a new algorithm for finding the shortest paths using pcnns

Chaos, Solitons and Fractals 33 (2007) 1220–1229

www.elsevier.com/locate/chaos

A new algorithm for finding the shortest paths using PCNNs q

Hong Qu *, Zhang Yi

Computational Intelligence Laboratory, School of Computer Science and Engineering,

University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China

Accepted 17 January 2006

Abstract

Pulse coupled neural networks (PCNNs), based on the phenomena of synchronous pulse bursts in the animal visualcortex, are different from traditional artificial neural networks. Caulfield and Kinser have presented the idea of utilizingthe autowave in PCNNs to find the solution of the maze problem. This paper which studies the performance of theautowave in PCNNs aims at applying it to optimization problems, such as the shortest path problem. A multi-outputmodel of pulse coupled neural networks (MPCNNs) is studied. A new algorithm for finding the shortest path problemusing MPCNNs is presented. Simulations are carried out to illustrate the performance of the proposed method.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Given a connected graph G = (V,E), a weight wij for each edge connected vi with vj, and a fixed vertex s in V, to findthe shortest paths from s to other nodes in V. This is the single-source shortest path problem. It is a classical problemwith diverse applications today, such as vehicle routing in transportation [1], traffic routing in communication networks[2], pickup and delivery system [3], website page searching in internet information retrieval system [4], and so on.

PCNNs is a result of research efforted on the development of artificial neuron model that was capable of emulatingthe behavior of cortical neurons observed in the visual cortices of animal [5–7]. According to the phenomena of syn-chronous pulse burst in the cat visual cortex, Eckhorn et al. [7] developed the linking field network. Johnson et al. intro-duced PCNNs based on the linking model [8,9]. Since Johnson’s work [10] in 1994, it has been an increased interestingin using the PCNNs for various applications, such as target recognition [11], image processing [12], motion matching[13], pattern recognition [14] and optimization [15,16].

It is well known that many combinatorial optimization problems, such as the well known travelling salesman prob-lem (TSP) [17–19], can be formulated as the shortest path problem. It is believed that neural network is an availablemethod for solving those problems [20,21]. Since the original work of Hopfield and Tank [22] in 1985, many researcheshave been aimed at applying the Hopfield neural networks to combinatorial optimization problems [22–26]. The majordrawbacks of Hopfield networks are listed as: (1) Invalidity of the obtained solutions. (2) Trial-and-error setting value

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.01.097

q This work was supported by National Science Foundation of China under Grant 60471055 and Specialized Research Fund for theDoctoral Program of Higher Education under Grant 20040614017.

* Corresponding author.E-mail addresses: [email protected] (H. Qu), [email protected] (Z. Yi).

mailto:[email protected]

mailto:[email protected]

H. Qu, Z. Yi / Chaos, Solitons and Fractals 33 (2007) 1220–1229 1221

process of the networks parameters. (3) Low computation efficiency. As a comparison, the spatiotemporal dynamics ofPCNNs provide a seminal computational capability for optimization problems. Caulfield and Kinser have present theidea of utilizing autowaves in PCNNs to find the solution of the maze problem [15].

This paper which presents a novel modified model of PCNNs : multi-output model of pulse coupled neural networks(MPCNNs), to solve the shortest path problem. Some conditions for the exploring of autowave in MPCNNs areexploited by mathematical analysis, which can guarantee the waves explore from the single ‘‘start’’ node to the multi‘‘end’’ nodes. The shortest paths from a single ‘‘start’’ node to multi destination nodes can be found simultaneously.

Section 2 presents a brief description of PCNNs. The multi-output model of pulse coupled neural networks(MPCNNs) is described in Section 3. Section 4 gives a method to solve the shortest paths problem using MPCNNs.Simulation’s results are given in Section 5. Finally, conclusions are drawn in Section 6.

2. PCNNs neuron model

A typical neuron of PCNNs consists of three parts: the receptive fields, the modulation fields, and the pulse gener-ator. The neuron receives input signals from other neurons and external sources through the receptive fields. The recep-tive fields can be divided into two channels: one is the feeding inputs and the other is the linking inputs. A typicalneuron’s model of PCNNs is shown in Fig. 1.

Suppose, N is the total number of iterations and n (n = 0, . . . ,N � 1) is the current iteration, then the receptive fieldscan be described by the following equations:

F ijðnÞ ¼ e�aF F ijðn� 1Þ þ V F

Xkl

Mij;klY klðn� 1Þ þ I ij; ð1Þ

LijðnÞ ¼ e�aL Lijðn� 1Þ þ V L

Xkl

W ij;klY klðn� 1Þ þ J ij; ð2Þ

where the (i, j) pair is the position of neuron. F and L are feeding inputs and linking inputs, respectively. Y is the pulseoutput, aF and aL are time constants for feeding and linking, VF and VL are normalizing constants, M and W are thesynaptic weights, Iij and Jij are external inputs.

The modulation fields generate the internal activity of each neuron, is modeled as follows:

U ijðnÞ ¼ F ijðnÞð1þ bLijðnÞÞ; ð3Þ

where b is strength of the linking.The pulse generator receives the result of total internal activity Uij and determines the firing events, can be modeled

as

Y ijðnÞ ¼ stepðU ijðnÞ � hijðn� 1ÞÞ ¼1; if U ijðnÞ > hijðn� 1Þ;0; otherwise,

�ð4Þ

hijðnÞ ¼ e�ahhijðn� 1Þ þ V hY ijðn� 1Þ; ð5Þ

where hij(n) represents the dynamic threshold of the neuron (i, j). ah and Vh are the time constant and the normalizationconstant, respectively. If Uij is greater than the threshold, the output of neuron (i, j) turns into 1, neuron (i, j) fires, then

0

1

ijY

V

ijUijF

...

+1

11Y

ijY

ijJ

nnY

...

11Y

ijY

nnY

ijL

Linking

Feeding

Step Function

Threshold Unit

Output

Modulation FieldReceptive Field Pulse Generator

1 + ij

-a

ij

βL

θ

θ

θ

Fig. 1. PCNNs neuron model.

1222 H. Qu, Z. Yi / Chaos, Solitons and Fractals 33 (2007) 1220–1229

Yij feedbacks to make hij rise over Uij immediately, then the output of neuron (i, j) turn into 0. Thus produces a pulseoutput. It is clear that the pulse generator is responsible for the modeling of the refractory period of spiking.

The basic idea to solve optimization problems is utilizing the autowave in PCNNs. In the next section, we propose amulti-output model of PCNNs which can mimic the autowaves in PCNNs and find the shortest paths from a single‘‘start’’ node to multi destination nodes simultaneously.

3. The multi-output model of pulse coupled neural networks (MPCNNs)

In this section, we will introduce the designation of multi-output model of pulse coupled neural networks(MPCNNs), in which the linear attenuated threshold is used. The reason for the dynamic threshold is designed tobe a linearly decreasing function rather than exponentially decreasing function is that such a function would makethe autowaves travel uniformly and ease the digital implementation of the network.

3.1. The design of MPCNNs

The proposed neuron’s model of MPCNNs is shown in Fig. 2. Each neuron i in MPCNNs has two outputs: Yi andMi. Yi produces the pulse output as that in original PCNNs, while Mi works as an indicator, which indicates the firedstate of neuron i. Mi is set to be ‘‘1’’ when neuron i fired and ‘‘0’’ if neuron i not fired. Yi and Mi in the network isdesigned as the following functions:

Y iðtÞ ¼ stepðUiðtÞ � hiðtÞÞ ¼1; if U iðtÞP hiðtÞ;0; otherwise,

i ¼ 1; 2; . . . ;N ;�

ð6Þ

MiðtÞ ¼1; if neuron i have fired,

0; if neuron i have not fired,i ¼ 1; 2; . . . ;N ;

�ð7Þ

where Ui(t) and hi(t) are the internal activity and threshold at time t, respectively. N is the total neuron’s number in thenetworks.

The threshold of neuron i (i = 1,2, . . . ,N) is designed as Eq. (8). It is clear that the threshold of neuron i is decreaselinearly respected to a constant value Dh.

hiðtÞ¼

Min fW ji�Dh;ðhiðt�DT Þ�DhÞg; if Miðt�DT Þ¼ 0 and Y jðt�DT Þ¼ 1 and W ji 6¼ 0; j¼ 1;2; . . . ;N

ðneuron j fired in t�DT and neuron i have not fired before t�DT Þ;V h; if Y iðt�DT Þ¼ 1; ðneuron i fired in t�DT Þ;hiðt�DT Þ�Dh; otherwise ðno neuron fired in t�DT Þ;

8>>><>>>:

ð8Þ

where Vh is the threshold whit a larger value. Wji is the linking strength from neuron j to neuron i. If there have nolinking connection between neuron j and neuron i, then Wji = 0.

The internal activity Ui of neuron i determines the firing events. It can be designed as a constant in MPCNNs, asshown in

U i ¼ C ði ¼ 1; 2; . . . ;NÞ and C is a constant. ð9Þ

Now, we have accomplished the design of MPCNNs by Eqs. (6)–(9). Some performances for the travelling of autowavesin MPCNNs will be discussed in the next subsection.

V

0

1Yi

M i

WJi

Ui

i

Y j

Fig. 2. The neuron’s model of MPCNNs.


3.2. Performance analysis of the travelling of autowaves in MPCNNs

To study the performances of the travelling of autowaves in MPCNNs, three calculation of crucial datas areinvolved. There are: (1) T 1

i : neuron i’s fired periods. (2) T 1ij: the periods from the time when neuron i fired to the time

neuron j (j = 1,2, . . . ,N, j 5 i) fired. (3) T1: the periods from the time when the first neuron fired to the the time whenthe last neuron fired. Before the calculations of those three data, some basic notions of the autowaves are introducedfirstly.

We define the single fired pass as a minimal continuous fired period, in which each neuron fired at least once. Thefirst neuron fired in a single fired pass is called the ‘‘source’’ of the autowave. On the contrary, the last neuron fired in asingle fired pass is called the ‘‘end’’ of the autowave. If each neuron fired only once in a single fired pass, then it can becalled the normal single fired pass. It is clearly that the autowave would explore from the ‘‘source’’, visiting each neurononly once, and reach to the ‘‘end’’ step by step in the normal single fired pass.

3.2.1. The calculation of T 1i

Suppose, after t1i � DT times iterations, neuron i fire at time t1

i , the output of neuron i at time t1i is

Y iðt1i Þ ¼ 1;

then

V h > U i P hiðt1i Þ.

In the next iteration, t ¼ t1i þ DT , the threshold and output of neuron i are shown in Eqs. (10) and (11), respectively,

according to Eqs. (6) and (8)

hiðt þ DT Þ ¼ V h; ð10ÞY iðt þ DT Þ ¼ stepðU i � hiðT þ DT ÞÞ ¼ stepðU i � V hÞ ¼ 0. ð11Þ

In the following iterations, the threshold will be attenuated with value Dh, the output will be remained to 0, until whenthe value of threshold decreased down to the level of Ui, supposed at time t2

i , then

T 1i ¼ t2

i � t1i ¼

V h � Ui

DhDT ¼ V h � C

DhDT . ð12Þ

3.2.2. The calculation of T 1ij

Suppose, neuron i fires at time t1i and neuron j fires at time t1

j , and there have k neuron fired from time t1i to t1

j , then

T 1ij ¼ t1

j � t1i ¼

minfP ijg � CDh

DT <ðk þ 1ÞW max � C

DhDT ðk 2 f1; 2; . . . ;N � 1gÞ; ð13Þ

where min{Pij} is the minimal path length from neuron i to neuron j, and Wmax is the maximal one of all the linkingstrength, this is

W max ¼ maxfW ijg; 0 < i; j < N ;

and N is the total neuron numbers.

3.2.3. The calculation of T1

Suppose, neuron 1 is the first fired neuron and neuron n is the last fired neuron, neuron 1 fired at time t11 and neuron

n fired at time t1n. The shortest path length from neuron 1 to neuron n is denoted as min{P1n}, then

T 1 ¼ t1n � t1

1 ¼minfP 1ng � C

DhDT <

NW max � CDh

DT . ð14Þ

With T 1i , T 1

ij and T1, the following theorem can be educed.

Theorem 1. Suppose, neuron start is the first fired neuron, N is the neuron’s number and Wmax is the maximal one of the

linking strengths in MPCNNs, if the following conditions:

ðIÞC P hstartð0Þ;C < hjð0Þ ¼ V h; j ¼ 1; 2; . . . ;N and j 6¼ start;

�ð15Þ

ðIIÞ V h > NW max ð16Þ


hold, then the autowaves travel from neuron start, visit each neuron in the network step by step, end to the last neuron, each

neuron fires only once.

Proof. When (I) holds, the outputs of the neurons at time t = 0 are shown as (17) according to Eq. (6).

Y ið0Þ ¼ stepðU ið0Þ � hið0ÞÞ ¼ stepðC � hið0ÞÞ ¼1; if i ¼ start;

0; otherwise

�ð17Þ

only neuron ‘start’ fires at time t = 0, so the autowaves travel starting from neuron ‘start’.When (II) holds,

T 1i > T 1 > T 1

ij ði; j ¼ 1; 2; . . . ;NÞ; ð18Þ

Eq. (18) would be satisfied according to (12)–(14). This is the sufficient condition for the form of the normal single firedpass. So the process of the firstly N neuron’s firing in MPCNNs form a normal single fired pass. This is also say that theautowaves of MPCNNs would explore from the ‘‘source’’, visiting each neuron only once, and reach to the ‘‘end’’ stepby step in the normal single fired pass.

This completes the proof of Theorem 1. h

This theorem presents a sufficient condition for MPCNNs to optimization problems. In next section, an arithmeticfor finding the shortest paths using MPCNNs will be discussed in detail.

4. The algorithm for solving the shortest path problems using MPCNNs

In Caulfield and Kinser’s work, PCNNs is structured in such a way that each point in the geometric maze figurecorresponds to a neuron in the network, and the autowave in the PCNNs travels from each neuron to its neighborhoodneuron(s) along the maze from iteration to iteration of the network. While in our works of MPCNNs, each neuron inthe network corresponds to a node in the graph, and each directed connection between neurons in the network corre-sponds to the directed edge between the nodes in the graph, the autowave travels along the connection between twoneurons.

The algorithm of discrete-time MPCNNs for finding the shortest paths in the graph is expressed in the followingsteps. The continuous-time MPCNN is the case of DT! 0.

// start: the starting neuron to fire;// N: the total number of neurons;// k: the current iterate;// l: the number of neurons that have fired;// RouteRecord[i, j]: a matrix to record the routes, RouteRecord[i, j] = 1 if the autowave travel from neruon i to neuron j,else RouteRecord[i, j] = 0;// PreNode(j): the neuron represent the parent of neuron j;// FiredNeuron: the neuron that fired in the pre-iteration;// Wmax: the maximal one of the linking strength.

(1) Initialize the network, set iterate number k = 0, and the number of neurons that have fired l = 0, and the record
of route RouteRecord[i, j] = 0, (i, j = 1,2, . . . ,N). Set PreNode(i) = 0 for "i = {1,2, . . . ,N}, and FiredNeuron = 0.
(2) Set

U i ¼ C; i ¼ 1; 2; . . . ;N ;

hstartð0Þ < C;

hið0Þ > C; i ¼ 1; 2; . . . ;N and i 6¼ start;

V h > NW max

8>>><>>>:then the neuron ‘start’ fires first, as shown in

Y startð0Þ ¼ 1;

Y ið0Þ ¼ 0;

�and

M startð0Þ ¼ 1;

Mið0Þ ¼ 0;

�ði ¼ 1; 2; . . . ;N and i 6¼ startÞ;

autowaves are generated and neuron ‘start’ is the source of the autowaves.


(3) For each neuron j

(a) Calculate hj(t) according to Eq. (8).(b) If W(FiredNeuron, j) < Theta(j), set PreNode(j) = FiredNeuron.(c) Calculate Yj(t) and Mj(t) according to Eqs. (6) and (7), respectively.(d) If Yj(t) == 1

Set FiredNeuron = j and l = l + 1.) Record the route, set RouteRecord[PreNode(j), j] = 1.

(i)
(ii
(e) Set k = k + 1.
(4) Repeat step 3 until all destination neurons in the network are in fired state.
This algorithm is a nondeterministic method, which guarantees the globally shortest path solution. The shortestpaths from a single ‘‘start’’ node to all multi destination nodes are searched simultaneously and parallelly by runningthe network only once.

5. Simulation results

To verify the theoretical results and the effectiveness of the discrete-time MPCNNs when it be applied to solve theshortest paths problem, several experiments have been carried out. These programs, which coded in MATLAB 6.5,were run in a compatible IBM PC, Pentium 4 2.66 GHz and 256 MB RAM.

The first experiment is based on a symmetric weighted graph with 10 nodes and 20 edges, shown as Fig. 3. The max-imal weight of all edges is: Wmax = 17. In this experiment, Vh is taken the value of 200 and Dh is set to be 1. The internalactives and the initial values of the thresholds are set to be

U i ¼ C ¼ 0 andhstartð0Þ < 0;

hið0Þ ¼ V h; if i 6¼ start.

�ð19Þ

Table 1 presents the results of the computer simulation. Obviously, all the paths found by the algorithm are globallyshortest paths. The finding process from the ‘‘start’’ node 1 to the others with the time increasing are shown in Fig. 4.

Theorem 1 have shown that a larger value of Vh should be required to guarantee the exploring of autowaves inMPCNNs. To test the relationship between the value of Vh and the exploring of autowaves in MPCNNs, eight simu-lations will be performed. The ‘‘start’’ node is set to be 1 in each simulation. Different values of Vh are taken in eachrunning. In this experiment, D h is also set to be 1. The initial value of the internal actives of each neuron Ui, and the theinitial value of the threshold of each neuron hi(0) is set to be the same value with the above experiment as shown in Eq.(19). The simulation results are given in Table 2.

The simulation results in Table 2 show that when Vh P 22, the resulted solutions are the globally solutions. This isalso shows that Vh > NWmax is the sufficient condition for the globally solutions of the shortest paths problem, but notthe necessary conditions.

Another simulation has been made to test the relations between the number of iterations and the value of Dh, alsobased on the graph shown in Fig. 3. Three case of the ‘‘start’’ node: 1, 5 and 10 are involved in this experiment, and all

1

9

87

6

5

4

32

10

5

7

6

8

2

12

17

13

7

15

4

12

10

9

11

9

3

14 16

7

Fig. 3. A symmetric weighted graph.

Table 1Result of MPCNN for the graph in Fig. 3

Start node The shortest path Iterations Globally solutions

1 1! 6! 5; 1! 6! 10; 1! 7! 2! 8! 3; 1! 9! 4 23 Yes2 2! 6! 5; 2! 7! 1; 2! 7! 10; 2! 8! 3; 2! 8! 4; 2! 8! 9 21 Yes3 3! 8! 2! 6; 3! 8! 2! 7! 1; 3! 8! 4; 3! 8! 9! 5; 3! 8! 10 23 Yes4 4! 8! 2; 4! 8! 3; 4! 8! 2! 7; 4! 9! 5; 4! 9! 6; 4! 9! 10 29 Yes5 5! 6! 1; 5! 6! 2; 5! 6! 7; 5! 9! 4; 5! 9! 8! 3; 5! 9! 10 23 Yes6 6! 1; 6! 2! 8! 3; 6! 5; 6! 7; 6! 9! 4; 6! 10 21 Yes7 7! 1! 9! 5; 7! 2! 8! 3; 7! 2! 8! 4; 7! 6; 7! 10 25 Yes8 8! 2! 6; 8! 2! 7! 1; 8! 3; 8! 4; 8! 9! 5; 8! 10 18 Yes9 9! 1! 7; 9! 4; 9! 5; 9! 6! 2; 9! 8! 3; 9! 10 21 Yes

10 10! 6! 1; 10! 7! 2; 10! 8! 3; 10! 9! 4; 10! 9! 5 22 Yes

11

5

3

410

9

87

6

2

11

5

3

410

9

87

2

1. 2.t = t = 6

62

11

5

3

410

9

8

2

3. t = 8

62

73

11

5

3

410

9

8

4. t = 11

62

73

4

2

11

3

410

8

6. t = 16

62

73

4

2

95 5

61

1

3

4

8

7. t = 17

62

73

4

2

95 5

6

107

11

3

4

8. t = 18

62

73

4

2

95 5

6

107

88

11

3

9. t = 21

62

73

4

2

95 5

6

107

88

49

11

10. t = 23

62

73

4

2

95 5

6

107

88

49

310

11

5

3

410

8

5. t = 13

62

73

4

2

95

T

T T

TTT

T T

T T

Fig. 4. The finding processing of the instance from ‘‘start’’ node 1 to others.


the initial conditions is set to the same value as in the first experiment, expect the value of Dh. The changing of thenumber of iterations with the different value of Dh are shown in Table 3.

Table 3 shows that a small value of Dh should be required to guarantee the globally solution of the shortest pathsproblem, while the number of iteration of algorithm will be increased with the decreasing of Dh.

We also apply our algorithm to finding the shortest paths from arbitrarily specified ‘‘start’’ node to all other nodes inrandomly generated undirected and symmetric graphs. The graphs are generated in such a way that the N nodes areuniformly distributed in a square with the edge length D, and each two nodes is connected with an edge if their distanceis no more than 30%D, the weight of the edges which connected the two nodes is just the Euclidean distance between thetwo nodes. A MPCNNs associated with the graph is constructed to find the shortest paths for each such generatedgraph from any specified ‘‘start’’ node to all of the other N � 1 nodes. Fig. 5 shows a generated graph with D = 50and N = 100.

The shortest paths searched by the new algorithm from three different ‘‘start’’ nodes to the others for the randomlygenerated graph are shown in Fig. 6.

Table 2Simulation results with the different value of Vh

Vh Fired sequence of neuron Spiking times Iterations times Globally solution

5 1! 6! 7! 2! 3! 4! 5! 8! 9! 10 10 6 No10 1! 6! 7! 2! 3! 4! 5! 8! 9! 10 10 11 No15 1! 6! 7! 2! 9! 3! 4! 5! 8! 10 10 16 No20 1! 6! 7! 2! 9! 5! 10! 8! 3! 4 10 21 No22 1! 6! 7! 2! 9! 5! 10! 8! 4! 3 10 24 Yes50 1! 6! 7! 2! 9! 5! 10! 8! 4! 3 10 24 Yes

170 1! 6! 7! 2! 9! 5! 10! 8! 4! 3 10 24 Yes300 1! 6! 7! 2! 9! 5! 10! 8! 4! 3 10 24 Yes

Table 3Simulation results with the different value of Vh

Start node Dh Iterations times: k Globally solution

1 0 < Dh 6 2.2 k P 12 YesDh > 2.2 k 6 12 No

5 0 < Dh < 40 k P 7 YesDh P 40 k < 7 No

10 0 < Dh 6 66 k P 4 YesDh > 67 k < 4 No

For all 0 < Dh 6 2.2 – YesDh > 2.2 – No

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

Fig. 5. A symmetric weighted graph randomly generated with N = 100 in a square of edge length D = 50.


0 20 40 600

5

10

15

20

25

30

35

40

45

50(a)

0 20 40 600

5

10

15

20

25

30

35

40

45

50(b)

0 20 40 600

5

10

15

20

25

30

35

40

45

50(c)

Fig. 6. (a), (b) and (c) show the shortest paths form differently specified ‘‘start’’ nodes to all others.


6. Conclusions

In this paper, we present a new arithmetic based on the multi-output pulse coupled neural networks (MPCNNs) tosolve the shortest path problems. Our works are based on the symmetric weighted graph. This is a nondeterministicmethod which would guarantee the globally solutions. The shortest paths from a single ‘‘start’’ node to all multi des-tination nodes are search simultaneously parallelly by running the network only once. The highly parallel computationof the network will result in fast finding of the shortest paths if the network is realized with VLSI. It is easy to extend theproposed MPCNNs to other case when the topology has some slight changes. The arithmetic also can be apply to manyreal time applications of shortest paths problems such as website page searching in internet information retrieval systemand traffic routing in communication network.

As shown in the paper, the discrete-time MPCNN is an iterative procedure in which the output from one iterationstimulates the iteration. In the case of continuous-time network, this is still the same if the time step is small enough.

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a new algorithm for finding the shortest paths using pcnns

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