Fixed channel assignment using new dynamic programming approach in cellular radio networks

Computers and Electrical Engineering 31 (2005) 303–333 www.elsevier.com/locate/compeleceng

Fixed channel assignment using new dynamic programming approach in cellular radio networks S. Alireza Ghasempour Shirazi

a,1

, Hamidreza Amindavar

b,*

a

b

ICT Faculty—Information and Communications Technology, P.O. Box 16315-746, Tehran, Iran Amirkabir University of Technology, Electrical Engineering Department, Hafez Avenue, 15914 Tehran, Iran Received 29 September 2003; received in revised form 5 April 2005; accepted 5 April 2005 Available online 25 August 2005

Abstract In this paper, we propose a new approach to the fixed channel assignment problem by modifying the dynamic programming technique. This new strategy extends the already known dynamic programming so that the channel assignment solutions can be obtained. There is no need to have an initial random solution for convergence. One of the questions in the fixed channel assignment is the minimum bandwidth, which is usually unknown; the new strategy can obtain this lower bound. Parallel processing can be implemented over the proposed algorithm. The existing fixed channel assignment methods do not have all these in one place. The performance of modified dynamic programming (MDP) is evaluated by computer simulation, applied to seven well-known benchmark problems on channel assignment. The channel assignment strategies results shows that required bandwidths of modified dynamic programming are closely match or sometimes better than the algorithms that we have investigated.
2005 Elsevier Ltd. All rights reserved.

*

1

Corresponding author. Tel.: +98 21 6454 3332; fax: +98 21 640 6469. E-mail addresses: [email protected] (S. Alireza Ghasempour Shirazi), [email protected] (H. Amindavar). Tel.: +98 21 465 4819; fax: +98 21 465 9665.

0045-7906/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compeleceng.2005.04.002

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1. Introduction The appearance of cellular mobile communication systems and their rapid growth due to the portability and the availability of these systems provided an important alternative in the field of wireless mobile communications. The increasing demand of new services in this field, however, is in contrast to the capacity constraints inherent in the current communication systems. Hence, the use of techniques, which are capable of ensuring that the frequency spectrum assigned for use in mobile communications will be better utilized, is gaining an ever-increasing importance. This makes the task of channel assignment more and more crucial [1]. The channel assignment problem (CAP) in this paper is based on a common model. The service area of the system is divided into a number of hexagonal cells. Every user is located in one cell. When a user requests a call in this system, a channel is assigned to that user to provide the communication service. This channel assignment must satisfy the electromagnetic compatibility (EMC) constraints to avoid the radio interference between channels. Three types of EMC constraints have usually been considered in CAP. 1. The co-channel constraint (CCC): The same channel cannot be assigned to cells that have a certain distance from each other. 2. The adjacent channel constraint (ACC): Adjacent channels cannot be assigned to adjacent cells simultaneously. In other words, any pair of channels in adjacent cells must have specified distance. Note that the distance indicates the difference in the channel domain. 3. The co-site constraint (CSC): Any pair of channels in a cell must have a specified distance. This distance is usually larger than necessary distance for ACC. The channel assignment problem (CAP) is then to allocate channel to every requested call in a cellular radio network subject to the above three EMC constraints such that the required bandwidth is minimized. Channel assignment is generally classified into fixed and dynamic. In fixed channel assignment (FCA), channels are nominally assigned to cells in advance according to the predetermined estimated traffic intensity. In dynamic channel assignment (DCA), channels are assigned dynamically as calls arrive. The latter method makes cellular systems more efficient particularly if the traffic distribution is unknown or changes with time, but has the disadvantage of requiring more complex control and is generally time consuming [2]. Normally, DCA gives better performance than FCA except under heavy traffic load condition, where FCA outperforms DCA [3]. Since heavy traffic load is expected in the future generation of cellular radio networks, an efficient FCA scheme that can provide high spectrum usage efficiency is desired. The FCA problem has been studied extensively for the past three decades. A comprehensive summary of the work done before 1980 can be found in [4]. Various extensions or combinations of the above two schemes have been discussed in the literature. The most basic are the hybrid channel assignment (HCA) and the borrowing channel assignment (BCA). In HCA, the set of channels of the cellular system is divided into two subsets, from which the one uses FCA and the other DCA. In BCA , the channel assignment is initially fixed. If there are incoming calls for a cell whose channels are all occupied, the cell borrows channels from its neighboring cells and thus call blocking is prevented [5,6].

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In the simplest form of the channel assignment problem, the co-channel constraint only is considered, and the problem is known to be equivalent to a graph coloring problem [4]. Since the graph coloring problem is known to be non-deterministic polynomial-complete (NP-complete) [7], therefore, CAP is also NP-complete. So, the calculation time and the computation complexity of searching for the optimum solution in the channel assignment problem grow exponentially with the problem size. The rest of the paper is organized as follows. In Section 2, we review existing solution approaches for the channel assignment problem. In Section 3, we formulate the channel assignment problem. In Section 4, we discuss two typical problems, in which dynamic programming has been extensively used. In Section 5, we introduce modified dynamic programming based on dynamic programming represented in Section 4. In Section 6, we assess the quality of MDP method by using it to solve seven well-known benchmark channel assignment problems and then compare the results with three of the existing channel assignment algorithms. Finally, we conclude our work in Section 7.

2. Previous work Many researchers have investigated the channel assignment problem using graph-theoretic methods, heuristic approaches, and various other optimization procedures. The algorithms that have been developed can be divided into two classes: non-iterative algorithms and iterative algorithms. Most of the non-iterative graph theoretical techniques are based on the sequential assignment of channels according to a heuristic ranking representing the local difficulty of assignment. In 1977, Zoellner and Beall [8] proposed an algorithm with considering co-channel constraint that ranks cells in node degree order or node coloring order and assigns frequencies using a frequency exhaustive strategy or a requirement exhaustive strategy. The four possible combinations of cell ordering procedure and assignment strategy yield four different versions of the algorithm. In 1978, Box [9] proposed a simple iterative technique for the channel assignment problem based on a ranking of the channel requirements of various cells in decreasing order of their assignment difficulty (order of assignment difficulty is a measure of how hard it is to find a compatible frequency to satisfy a given channel requirement). In his algorithm, the cells are first sorted in alphabetical order and the channels are sequentially assigned until a denial occurs. Subsequently, the cells are resorted in the order of increasing difficulty, and a reassignment is made until no further denial occurs. The assignment order is changed when denials occur, and frequencies are assigned to each cell based on the assignment difficulty during each iteration. In 1980, Hale [4] presented a wide collection of different versions of channel assignment problems in radio and television fields. In 1982, Gamst and Rave [10] summarized four existing sequential approximation algorithms. The first algorithm has four different versions by combining two different assignment strategiesthe frequency exhaustive assignment and the requirement exhaustive assignment, and two different ordering strategies-the node degree ordering and the node color ordering. The second algorithm repeatedly assigns frequencies according to the assignment difficulty of requirements.

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The third algorithm uses the heuristic geometric principle of maximum overlap of denial areas. It states that a frequency should be assigned to cell whose denial area has the maximum overlap with the existing denial area of that frequency. The fourth algorithm is based on the graph theory, where the clique number plays a key role. In 1982, Gamst [11] developed a theory on the optimal distance (in frequency terms) among adjacent channels for a homogeneous system of hexagonal cells. In 1986, Gamst [12] proposed procedures to generate lower bounds for a class of channel assignment problems. In 1989, Sivarajan et al. [13] developed eight algorithms based on ranking calls using row wise ordering or column wise ordering and ranking cells using node degree ordering or node color ordering and assigning frequencies using frequency exhaustive strategy or requirement exhaustive strategy. In 1991, Sengoku et al. [14] formulated a channel offset system design using a graph theoretical concept in a cellular mobile system. From the graph theoretical considerations, they investigated an optimal channel offset scheme in a cellular mobile system and gave an upper and a lower bound for overall spectral bandwidth. In 1991, Gamst [15] presented some graph theoretical planning techniques which had been employed in the design of the German D1 network, a GSM PLMN operated by the Deutsche Bundespost Telekom. Those techniques dealt with the determination of base station identity code (BSIC), hopping sequence number (HSN) and location area code (LAC). In 1994, Kim and Kim [16] proposed an efficient two phase optimization procedure for the channel assignment problem based on the notion of frequency reuse pattern. Their method for nine channel assignment benchmark problems showed some improvement compared to the channel assignment algorithms described in [13]. In 1994, Ko [17] presented a frequency exhaustive insertion strategy and a frequency selective insertion strategy for FCA in cellular networks. In his paper, with the insertion strategies, an improvement of over 10%; had been shown in many cases in comparison with the traditional frequency exhaustive strategy. The presented algorithms using the insertion strategies had yielded optimal assignments in some cases and had shown better performance than many other FCA algorithms in configuring real GSM systems. In 1994, Kishi et al. [18] proposed a unified approach for channel assignment of any macro and micro cellular networks having irregular cell shapes and non-uniform traffic distribution. Their approach coped with network evolution such as growth of traffic and service area as well as initial channel assignment and it also was effective for spectrum conservation in flexible assignment for time variant traffic and provided a guideline on frequency utilization in dynamic assignment. They presented efficient algorithms based on combinatorial optimization, whose excellent performance was demonstrated by simulation results. Their algorithms provide optimum or quasi-optimum solution with the computing time of polynomial order. In 1996, Wang and Rushforth [19] proposed an efficient two phase adaptive local search algorithm for the channel assignment problem which is a special case of simulated annealing approach. Their algorithm in phase I, searches the neighborhood of the current solution point Xp in the feasible region R for a better solution. At the next point, if a solution Xp+1 is found such that the function f(Xp+1) < f(Xp) then Xp+1 becomes the new solution point. Also in phase II, it searches the neighborhood of the current solution point Xp in the feasible region R for a

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better solution. At the next point, if a solution Xp+1 is found such that the function f(Xp+1) 6 f(Xp) then Xp+1 becomes the new solution point. In 1996, Jordan and Schabe [20] proposed some new metrics for measuring the performance of various channel assignment techniques. In 1997, Leese [21] considered the channel assignment problem on a lattice of hexagonal cells, with the allowed assignments generated by regular tilling of a single poly-hexagonal. He investigated the interplay between the co-channel and adjacent channel separations without restricting the assignments unnecessarily. In his approach, the possible co-channel lattices are carefully classified, and then an algorithm is developed, which assigns on any co-channel lattice so that the adjacent channel separation is maximized. In 1997, Tcha et al. [22] proposed a new lower bound on the number of frequencies required to meet the frequency demands in a cellular radio network and thus improved previous results given by Gamst [12] by devising a procedure of frequency insertion, which makes the best of unexpected frequency spaces between the assigned frequencies. They claimed that their lower bound have much wider and easier real world applicability due to its relaxed prerequisite condition. In 1997, Sen et al. [23] presented new lower and upper bounds in a particular homogeneous environment with the same number of calls in each cell. In 1999, Rouskas et al. [24] considered the problem of minimizing the span of frequencies required to satisfy a certain demand in a cellular radio network under certain interference constraints and presented a new iterative algorithm. Their algorithm has the ability to react to variations of the traffic demand as more and more channels are being assigned to cell requirement. Also in their paper, allocations of channels to cells are made with a method that borrows insight from the theory of convex maximization. That method is, however, equivalent to simple and fast heuristics when selecting proper values for its parameters. In 2001, Battiti et al. [25] investigated the problem of assigning channels to the cells of a cellular radio network so as to avoid interference and minimize the number of channels used. They formulated the problem as a generalization of the graph-coloring problem and considered the saturation degree heuristic, which was already successfully used for code assignment in packet radio networks. They gave a new version of this heuristic technique for cellular radio networks, called randomized saturation degree (RSD) that based on node ordering and randomization and improved the solution given by RSD by means of a local search technique. In 2001, Chakraborty [26] proposed an efficient heuristic algorithm for channel assignment problem in cellular radio networks that generate a population of random valid solutions of the problem, during which the optimization criterion of minimizing the bandwidth is not given any attention. Finally, if the best of all solutions is selected, there is a high probability that the optimum or near optimum solution is obtained. In 2002, Fernando and Fapojuwo [27] proposed a new channel assignment algorithm, called the viterbi liked algorithm (VLA) to solve the channel assignment problem in cellular radio networks. The basic idea of the proposed algorithm is step-by-step (sequential) channel assignment with the objectives of minimum bandwidth required at every step, subject to adjacent channel and co-channel constraints. Neural network algorithms have been proposed several researchers to solve the channel assignment problem. In 1990, Kunz [28] investigated a practical channel assignment scheme with three constraints including adjacent channel constraint.

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In 1991, Kunz [29] used Hopfield and TankÕs model for each neuron in the network, and creating one neuron for each channel i at each cell. An energy cost function representing co-channel and co-site constraints and channel demand is formulated and is then minimized by the neural network algorithm. In 1992, Funabiki and Takefuji [30] improved KunzÕs results by developing a neural network parallel algorithm incorporating co-channel and adjacent channel and co-site constraints. The proposed parallel algorithm is based on a two-dimensional artificial neural network composed of nm processing elements for an n-cell m-channel problem. In 1994, Chan et al. [31] used a feed forward neural network, which had a learning process prior to actual channel assignment and considered only co-channel constraint. For the learning process, they used training data that was dependently obtained by other assignment methods. The performance of their algorithm is totally dependent on the used training data. In 1995, Lochtie and Mehler [32] proposed a parallel algorithm which locates the solutions within a valid subspace situated in the unit hypercube whilst minimizing an energy function. Their algorithm had been extended from their previous work [33] to handle planning problems with arbitrary adjacent channel interference by the introduction of a modified connection matrix. In 1997, Kim et al. [34] proposed a new channel assignment algorithm using a modified discrete Hopfield neural network in which the channel assignment problem is formulated as an energy minimization problem. Also, a new technique to escape local minima is introduced. In their algorithm, an energy function is derived, and the appropriate interconnection weights between the neurons are designed in such a way that each neuron receives inhibitory support if the constraint conditions are violated and receives excitatory support if the constraint conditions are satisfied. They considered co-channel, adjacent channel, and co-site constraints. In 1997, Smith and Palaniswami [35] formulated a new non-linear integer programming representation of the FCA problem. They proposed two different neural networks for solving this problem. The first neural network was an improved Hopfield neural network, which resolves the issues of infeasibility, and poor solution quality, which have plagued the reputation of Hopfield network. The second neural network was a new self-organizing neural network, which is able to solve FCA problem and many other practical optimization problems due to its generalizing ability. In 1997, Sung and Wong [36] proposed a generalized sequential packing (GSP) algorithm with two variations for CAP, and a lower bound on the minimum number of channels. In 2000, Funabiki et al. [37] proposed a three-stage algorithm of combining sequential heuristic methods into a parallel neural network for the channel assignment problem in cellular mobile communication systems. The three-stage algorithm consists of 1. the regular interval assignment stage, 2. the greedy assignment stage, and 3. the neural network assignment stage. In the first stage, the calls in a cell determining the lower bound on the total number of channels are assigned channels at regular intervals. In the second stage, the calls in a cell with the largest degree and its adjacent cells are assigned channels by a greedy heuristic method. In the third stage, the calls in the remaining cells are assigned channels by a binary neural network.

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A much more powerful approach to cope with the problem of local minima consists in the application of genetic algorithms. Consequently, some authors proposed to minimize the cost function this way. In 1994, Cuppini [38] proposed a genetic algorithm for the channel assignment problem based on a binary solution representation. In 1995, Dorne and Hao [39] presented a study of genetic algorithms for the channel assignment problem in cellular radio networks. They took a progressive approach. First, they studied separately the different components of genetic algorithms in order to understand the interest of each of them for CAP; then, they designed hybrid genetic algorithms, which integrate efficient techniques (local search, constraint programming) into genetic operators. They considered adjacent channel and co-site constraints. In 1996, Lai and Coghill [40] proposed a genetic algorithm to solve channel assignment problem. Their algorithm avoids many of the shortcomings exhibited by local search techniques on different search spaces. A major advantage of their method, besides its elegance and simplicity, is the large speed-ups through implicit parallelism. In addition, new rules may easily be added without imposing the extra computational burden of checking for their consistency with the existing rules. In 1996, Kim et al. [41] proposed genetic algorithms for channel assignment problem in cellular radio networks. They represented possible channel assignments as a string of channel numbers. In 1998, Ngo and Li [42] formulated the problem of channel assignment by assuming a given channel span. Their objective was to obtain a conflict free channel assignment among the cells, which satisfies both the electromagnetic compatibility (EMC) constraints and traffic demand requirements. They proposed an approach based on a modified genetic algorithm. Their approach consists of a genetic fix algorithm that generates and manipulates individuals with fixed size and a minimum separation-encoding scheme that eliminates redundant zeros in the solution representation. Using these two strategies, the search space can be reduced substantially. In 1998, Smith [43] proposed a genetic algorithm approach to the channel assignment problem in cellular telephone networks. She used an alternative representation of the solution, which, together with appropriate definitions of the crossover and mutation operators, enables feasibility of the solutions to be guaranteed. Furthermore, her definition of these operators helps to provide insights into the unique roles that they play in the optimization process. In 1999, Beckmann and Kilat [44] proposed a new powerful approach to the channel assignment problem by combining the frequency exhaustive strategy and genetic algorithm. The power of their method bases especially on the ability to cope with the problem of local minima during the optimization process. Additionally, their algorithm always produces solutions which do not violate any given interference constraints. Over recent years, simulated annealing has been used to solve the channel assignment problem. In simulated annealing, a cost function representing co-channel, adjacent channel and co-site constraints and the channel demand is formulated. The cost function reaches its minimum of zero if all constraints are satisfied. In 1993, Duque-Anto´n et al. [45] proposed simulated annealing for practical radio network planning. One major benefit of their approach consists in the enhanced flexibility it gives to the engineer.

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In 1993, Mathar and Mattfeldt [46] investigated algorithms based on simulated annealing to solve the channel assignment problem for cellular radio networks. The block probability of a network is chosen as the optimization criterion. In 1995, Hurley and Smith [47] proposed a simulated annealing algorithm for the channel assignment problem subject to co-channel, adjacent channel and co-site constraints.

3. Channel assignment problem formulation CAP in this paper follows the problem formulation by Gamst and Rave [10] as in existing papers [9,12,13,16,19,29,30,40,45,46,48]. In 1982, Gamst and Rave [10] defined the general form of the channel assignment problem in an arbitrary inhomogeneous cellular radio network. In their definition, the electromagnetic compatibility constraints in an n-cell network are described by a n · n symmetric matrix which is called compatibility matrix C. Each non-diagonal element cij in C represents the minimum separation distance in the frequency domain between a frequency assigned to cell #i and a frequency to cell #j. The co-channel constraint is represented by cij = 1, and the adjacent channel constraint is represented by cij = 2. cij = 0 indicates that cells #i and #j are allowed to use the same frequency. Each diagonal element cii in C represents the minimum separation distance between any two frequencies assigned to cell #i, which is the co-site constraint, where cij P 1 is always satisfied. The channel requirements for each cell in an n-cell network are described by an n-element vector, which is called demand vector D. Each element di in D represents the number of frequencies to be assigned to cell #i. When fik indicates the kth frequency assigned to cell #i, the electromagnetic compatibility constraints are represented by jfik  fj‘ j P cij ;

i ¼ 1; . . . ; n; j ¼ 1; . . . ; n; k ¼ 1; . . . ; d i ; ‘ ¼ 1; . . . ; d j ; i 6¼ j; k 6¼ ‘.

ð1Þ

The channel assignment problem in cellular radio network is to find a conflict-free channel assignment, i.e., the fikÕs, such that the bandwidth required by the system, that is maxi,k{fik}, is minimized. In addition to constraint matrix C and demand vector D, we also consider another important parameter called lower bound (lb) in the formulation of channel assignment problem. Parameter lb determines minimum value of the maximum fik for all i and k, so that no interference is caused (i.e., lb = min{maxi,k{fik}}). This means if fikÕs can take values between 1 and lb, the values of fikÕs will not violate any constraints and a conflict-free channel assignment will be obtained. In fact, if we define the difference of the minimum fik and the maximum fik as bandwidth or span, lb will indicate the minimum of the necessary bandwidth for channel assignment and if smaller bandwidth is used, interference will be caused certainly and some constraints will be violated. Consider a channel assignment problem in a four-cell network in (2). Eq. (2) shows the constraint matrix C and the demand vector D. For example, a frequency within distance 4 from the frequency assigned to cell #1 cannot be assigned to cell #2 because of c12 = c21 = 4. Also any two frequencies assigned to cell #4 must have at least distance 5 because of c44 = 5. The minimum number of total frequencies in this problem is 11 because cell #4 requires at least 11 (1 + 5 · 2), frequencies. In this example, lb is equal to 11 and bandwidth is equal to 10 (11  1).

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2

3

5

4

0

0

64 6 C1 ¼ 6 40

5

0

0 1

5 2

17 7 7; 25

0

5

2 3 1 617 6 7 D1 ¼ 6 7. 415

311

ð2Þ

3

4. Dynamic programming Dynamic programming (DP) is an extensively studied and widely used tool in operation research for solving sequential decision problems [49]. We discuss two typical problems in which dynamic programming has been extensively used. The first problem is an optimal path problem that can be stated as follows. Consider a set of points labelled from 1 to N. Associated with every pair of points (i, j) is a non-negative cost f(i, j) that represents the cost of moving directly from the ith point to the jth point in one step. The problem is to find the minimum cost, as well as the corresponding sequence of moves from point i in the set to another point j using as many steps as needed. This problem is illustrated in Fig. 1. Since the sequence of moves has an unspecified number of transitions (steps), from one point to another, we call this an asynchronous sequential decision problem. Using traditional terminology, we call the decision rule for determining the next point to be visited after point i a ‘‘policy’’. Since the policy determines the sequence of points traversed from the (fixed) originating point 1 to the destination point i, the cost is therefore completely defined by the policy and the destination point i. The question is what policy leads to the minimum cost, moving from point 1 to point i. we denote this minimum cost by u(1, i). The principle of optimality, which is the basis of a class of computational algorithms for the above optimization problem, is according to Bellman [50]. An optimal policy has the property that, whatever the initial state and decision are the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.

Fig. 1. The optimal path problem—Finding the minimum cost path from point 1 to point i in as many moves as needed.

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To put BellmanÕs principle of optimality into a functional equation suitable for computational algorithms, consider first moving from the initial point 1 to an intermediate point j in one or more steps. The minimum cost, as defined, is u(1, j). Since moving from point j to point i in one step incurs a cost f(j, i), the optimal policy, which determines which intermediate point (j) to pass through (should one exist) satisfies the following equation: uð1; iÞ ¼ min½uð1; jÞ þ fðj; iÞ .

ð3Þ

j

Generalizing (3) to the case in which we are interested in obtaining the optimal sequence of moves and the associated minimum cost from any point i to any other point j, we have uði; jÞ ¼ min½uði; ‘Þ þ fð‘; jÞ ;

ð4Þ

‘

where u(i, j) is the minimum cost from i to j in as many steps as necessary. Eq. (4) implies that any partial, consequence of moves of the optimal sequence from i to j must also be optimal, and that any intermediate point must be the optimal point linking the optimal partial sequences before and after that point. To actually determine the minimum cost path between points i and j, in any number of steps, the following simple dynamic program would be used: u1 ði; ‘Þ ¼ fði; ‘Þ; u2 ði; ‘Þ ¼ min½u1 ði; kÞ þ fðk; ‘Þ ;

k; ‘ ¼ 1; 2; . . . ; N;

u3 ði; ‘Þ ¼ min½u2 ði; kÞ þ fðk; ‘Þ ;

k; ‘ ¼ 1; 2; . . . ; N;

k k

.. . us ði; ‘Þ ¼ min½us1 ði; kÞ þ fðk; ‘Þ ; k

k; ‘ ¼ 1; 2; . . . ; N;

uði; ‘Þ ¼ min us ði; ‘Þ; 16s6S

where us(i, ‘) is the s-step best path from point i to point ‘, and S is the maximum number of steps allowed in the path. A second dynamic programming problem is the synchronous sequential decision problem, which differs from the asynchronous one in the regularity of the decision process. In terms of the optimal path problem, the objective now is to find the optimal sequence of a fixed number, say M, of moves, starting from point i an ending at point k, and the associated minimum cost, uM(i, k). The regularity of the problem can best be explained with the Trellis structure in Fig. 2. The N points are plotted vertically and the M transitions progress horizontally to the right in Fig. 2. Since there are N possible moves for each point, at every moment, the total number of one-step moves is thus N2. Furthermore, the total number of sequences of moves, which will be called ‘‘paths’’, connecting point i at the beginning of the move and point k at the end of Mth move, is NM1. The principle of optimality is equally applicable in this case. After the mth move, m < M, the path can end up at any point ‘, ‘ = 1, 2, . . . , N, with the associated minimum cost um(i, ‘). Suppose the (m + 1)th move is to go to point j. Then, similar to (3), um+1(i, j) satisfies

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Fig. 2. A Trellis structure that illustrates the problem of finding the optimal path from point i to point j in M steps.

umþ1 ði; jÞ ¼ min½um ði; ‘Þ þ fð‘; jÞ .

ð5Þ

‘

Eq. (5) describes a recursion that allows the optimal path search to be conducted incrementally, in a progressive manner. Although there are N possible moves that end at point ‘, the optimality principle indicates that only the best move is necessary to be considered according to (5). The algorithm can be summarized as follows: 1. Initialization u1 ði; jÞ ¼ fði; jÞ;

ð6Þ

n1 ðjÞ ¼ i;

ð7Þ

1 6 j 6 N.

2. Recursion umþ1 ði; jÞ ¼ min ½um ði; ‘Þ þ fð‘; jÞ ; 16‘6N

nmþ1 ðjÞ ¼ arg min ½um ði; ‘Þ þ fð‘; jÞ ; 16‘6N

1 6 m 6 M  2; 1 6 j 6 N ; 1 6 m 6 M  2; 1 6 j 6 N .

ð8Þ ð9Þ

3. Termination uM ði; kÞ ¼ min ½uM1 ði; ‘Þ þ fð‘; kÞ ;

ð10Þ

nM ðkÞ ¼ arg min ½uM1 ði; ‘Þ þ fð‘; kÞ .

ð11Þ

16‘6N

16‘6N

4. Path backtracking Optimal path ¼ ði; i1 ; i2 ; . . . ; iM1 ; kÞ

ð12Þ

where im ¼ nmþ1 ðimþ1 Þ; with iM = k.

M 1PmP1

ð13Þ

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5. Modified dynamic programming We provide the necessary modifications in synchronous version of dynamic programming to make it applicable for the channel assignment problem. The new algorithm is named modified dynamic programming (MDP). The modifications are as follows: 1. i = 1. 2. Initialization u1 ði; jÞ ¼ fði; jÞ;

j ¼ 1; 2; . . . ; N ;

ð14Þ

n1 ðjÞ ¼ i.

ð15Þ

3. Recursion and termination " umþ1 ði; jÞ ¼ min um ði; ‘Þ þ fð‘; jÞ þ 16‘6N

1 X

# fðnk ð‘Þ; jÞ ;

k¼m1

"

nmþ1 ðjÞ ¼ arg min um ði; ‘Þ þ fð‘; jÞ þ 16‘6N

1 X

1 6 j 6 N; 1 6 m 6 M  1;

ð16Þ

# fðnk ð‘Þ; jÞ ;

1 6 j 6 N ; 1 6 m 6 M  1.

k¼m1

ð17Þ 4. Path backtracking jth path ¼ ði0 ; i1 ; . . . ; iM1 ; iM Þ;

1 6 j 6 N;

ð18Þ

where im ¼ nmþ1 ðimþ1 Þ;

M  1 P m P 0;

ð19Þ

with i0 = i, iM = j, then optimal paths are a group of jth Paths (j = 1, 2, . . . , N) for which the condition max06k6M(ik) 6 lb is satisfied. 5. Repeat steps 2–5 for i = 2, 3, . . . , N. According to CAP formulation which is mentioned in Section 3, we have a channel assignment problem composed of an n-cell network which itÕs constraint matrix Cn·n, demand vector Dn·1 and lb are given. Since MDP uses Trellis structure (Fig. 2), therefore, we should consider a column in the Trellis structure for each channel and because element dP i of the demand vector D shows the number of n the necessary channels in cell i, therefore, term i¼1 d i shows the total number of the required channels in a channel assignment problem. Consequently, the Trellis structure should have Pn d columns. On the other hand, i¼1 i Pas mentioned in Section 4, because this structure has M + 1 columns, so, we have M þ 1 ¼ ni¼1 d i . Thus, M variables {u1, u2, . . . , uM} and M variables {n1, n2, . . . , nM} should be calculated. For example, if in a 4-cell network, the number of the channels in each cell is specified by the demand vector DT = (1, 1, 1, 3), then 4 X d i ¼ 1 þ 1 þ 1 þ 3 ¼ 6 ¼ M þ 1 ) M ¼ 5; i¼1

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i.e., the Trellis structure should have six columns. In this case, we have totally six channels which we show them by six variables (f11, f21, f31, f41, f42, f43). Because each column shows a channel, therefore, from left to right in the Trellis structure, channels of the first cell are located first and channels of the second cell come after and so on. In the previous example, from left to right, the first column shows channel 1 in cell 1 (f11), the second column shows channel 1 in cell 2 (f21), the third column shows channel 1 in cell 3 (f31), the fourth column shows channel 1 in cell 4 (f41), the fifth column shows channel 2 in cell 4 (f42), the sixth column shows channel 3 in cell 4 (f43). In accordance to the Trellis structure in Fig. 2, each channel (fik) can take values from 1 to N and also with regard to CAP formulation and definition of parameter lb, each channel can take values from 1 to lb, therefore, N = lb, i.e., in each column of the Trellis structure lb nodes or values exist. Function f(k, l) is used for applying adjacent channel, co-channel and co-site constraints to channel assignment problem. This function should be defined in such a way that prevents choosing of frequencies that cause interference. According to (1), we define this function as following:  jk  ‘j P cij ; jk  ‘j  cij ; ð20Þ fðk; ‘Þ ¼ ðcij  jk  ‘jÞ  lb; jk  ‘j < cij . In fact with this definition, the value of function f(k, ‘) is greater or equal to zero and this function compares the absolute value of frequency difference between two channels in cell i and cell j; i.e., (jk  ‘j), with cij. If this absolute value (jk  ‘j) is greater or equal to cij, it means these two channels do not have any interference with each other and jk  ‘j  cij is allocated to f(k, ‘). If the absolute value is smaller than cij, it means these two channels have interference with each other, consequently, the bigger value (cij  jk  ‘j) · lb is allocated to f(k, ‘). So that according to (16), h i P1 when finding the minimum of um ði; ‘Þ þ fð‘; jÞ þ k¼m1 fðnk ð‘Þ; jÞ , summation of these three terms becomes a big value and because of causing interference the corresponding frequency is not selected. The most important modification that should be applied to dynamic programming formulation to transform it to MDP and also make DP usable for channel assignment problems is addition of P term 1k¼m1 fðnk ð‘Þ; jÞ. By adding this term, frequency interference of the current channel with all previous channels are calculated and added together. For calculating the frequency of each channel, function f(k, ‘) and path backtracking formula are used. 6. Simulation results To test MDP algorithm and compare its performance with the three existing approaches (heuristic method proposed by Sivarajan et al. [13], neural network parallel algorithm proposed by Funabiki and Takefuji [30] and adaptive local search algorithm proposed by Wang and Rushforth [19]), we used seven well-known benchmarks in channel assignment problems. We have implemented the MDP algorithm in C language. This program was run on a PC with a Pentium 4 (3.2 GHz) CPU.

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Problem 1 is given in most references ([13,19,30,26], etc.) as a simple example and it includes a 4-cell network with 6 channels. Constraints matrix and the demand vector for problem are shown in (2) and lb value for this problem is 11. Problem 2 is a practical channel assignment problem taken from [29,30], in Helsinki, Finland that is composed from a 25-cell network with 167 channels. Constraint matrix and demand vector of this problem are given in (21) and its lb value is 73. 2

2 1 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0

6 61 6 6 61 6 6 60 6 6 61 6 6 60 6 6 61 6 6 61 6 6 61 6 6 61 6 6 60 6 6 61 6 6 C2 ¼ 6 61 6 61 6 6 61 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 4

2 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 2 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 2 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 0 0 0 0 0 1 0 1 0 1 1 1 0 1 1 1 1 2 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 2 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 2 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 2 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 2 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 1 1 2 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 2 1 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1

3

2 3 10 7 07 7 6 7 6 11 7 7 7 6 07 6 7 7 7 6 9 7 6 7 17 7 6 7 657 7 7 6 07 6 7 7 7 6 9 7 6 7 07 7 6 7 647 7 7 6 07 6 7 7 7 6 5 7 6 7 07 7 6 7 677 7 7 6 17 6 7 7 7 6 4 7 6 7 07 7 6 7 687 7 7 6 17 6 7 7 687 7 6 7 07 6 7 7 697 7 6 7 7 0 7; D2 ¼ 6 7 6 10 7 7 6 7 07 6 7 7 677 7 6 7 7 07 6 7 677 7 6 7 07 6 7 7 667 7 6 7 6 7 07 7 647 7 6 7 6 7 07 7 657 7 6 7 6 7 07 7 657 7 6 7 7 6 7 07 677 7 6 7 6 7 07 7 667 7 6 7 7 6 7 17 6 45 7 7 6 7 6 7 17 7 677 7 4 5 17 5 5 2 ð21Þ

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317

Problems 3 and 4 are 21-cell networks with 481 channels which their cell structure is shown in Fig. 3. These problems are taken from [13,30]. Constraint matrices and demand vectors of these problems are given in (22) and (23), respectively. lb value for problem 3 is 381 and for problem 4 is 533. 2

5 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0

61 6 6 61 6 6 60 6 60 6 6 61 6 6 61 6 6 61 6 61 6 6 60 6 6 C3 ¼ 6 0 6 60 6 6 60 6 6 61 6 61 6 6 61 6 6 60 6 60 6 6 60 6 6 40

3

2

8

3

6 25 7 5 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 07 7 6 7 7 6 7 7 687 1 5 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 07 6 7 7 6 7 687 1 1 5 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 07 7 6 7 7 687 0 1 1 5 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 07 6 7 7 6 7 7 6 15 7 0 0 0 0 5 1 1 0 0 0 0 1 1 1 0 0 0 0 0 07 6 7 7 6 7 6 18 7 1 0 0 0 1 5 1 1 0 0 0 1 1 1 1 0 0 1 0 07 7 6 7 7 6 7 1 1 0 0 1 1 5 1 1 0 0 0 1 1 1 1 0 1 1 07 6 52 7 7 6 7 7 6 77 7 1 1 1 0 0 1 1 5 1 1 0 0 0 1 1 1 1 1 1 17 6 7 7 6 7 7 6 28 7 1 1 1 1 0 0 1 1 5 1 1 0 0 0 1 1 1 0 1 17 6 7 7 6 7 0 1 1 1 0 0 0 1 1 5 1 0 0 0 0 1 1 0 0 1 7; D3 ¼ 6 13 7 7 6 7 6 15 7 0 0 1 1 0 0 0 0 1 1 5 0 0 0 0 0 1 0 0 07 7 6 7 7 6 7 7 6 31 7 0 0 0 0 1 1 0 0 0 0 0 5 1 1 0 0 0 0 0 07 6 7 7 6 7 0 0 0 0 1 1 1 0 0 0 0 1 5 1 1 0 0 1 0 07 6 15 7 7 6 7 7 6 36 7 1 0 0 0 1 1 1 1 0 0 0 1 1 5 1 1 0 1 1 07 6 7 7 6 7 7 6 57 7 1 1 0 0 0 1 1 1 1 0 0 0 1 1 5 1 1 1 1 17 6 7 7 6 7 6 28 7 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 5 1 1 1 17 7 6 7 7 687 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 5 0 1 17 6 7 7 6 7 7 6 10 7 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 5 1 17 6 7 7 6 7 4 13 5 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 5 15

0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 5

8 ð22Þ

1 7

6 13

14

3

2 9

8 15

16 19

10

11

12

18

17 20

5

4

21

Fig. 3. The 21-cell network structure problems 3–7.

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2

2

7 6 6 61 6 6 61 6 6 6 60 6 6 6 60 6 6 61 6 6 6 61 6 6 6 61 6 C4 ¼ 6 61 6 6 6 60 6 6 6 60 6 6 60 6 6 6 60 6 6 6 61 6 6 61 6 4 1

1 1 0 0 1 1 1 1 0 0 0 0 7 1 1 0 0 1 1 1 1 0 0 0 1 7 1 1 0 0 1 1 1 1 0 0 1 1 7 1 0 0 0 1 1 1 1 0 0 1 1 7 0 0 0 0 1 1 1 0 0 0 0 0 7 1 1 0 0 0 0 1 1 0 0 0 1 7 1 1 0 0 0 1 1 1 0 0 1 1 7 1 1 0 0 0 1 1 1 0 0 1 1 7 1 1 0 0 1 1 1 1 0 0 1 1 7 1 1 0 0 1 1 1 0 0 0 1 1 7 1 0 0 0 1 1 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 7 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0

8

3

6 7 6 25 7 6 7 6 7 6 7 687 6 7 3 6 7 1 1 1 0 0 0 0 0 6 7 7 687 7 6 7 0 1 1 1 0 0 0 07 6 7 7 687 7 6 7 7 6 7 0 0 1 1 1 0 0 07 6 7 7 6 15 7 7 6 7 7 6 7 0 0 0 1 1 0 0 07 6 7 7 6 18 7 7 6 7 6 7 0 0 0 0 1 0 0 07 7 6 7 7 6 52 7 6 7 1 1 0 0 0 0 0 07 7 6 7 7 6 77 7 7 6 7 7 6 7 1 1 1 0 0 1 0 07 6 7 7 6 28 7 7 6 7 6 7 1 1 1 1 0 1 1 07 7 6 7 7; D4 ¼ 6 13 7 6 7 0 1 1 1 1 1 1 17 7 6 7 7 6 15 7 7 6 7 7 6 7 0 0 1 1 1 0 1 17 6 7 7 6 31 7 7 6 7 6 7 0 0 0 1 1 0 0 17 7 6 7 7 6 15 7 7 6 7 0 0 0 0 1 0 0 07 6 7 7 6 36 7 7 6 7 7 6 7 1 1 0 0 0 0 0 07 6 7 7 6 57 7 7 6 7 7 6 7 7 1 1 0 0 1 0 0 7 6 7 7 6 28 7 7 6 7 1 7 1 1 0 1 1 07 6 7 5 687 6 7 6 7 1 1 7 1 1 1 1 1 6 7 6 10 7 6 7 6 7 6 7 6 13 7 4 5 8 ð23Þ

Problem 5 is a 21-cell network with 470 channels. Its cell structure is shown in Fig. 3. This problem is taken from [30]. Its constraint matrix and demand vector are shown in (24) and its lb value is 221.

S.A.G. Shirazi, H. Amindavar / Computers and Electrical Engineering 31 (2005) 303–333

2

5 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0

6 61 6 6 6 61 6 6 6 60 6 6 60 6 6 6 61 6 6 6 61 6 6 6 61 6 6 61 6 6 6 60 6 6 6 C5 ¼ 6 0 6 6 60 6 6 6 60 6 6 6 61 6 6 61 6 6 6 61 6 6 6 60 6 6 60 6 6 6 60 6 6 6 60 4

3

319

2

5

3

7 6 7 657 5 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 07 7 6 7 7 6 7 7 6 7 7 657 1 5 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 07 6 7 7 6 7 7 6 7 687 1 1 5 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 07 7 6 7 7 6 7 6 12 7 0 1 1 5 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 07 7 6 7 7 6 7 7 6 7 7 6 25 7 0 0 0 0 5 1 1 0 0 0 0 1 1 1 0 0 0 0 0 07 6 7 7 6 7 7 6 7 6 30 7 1 0 0 0 1 5 1 1 0 0 0 1 1 1 1 0 0 1 0 07 7 6 7 7 6 7 7 6 7 1 1 0 0 1 1 5 1 1 0 0 0 1 1 1 1 0 1 1 07 6 25 7 7 6 7 7 6 7 7 6 30 7 1 1 1 0 0 1 1 5 1 1 0 0 0 1 1 1 1 1 1 17 6 7 7 6 7 7 6 7 7 6 40 7 1 1 1 1 0 0 1 1 5 1 1 0 0 0 1 1 1 0 1 17 6 7 7 6 7 7 6 7 0 1 1 1 0 0 0 1 1 5 1 0 0 0 0 1 1 0 0 1 7; D5 ¼ 6 40 7 7 6 7 7 6 7 7 6 45 7 0 0 1 1 0 0 0 0 1 1 5 0 0 0 0 0 1 0 0 07 6 7 7 6 7 7 6 7 7 6 20 7 0 0 0 0 1 1 0 0 0 0 0 5 1 1 0 0 0 0 0 07 6 7 7 6 7 7 6 7 0 0 0 0 1 1 1 0 0 0 0 1 5 1 1 0 0 1 0 07 6 30 7 7 6 7 7 6 7 6 25 7 1 0 0 0 1 1 1 1 0 0 0 1 1 5 1 1 0 1 1 07 7 6 7 7 6 7 7 6 7 7 6 15 7 1 1 0 0 0 1 1 1 1 0 0 0 1 1 5 1 1 1 1 17 6 7 7 6 7 7 6 7 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 5 1 1 1 17 6 15 7 7 6 7 7 6 7 7 6 30 7 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 5 0 1 17 6 7 7 6 7 7 6 7 7 6 20 7 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 5 1 17 6 7 7 6 7 7 6 7 7 6 20 7 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 5 1 5 4 5

0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 5

25 ð24Þ

Problem 6 which is taken from [13,30] is a 21-cell network with 470 channels. Its cell structure is shown in Fig. 3. Constraint matrix and demand vector for this problem are given in (25) and lb value for this problem is 309.

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2

7 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0

6 6 61 6 6 61 6 6 6 60 6 6 60 6 6 6 61 6 6 6 61 6 6 61 6 6 6 61 6 6 60 6 6 6 C6 ¼ 6 60 6 6 60 6 6 60 6 6 6 61 6 6 61 6 6 6 61 6 6 6 60 6 6 60 6 6 6 60 6 6 60 6 4 0

3

2

5

3

7 6 7 7 6 7 7 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 07 657 7 6 7 7 6 7 7 657 1 7 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 07 6 7 7 6 7 7 6 7 7 687 1 1 7 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 07 6 7 7 6 7 7 6 12 7 0 1 1 7 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 07 6 7 7 6 7 7 6 7 6 25 7 0 0 0 0 7 1 1 0 0 0 0 1 1 1 0 0 0 0 0 07 7 6 7 7 6 7 7 6 7 1 0 0 0 1 7 1 1 0 0 0 1 1 1 1 0 0 1 0 07 6 30 7 7 6 7 7 6 7 7 6 25 7 1 1 0 0 1 1 7 1 1 0 0 0 1 1 1 1 0 1 1 07 6 7 7 6 7 7 6 7 6 30 7 1 1 1 0 0 1 1 7 1 1 0 0 0 1 1 1 1 1 1 17 7 6 7 7 6 7 7 6 40 7 1 1 1 1 0 0 1 1 7 1 1 0 0 0 1 1 1 0 1 17 6 7 7 6 7 7 6 7 7 7 0 1 1 1 0 0 0 1 1 7 1 0 0 0 0 1 1 0 0 1 7; D6 ¼ 6 6 40 7 7 6 7 7 6 7 6 45 7 0 0 1 1 0 0 0 0 1 1 7 0 0 0 0 0 1 0 0 07 7 6 7 7 6 7 7 6 20 7 0 0 0 0 1 1 0 0 0 0 0 7 1 1 0 0 0 0 0 07 6 7 7 6 7 7 6 7 6 30 7 0 0 0 0 1 1 1 0 0 0 0 1 7 1 1 0 0 1 0 07 7 6 7 7 6 7 6 25 7 1 0 0 0 1 1 1 1 0 0 0 1 1 7 1 1 0 1 1 07 7 6 7 7 6 7 7 6 7 7 6 15 7 1 1 0 0 0 1 1 1 1 0 0 0 1 1 7 1 1 1 1 17 6 7 7 6 7 7 6 7 6 15 7 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 7 1 1 1 17 7 6 7 7 6 7 6 30 7 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 7 0 1 17 7 6 7 7 6 7 7 6 7 6 20 7 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 7 1 17 7 6 7 7 6 7 7 6 20 7 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 7 17 6 7 5 4 5 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 7 25 ð25Þ

Problem 7 that is taken from [13,51,52] is one of the practical Philadelphia benchmark problems and is composed of a 21-cell network with 481 channels. Its cell structure is given in Fig. 3. Constraint matrix and demand vector for this problem are shown in (26) and its lb value is 414.

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2

5 2 1 1 0 1 2 2 1 1 0 0 1 1 1 1 1 0 1 1 0

62 6 6 61 6 61 6 6 60 6 61 6 6 62 6 62 6 6 61 6 6 61 6 C7 ¼ 6 60 6 60 6 61 6 6 61 6 61 6 6 61 6 61 6 6 60 6 61 6 6 41

3

321

2

8

3

6 25 7 5 2 1 1 1 1 2 2 1 1 0 0 1 1 1 1 1 1 1 17 7 6 7 7 6 7 687 2 5 2 1 0 1 1 2 2 1 1 0 0 1 1 1 1 1 1 17 7 6 7 7 687 1 2 5 2 0 0 1 1 2 2 1 0 0 0 1 1 1 0 1 17 6 7 7 6 7 687 1 1 2 5 0 0 0 1 1 2 2 0 0 0 0 1 1 0 0 17 7 6 7 7 6 15 7 1 0 0 0 5 2 1 1 0 0 0 2 2 1 1 0 0 1 0 07 6 7 7 6 7 7 6 18 7 1 1 0 0 2 5 2 1 1 0 0 1 2 2 1 1 0 1 1 07 6 7 7 6 52 7 2 1 1 0 1 2 5 2 1 1 0 1 1 2 2 1 1 1 1 17 6 7 7 6 7 7 6 77 7 2 2 1 1 1 1 2 5 2 1 1 0 1 1 2 2 1 1 1 17 6 7 7 6 7 1 2 2 1 0 1 1 2 5 2 1 0 0 1 1 2 2 1 1 17 6 28 7 7 6 7 7 7 1 1 2 2 0 0 1 1 2 5 2 0 0 0 1 1 2 0 1 1 7; D7 ¼ 6 6 13 7 7 6 7 0 1 1 2 0 0 0 1 1 2 5 0 0 0 0 1 1 0 0 17 6 15 7 7 6 7 6 31 7 0 0 0 0 2 1 1 0 0 0 0 5 2 1 1 0 0 1 0 07 7 6 7 7 6 7 1 0 0 0 2 2 1 1 0 0 0 2 5 2 1 1 0 1 1 07 6 15 7 7 6 7 6 36 7 1 1 0 0 1 2 2 1 1 0 0 1 2 5 2 1 1 2 1 17 7 6 7 7 6 7 6 57 7 1 1 1 0 1 1 2 2 1 1 0 1 1 2 5 2 1 2 2 17 7 6 7 7 6 28 7 1 1 1 1 0 1 1 2 2 1 1 0 1 1 2 5 2 1 2 27 6 7 7 6 7 687 1 1 1 1 0 0 1 1 2 2 1 0 0 1 1 2 5 1 1 27 7 6 7 7 6 10 7 1 1 0 0 1 1 1 1 1 0 0 1 1 2 2 1 1 5 2 17 6 7 7 6 7 4 13 5 1 1 1 0 0 1 1 1 1 1 0 0 1 1 2 2 1 2 5 25

0 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 2 2 1 2 5

8 ð26Þ

Table 1 shows the results produced by MDP algorithm and three existing approaches for the seven stated problems. First and second columns of this table show problem number and number of cells, respectively. Total number of channels, constraint matrix C, demand vector D and lb for Table 1 Results obtained by four channel assignment algorithms for seven well-known benchmark channel assignment problems Problem number

Number of cells

Total number of channels

C

D

lb

MDP

[13]

[30]

[19]

Number of solutions

Average calculation time per solution (s)

1 2 3 4 5 6 7

4 25 21 21 21 21 21

6 167 481 481 470 470 481

C1 C2 C3 C4 C5 C6 C7

D1 D2 D3 D4 D5 D6 D7

11 73 381 533 221 309 414

11 73 381 533 221 309 427

11 – 381 533 – 310 460

11 73 381 533 221 309 –

11 73 381 533 221 309 433

17 781 2396 17,426 20 1979 5

0.01 0.5 40 55 33 47 45

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each problem are given in the third, fourth, fifth and sixth columns, respectively. The seventh, eighth, ninth and tenth columns show the maximum frequency obtained (maxi,k fik) by MDP algorithm, heuristic method [13], neural network algorithm [30] and adaptive local search algorithm [19], respectively. In the 11th column, number of obtained solutions by MDP algorithm for each problem is given and maximum frequency in these solutions is equal to lb. The 12th column shows average time needed for calculating one solution by MDP algorithm. As shown in Table 1, the maximum frequency (required bandwidth) obtained by MDP method is smaller or equal to the maximum frequency obtained by the other three methods. According to column 11 in Table 1, the MDP algorithm finds up to tens of thousands of solutions in some problems while neural network parallel algorithm [30] and adaptive local search algorithm [19] find only one solution and heuristic method [13] finds at most eight different solutions. This is one of the advantages of the MDP method. Neural network parallel [30] and adaptive local search [19] methods produce an initial random solution and then converge to an optimum or near optimum solution, i.e., they are dependent on an initial solution and they may not converge to an optimum solution while the MDP method is independent of any initial solution. This is another advantage of MDP algorithm. Another advantage of this method is that if the chosen value for lb is greater than the actual value for the lb (e.g., the chosen value for lb is 12 while the actual value of lb is 11 like problem 1), the MDP approach will find solutions that their maximum frequency is equal to the actual value for lb, i.e., the MDP algorithm is robust. Table 2 Derived channel assignment solution for problem 1 Cell number 1

2

3

4

1

5

4

1 6 11

Table 3 Derived channel assignment solution for problem 2 Cell number 1

2

3

4

5

1 4 6 8 10 12 14 16 18 20

19 21 23 25 27 29 31 33 35 37 39

22 24 26 28 30 32 34 36 38

15 3 17 5 19 7 21 9 23 11 13 15 17 40

6

7

8

9

10

11 12 13

14 15 16

17 18 19

20 21 22

23 24 25

23 25 37 39

5 7 9 11 13

3 40 42 44 46 48 50

29 31 33 35

41 43 45 47 49 51 53 55

12 14 16 18 20 25 27 37

42 44 46 48 50 70 72

37 39 41 43

40 43 55 62 64 66 68

5 4 1 7 6 3 9 8 28 11 10 30 13 22 32 24 26

52 54 56 58 60 62 64 66 68

2 57 59 61 63 65 67 69 71

26 28 30 32 34 36 38

22 24 29 31 33 35

45 47 49 51 53

52 54 56 58 60

24 29 31 33 35 39

17 19 21 23

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323

Table 4 Derived channel assignment solution for problem 3 Cell number 1

2

3

4

5

6

7

8

10 15 20 25 30 35 40 45

44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164

128 133 138 143 148 153 158 163

92 97 102 107 112 117 122 127

58 63 68 73 78 83 88 93

58 42 43 63 47 50 68 52 55 73 57 60 78 62 65 83 67 70 88 72 75 93 77 80 98 82 85 103 87 90 108 92 95 113 97 100 118 102 105 123 107 110 128 112 115 117 120 122 125 127 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270

9

10

11

12 13

14

15

16

17

18

19

20

21

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216 221 226

147 152 157 162 167 172 177 182 187 192 197 202 207 212 217 222 227 232 237 242 247 252 257 262 267 272 277 282

85 90 95 100 105 110 115 120 125 130 135 140 145

14 19 24 29 34 39 44 49 54 59 64 69 74 79 84

94 99 104 109 114 119 124 133 138 143 148 153 158 163 168

19 24 29 34 39 48 53 132 137 142 147 152 157 162 167 172 177 182 187 192 197 202 207 212 217 222 227 232 237 242 247 252 257 262 267 272

38 58 63 68 73 78 83 88 93 98 103 108 113 118 123 169 174 179 184 193 198 203 208 213 218 223 228 233 238 243 248 253 258 263 268 273 278 283 288 293 298 303 308 313 318 323

52 57 62 67 72 77 82 87 168 173 178 183 188 194 199 204 209 214 219 224 229 234 239 244 249 254 259 264

129 134 139 144 149 154 159 164

25 30 35 40 189 269 274 279 284 289

107 112 117 122 128 133 138 143 148 153 158 163 287

74 119 124 132 137 142 165 170

9 14 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216

(continued on next page)

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Table 4 (continued) Cell number 1

2

3

4

5

6

7

8

9

275 280 285 290 295 300

231 236 241 246 251 256 261 266 271 276 281 286 291 296 301 306 311 316 321 326 331 336 341 346 351 356 361 366 371 376 381

10

11

12

13

14

15

16

17

18

19

20

21

328 333 338 343 348 353 358 363 368 373 378

We can use this property to find the actual value for lb in a specific problem, in such a way that we run MDP with a given lb and then decrease lb value by 1. If MDP finds any solution, this means the given lb is not the actual value for the lb of that problem. We continue decreasing lb value until MDP does not find any solution. Therefore, the minimum value of lb for which MDP gives us a solution, is the actual value of lb for that problem. This is the fourth advantage of MDP. Another advantage of MDP algorithm is that not only it can be run on a personal computer also we can run it on a computer with several parallel processors. This means MDP has a parallel processing property. In fact, steps 2–5 in MDP method correspond to Eqs. (14)–(19) for different values of i can be done independently by different processors. By using this property, the computation time will be reduced by the number of processors, i.e., if a computer has two parallel processors, computation time will be reduced by half.

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325

Table 5 Derived channel assignment solution for problem 4 Cell number 1

2

5 53 12 60 19 67 26 74 33 81 40 88 47 95 54 102 109 116 123 130 137 144 151 158 165 172 179 186 193 200 207 214 221

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

171 178 185 192 199 206 213 220

121 128 135 142 149 156 163 170

73 80 87 94 101 108 115 122

73 80 87 94 101 108 115 122 129 136 143 150 157 164 171

51 58 65 72 79 86 93 100 107 114 121 128 135 142 149 156 163 170

9 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 182 189 196 203 210 217 224 231 238 245 252 259 266 273 280 287 294 301 308 315 322 329 336 343

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190 197 204 211 218 225 232 239 246 253 260 267 274 281 288 295

76 83 90 97 104 111 118 129 136 143 150 157 164 177 184 191 198 205 212 219 228 235 242 249 256 263 270 277

161 168 175 147 154 182 140 189 133 196 126 203 210

107 125 132 139 146 153 160 167 174 181 188 195 202 209 216

69 76 83 90 97 104 111 118 125 132 139 146 153 160 167 174 181 188 195 202 209 216 223 230 237 244 251 258 265 272 279

124 131 138 145 152 159 166 173 180 187 194 201 208 215 222

25 32 39 52 59 66 75 82 89 96 103 110 117 178 185 192 199 206 213 220 227 234 241 248 255 263 270 277 284 291 298 305 312 319 326 333

6 13 20 27 34 41 48 55 62 73 80 87 94 101 108 115 125 132 139 146 153 160 167 174 181 188 195 202 209 216 229 236 243 250 257 264 271 278 285 292 299 306 313

79 86 93 100 107 114 124 131 138 145 152 159 166 173 180 187 194 201 208 215 222 230 237 244 251 258 268 275

151 158 165 172 179 186 193 200

144 151 158 165 172 179 186 193 200 207

109 121 128 135 142 149 156 163 171 214 221 233 240

11 137 170 178 185 192 199 206

(continued on next page)

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Table 5 (continued) Cell number 1

2

3

4

5

6

7

8

9

350 357 364 371 378 385 392 399 406

302 309 316 323 330 337 344 351 358 365 372 379 386 393 400 407 414 421 428 435 442 449 456 463 470 477 484 491 498 505 512 519 526 533

10

11

12

13

14

15

16

17

18

19

20

21

320 327 334 341 348 355 362 369 376 383 390 397 404 411

Consequently, we can conclude that the MDP method has a better performance in comparison to the other three methods. Tables 2–8 contain the derived channel assignment solutions for problems 1–7, respectively.

7. Conclusion In this paper, we proposed a new and efficient algorithm to solve channel assignment problems in cellular radio networks that is based on synchronous version of DP and has a Trellis

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Table 6 Derived channel assignment solution for problem 5 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

56 61 66 71 76

75 80 85 90 95

96 101 106 111 116

82 87 92 97 102 107 112 117

83 88 93 98 103 108 113 118 123 128 133 138

18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138

19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164

42 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137 142 147 152 157 162

43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138 143 148 153 158 163 168 173 178 183 188

4 9 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164 169 174 179 184 189

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216 221

61 66 71 76 81 86 91 96 101 107 112 117 122 127 132 137 142 147 152 157

35 40 45 50 55 60 65 70 75 80 85 90 95 100 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181

13 26 31 36 41 46 51 105 110 115 120 125 130 135 140 145 150 155 160 167 172 177 182 187 192

17 22 27 32 37 81 86 91 165 170 175 180 185 190 195

56 61 66 71 76 121 126 131 136 141 146 151 156 161 166

18 23 28 33 42 47 52 57 62 67 72 77 122 127 132 137 142 147 152 157 162 167 172 177 182 187 192 197 202 207

2 7 12 18 23 28 33 38 96 101 169 174 179 184 189 194 199 204 209 214

30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 106 111 116 171 176

1 11 16 21 26 31 36 41 46 51 82 87 92 97 102 107 112 117 181 186 191 196 201 206 211

328

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Table 7 Derived channel assignment solution for problem 6 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

11 18 25 32 39

38 45 52 59 66

65 72 79 86 93

94 101 108 115 122 129 136 143

67 74 81 88 95 102 109 116 123 130 137 144

67 74 81 88 95 102 109 116 123 130 137 144 151 158 165 172 179 186 193 200 207 214 221 228 235

33 40 47 54 61 68 75 82 89 96 103 110 117 124 131 138 145 152 159 166 173 18 187 194 201 208 215 222 229 236

35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 182 189 196 203

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190 197 204

5 12 19 26 33 40 47 54 61 68 75 82 89 96 103 110 117 124 131 138 145 152 159 166 173 180 187 194 201 208 215 222 229 236 243 250 257 264 271 278

6 13 20 27 34 41 48 55 62 69 76 83 90 97 104 111 118 125 132 139 146 153 160 167 174 181 188 195 202 209 216 223 230 237 244 251 258 265 272 279

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190 197 204 211 218 225 232 239 246 253 260 267 274 281 288 295 302 309

66 73 80 87 94 101 108 115 122 129 136 143 150 157 164 171 178 185 192 199

30 37 44 51 58 65 72 79 86 93 100 107 114 121 128 135 142 149 156 163 170 177 184 191 198 205 212 224 231 238

62 69 76 83 90 97 104 111 118 125 132 139 146 153 160 167 174 181 188 195 202 209 216 223 230

60 73 80 87 94 101 108 115 122 129 136 143 150 233 240

81 88 95 102 109 116 123 130 137 144 151 158 165 172 179

28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 182 189 196 203 210 217 224 231

24 31 38 45 52 59 67 74 157 164 171 178 185 192 199 211 218 225 232 241

65 72 79 86 93 100 107 114 121 128 135 142 149 156 163 170 177 184 191 198

4 11 18 25 32 39 46 53 66 186 193 200 207 214 221 228 235 242 249 256 263 270 277 284 291

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329

Table 8 Derived channel assignment solution for problem 7 Cell number 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

221 227 233 239 245 251 257 263

119 124 129 134 139 144 149 154 159 164 169 174 179 184 189 194 199 204 209 214 340 347 354 361 368

375 382 389 396 403 410 417 424

342 349 356 363 370 377 384 391

217 223 229 235 241 247 253 259

219 225 231 237 243 249 255 261 267 273 279 285 291 297 303

269 275 281 287 293 299 305 311 317 323 329 335 342 349 356 363 370 377

3 8 13 18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 218 224 230 236 242 248 254 260 266 272 278 284 290 296 302 308 314 320 326 332 338 345 352 359 366 373 380

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216 222 228

4 9 14 19 99 104 109 114 219 225 231 237 243 249 255 261 267 273 279 285 291 297 303 309 315 321 327 333

337 344 351 358 365 372 379 386 393 400 407 414 421

220 226 232 238 244 250 256 262 268 274 280 286 292 298 304

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151

185 190 195 200 205 210 215 358 365 372 379 386 393 400 407

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180

98 103 108 113 118 123 128 133 138 143 148 153 158 163 168 173 178 183 188 193 198 203 208 213 220 226 232 238 244 250 256 262 268 274 280 286 292 298 304 310 316 322 328 334 340 347

4 29 34 39 44 49 54 59 64 69 74 79 84 89 94 339 346 353 360 367 374 381 388 395 402 409 416 423

221 227 233 239 245 251 257 263

295 301 307 313 319 325 331 337 344 351

217 223 229 235 241 247 253 259 265 271 277 283 289

269 275 281 287 293 299 305 311

(continued on next page)

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Table 8 (continued) Cell number 1

2

3

4

5

6

7

8

9

387 394 401 408 415 422

234 240 246 252 258 264 270 276 282 288 294 300 306 312 318 324 330 336 343 350 357 364 371 378 385 392 399 406 413 420 427

10

11

12

13

14

15

16

17

18

19

20

21

354 361 368 375 382 389 396 403 410 417 424

structure. We modified DP by defining function f(k, ‘) as (20) to apply co-channel, adjacent channel and co-site constraints on channel assignment problem and adding term P1 k¼m1 fðnk ð‘Þ; jÞ that represents frequency interference of the current channel with all previous channels. Consequently, MDP can find some optimum or near optimum solutions for a fixed channel assignment problem with no interference and the least bandwidth in a cellular structure. MDP differs from the other three methods in its independence from any initial solution requirements, also it can determine more than one valid solution and it is able to find the actual value of lb for a specific problem. We compared the performance of MDP against the other three methods used in channel assignment problem and showed through simulations that MDP achieves better solutions with the least necessary bandwidth that does not violate any electromagnetic compatibility constraints.

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Acknowledgements Authors appreciate the comments by the reviewers to enhance the quality of this paper. Authors also express their gratitude for the coordination of the paper by Professor M. Jamshidi and the editorial board at the Journal of Computers and Electrical Engineering. We also would like to express our appreciation to Mrs. S. Dadras for the preparation of the final manuscript.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21] [22]

Asha M. Cellular radio analog and digital systems. Norwood, MA: Artech House; 1994. Jakes WC. Microwave mobile communication. New York: Wiley; 1974. Raymond PA. Performance analysis of cellular networks. IEEE Trans Commun 1991;39(12):1787–93. Hale WK. Frequency assignment: theory and applications. Proc IEEE 1980;68(12):1497–514. Lee WCY. Mobile cellular telecommunications systems. New York: McGraw-Hill; 1989. Okada K, Kubota F. On dynamic channel assignment in cellular mobile radio systems. In: Proceeding IEEE international symposium on circuits and systems, Singapore, June 11–14, 1991. p. 938–41. Garey MR, Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. New York: W.H. Freeman; 1979. Zoellner JA, Beall CL. A breakthrough in spectrum conserving frequency assignment technology. IEEE Trans Electromag Compat 1977;EMC-19:313–9. Box F. A heuristic technique for assigning frequencies to mobile radio nets. IEEE Trans Veh Technol 1978;VT27(May):57–64. Gamst A, Rave W. On frequency assignment in mobile automatic telephone systems. In: Proceeding of IEEE global communication conference (GLOBECOMÕ82), Miami, USA, November 29–December 2, 1982. p. 309–15. Gamst A. Homogeneous distribution of frequencies in a regular hexagonal cell system. IEEE Trans Veh Technol 1982;VT-31(August):132–44. Gamst A. Some lower bounds for a class of frequency assignment problems. IEEE Trans Veh Technol 1986;VT35(February):8–14. Sivarajan KN, McEliece RJ, Ketchum JW. Channel assignment in cellular radio. In: Proceeding of 39th IEEE vehicular technology society conference, May 1989. p. 846–50. Sengoku M, Tamura H, Shinoda S, Abe T, Kajitani Y. Graph theoretical considerations of channel offset systems in a cellular mobile system. IEEE Trans Veh Technol 1991;40(2):405–11. Gamst A. Application of graph theoretical methods to GSM radio network planning. Proc IEEE Int Symp Circ Syst 1991;2(June):942–5. Kim S, Kim S-L. A two-phase algorithm for frequency assignment in cellular mobile systems. IEEE Trans Veh Technol 1994;43(3):542–8. Ko T-M. A frequency selective insertion strategy for fixed channel assignment. In: Proceedings of ICCC regional meeting on wireless computer networks, Den Haag, Netherlands, September 18–23, 1994. Kishi Y, Mizuike T, Watanabe F. A unified approach for frequency assignment of cellular mobile networks. In: Third annual international conference on universal personal communications, San Diego, CA, September 27– October 1, 1994. p. 563–7. Wang W, Rushforth CK. An adaptive local-search algorithm for the channel-assignment problem (CAP). IEEE Trans Veh Technol 1996;45(3):459–66. Jordan S, Schwabe EJ. Worst-case performance of cellular channel assignment policies. Wirel Netw 1996;2:265–75. Leese RA. A unified approach to the assignment of radio channels on a regular hexagonal grid. IEEE Trans Veh Technol 1997;46(4):968–80. Tcha D-W, Chung Y-J, Choi T-J. A new lower bound for the frequency assignment problem. IEEE/ACM Trans Netw 1997;5(1):34–9.

332

S.A.G. Shirazi, H. Amindavar / Computers and Electrical Engineering 31 (2005) 303–333

[23] Sen A, Roxborough T, Medidi S. Upper and lower bounds of a class of channel assignment problems in cellular networks. In: Proceedings of IEEE INFOCOMÕ98, San Francisco, CA, March 29–April 2, 1998, p. 1284–91. [24] Rouskas AN, Kazantzakis MG, Anagnostou ME. Minimization of frequency assignment span in cellular networks. IEEE Trans Veh Technol 1999;48(3):873–82. [25] Battiti R, Bertossi Alan, Cavallaro Daniela. A randomized saturation degree heuristic for channel assignment in cellular radio networks. IEEE Trans Veh Technol 2001;50(2):364–74. [26] Chakraborty G. An efficient heuristic algorithm for channel assignment problem in cellular radio networks. IEEE Trans Veh Technol 2001;50(6):1528–39. [27] Fernando XN, Fapojuwo AO. A viterbi-like algorithm with adaptive clustering for channel assignment in cellular radio networks. IEEE Trans Veh Technol 2002;51(1):73–87. [28] Kunz D. Practical channel assignment using neural networks. In: Proceeding of 40th IEEE vehicular technology conference, Orlando, FL, May 1990. p. 652–5. [29] Kunz D. Channel assignment for cellular radio using neural networks. IEEE Trans Veh Technol 1991;40(1):188–93. [30] Funabiki N, Takefuji Y. A neural network parallel algorithm for channel assignment problems in cellular radio networks. IEEE Trans Veh Technol 1992;41(4):430–7. [31] Chan PTH, Palaniswami M, Everitt D. Neural network-based dynamic channel assignment for cellular mobile communication systems. IEEE Trans Veh Technol 1994;43(May):279–88. [32] Lochtie GD, Mehler MJ. Channel assignment using a subspace approach to neural networks. IEE Conference Publication, vol. 407; 1995. p. 296–300. [33] Lochtie GD, Mehler MJ. Subspace approach to channel assignment in mobile communications. IEE Proc Commun 1994;142:179–85. [34] Kim J-S, Park SH, Dowd PW, Nasrabadi NM. Cellular radio channel assignment using a modified Hopfield network. IEEE Trans Veh Technol 1997;46(4):957–67. [35] Smith KA, Palaniswami M. Static and dynamic channel assignment using neural networks. IEEE J Select Areas Commun 1997;15(2):238–49. [36] Sung CW, Wong WS. Sequential packing algorithm for channel assignment under cochannel and adjacent channel interference constraint. IEEE Trans Veh Technol 1997;46(3):676–86. [37] Funabiki N, Okutani N, Nishikawa S. A three-stage heuristic combined neural network algorithm for channel assignment in cellular mobile systems. IEEE Trans Veh Technol 2000;49(2):397–403. [38] Cuppini M. A genetic algorithm for channel assignment problems. Eur Trans Telecommun Relat Technol 1994;5(2):285–94. [39] Dorne R, Hao J-K. An evolutionary approach for frequency assignment in cellular radio networks, 1995. [40] Lai WK, Coghill GG. Channel assignment through evolutionary optimization. IEEE Trans Veh Technol 1996;45(1). [41] Kim J-S, Park SH, Dowd PW, Nasrabadi NM. Channel assignment in cellular radio using genetic algorithms. Int J Wirel Personal Commun 1996;3(3):273–86. [42] Ngo CY, Li VOK. Fixed channel assignment in cellular radio networks using a modified genetic algorithm. IEEE Trans Veh Technol 1998;47(1):163–72. [43] Smith KA. A genetic algorithm for the channel assignment problem. IEEE Global Commun Conf 1998;4:2013–8. [44] Beckmann D, Killat U. A new strategy for the application of genetic algorithms to the channel-assignment problem. IEEE Trans Veh Technol 1999;48(4):1261–9. [45] Duque-Anto´n M, Kunz D, Ru¨ber B. Channel assignment for cellular radio using simulated annealing. IEEE Trans Veh Technol 1993;42(1):14–21. [46] Mathar R, Mattfield J. Channel assignment in cellular radio networks. IEEE Trans Veh Technol 1993;42(4):647–56. [47] Hurley S, Smith DH. Fixed spectrum frequency assignment using natural algorithms. In: Proceedings IEE/IEEE international conference on genetic algorithms in engineering systems (GALESIAÕ95), Sheffield, UK, September 1995. p. 373–8. [48] Zhang M, Yum T-SP. The nonuniform compact pattern allocation algorithm for cellular mobile systems. IEEE Trans Veh Technol 1991;40(2):387–91.

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[49] Bertsekas DP. Dynamic programming and optimal control, vols. 1 and 2, 2nd ed.. Belmont, MA: Athena Scientific; 2001. [50] Bellman RE. Dynamic programming. Princeton (NJ): Princeton University Press; 1957. [51] Anderson G. A simulation study of some dynamic assignment algorithm in a high capacity mobile telecommunications system. IEEE Trans Commun 1973;COM-21(11):1294–301. [52] http://fap.zib.de/problems/Philadelphia. Seyyed Alireza Ghasempour Shirazi received the B.Sc. degree from Shiraz University, Shiraz, Iran, in 1998, and the M.Sc. degree from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2001 both in communication engineering. Since 2001 he joined Information and Communication Technology (ICT) faculty as an academic member and the chief of the research group. His research interests are channel assignment, resource management, optimization techniques and their applications in communication engineering, and wireless communications.

Hamidreza Amindavar is a professor at Amirkabir University of Technology, Department of Electrical Engineering since 1993. His research interests include digital communications, multiuser detection, digital signal and image processing, fault tolerant computing.