. 72-81. S. L. Lee, and S. Chang, “Neural networks for routing of communication networks with unreliable components,” IEEE Trans. Neural Networks, vol. 4, 1993, pp. 854-863. M. K. M. Ali, and F. Kamoun, “Neural networks for shortest path computation and routing in computer networks, “ IEEE Trans. Neural Networks, vol. 4, 1993, pp. 854-863. J.-S. Lin, M. -S. Liu, and N.-F. Huang, “The shortest path computation in MOSPF protocol using an annealing Hopfield neural network with a new cooling schedule,” Information Sciences, vol. 129, 2000, pp. 17-30. Di Caro, G., and M. Dorigo, “Ant Net: A mobile agents approach to adaptive routing,” (Tech.Rep.97-12). Univeraite Libre de Bruxlles, IRIDIA, 1997. 230
90
50
17
3 60
[16]
16
61
150
35 30
20
136
58
144 12
Destination
24
18 54
13 220
16
72
61
7 25
6
72
22
14
88
32 27
220
40
20 8
62
15
120
250
2
19
14
194 60
9
89
77
110
10
45
65
[15]
40
29 1
[14]
147
43
Source
[13]
32
4 30
[12]
130
11 15
5
[11]
26
71 161
17
Figure 1. Network topology
Proceedings of the Fourth Annual ACIS International Conference on Computer and Information Science (ICIS’05) 0-7695-2296-3/05 $20.00 © 2005 IEEE Authorized licensed use limited to: CBNU. Downloaded on March 30,2010 at 10:09:11 EDT from IEEE Xplore. Restrictions apply.
100
100
90
90 80 successful percentage
successful percentage
80
70
o: U =0.2
60
+: U =0.4 50
*: U =0.6
30
50
100
150 200 the number of ants
o: U =0.2
50
+: U =0.4 *: U =0.6
' : U =0.8
30
: U =1 0
60
40
' : U =0.8
40
70
250
20
300
: U =1 0
50
100
150 200 the number of ants
250
300
(a) (b) Figure 2. Successful percentage for the different number of ants with distinct persistence by the conventional ant system algorithm with the Roulette wheel selection: (a) CM model, and (b) DM model. 100
100 90
95
80 successful percentage
successful percentage
90 85 80 75
U =0.8 x:w=0.98, U =0.99 +:w=0.998, U =0.8 :w=0.998, U =0.99 o:w=0.98,
70 65 60
0
50
100
150 200 the number of ants
70 60 50 40
U =0.8 U =0.99 +:w=0.998, U =0.8 :w=0.998, U =0.99
o:w=0.98,
30
x:w=0.98,
20 10
250
0
300
0
50
100
(a)
150 200 the number of ants
250
300
(b)
Figure 3. Successful percentage for the different number of ants with distinct persistence and w by the proposed annealed ant system algorithm with the Roulette wheel selection: (a) CM model, and DM model. 100
100
98
90 80
94
the number of success
successful percentage
96
92 90 88
o: U =0.2
86
+: U =0.4 *: U =0.6
84
80
50
100
150 200 the number of ants
250
50
o: U =0.2
40
+: U =0.4 *: U =0.6
' : U =0.8
20
: U =1 0
60
30
' : U =0.8
82
70
300
10
: U =1 0
50
100
150 200 the number of ant
250
300
(a) (b) Figure 4 Successful percentage for the different number of ants with distinct persistence by the conventional ant system algorithm in (a) CM model, and (b) DM model.
Proceedings of the Fourth Annual ACIS International Conference on Computer and Information Science (ICIS’05) 0-7695-2296-3/05 $20.00 © 2005 IEEE Authorized licensed use limited to: CBNU. Downloaded on March 30,2010 at 10:09:11 EDT from IEEE Xplore. Restrictions apply.
