Why higher order constraints are necessary to

Why higher order constraints are necessary to model frequency assignment problems Peter Jeavons, Nick Dunkin, Joe Bater Department of Computer Science, Royal Holloway, University of London email: [email protected]

Abstract. The frequency assignment problem is an impor- 2 The frequency assignment problem tant practical problem in telecommunications which is often modelled as a binary constraint satisfaction problem. We give In general, a radio communications network consists of a set a number of examples to show that important features of the of transmitters and receivers in a given region, where each problem cannot in fact be expressed using binary constraints transmitter has the following properties: between pairs of transmitters. Hence we argue that this prob- (a) A position in the region. lem should be modelled as a higher-order constraint satisfac- (b) An output signal which has an associated tion problem, and we discuss ways in which these constraints { frequency spectrum, might be speci ed. { power, { directional distribution. Each receiver responds to the signals from each of the trans1 Introduction mitters, to a greater or lesser extent. Some of these signals are generally \wanted" or \carrier" signals, which are used The frequency assignment problem is a problem of great im- for communication, and others are regarded as \interference". portance to the telecommunications industry and the mil- The aim in a frequency assignment problem is to assign values itary. It is often modelled as a binary constraint satisfac- to some or all of the properties of the transmitters in order to tion problem (CSP) where the constraints restrict the fre- satisfy given criteria for the overall behaviour of the system. quencies that can be assigned to certain pairs of transmit- A secondary aim is to minimise the amount of frequency specters [7, 12, 13, 14, 15, 18]. In fact, the problem is often con- trum used by the system, as the radio frequency spectrum is sidered to be a generalised form of the Graph Colouring a limited natural resource, which is increasingly in demand, and its usage is tightly controlled by regulatory authorities. problem [6, 7, 8, 16, 23]. We argue below that in many problem instances a CSP model For the frequency assignment problems we consider in this pawhich considers only pairwise interactions between transmit- per we make the following standard simplifying assumptions, ters will be unable to express the requirements of the problem which are appropriate for systems such as mobile telephone adequately, and will either allow poor quality solutions or else networks [1, 2]. require too many frequency channels, and so result in an in- (a) The positions of transmitters are xed in the region. ecient use of radio spectrum. (b) The output signal from each transmitter has A number of other studies have raised concerns about the ad{ a single frequency channel, to be assigned from a set of equacy of the standard binary CSP model [7, 9, 17], but the uniformly spaced channels, main solution proposed by these studies has been to increase { a xed power (constant across all transmitters), the exibility and expressive power of the model by allowing { a xed and uniform directional distribution. the use of both hard and soft binary constraints. This means that the problem is reformulated as an optimisation problem, (c) The receivers must be able to function at any point in the or partial CSP. This greatly adds to the complexity of the region. solution process, but it does not, in our view, overcome the (d) The desired carrier signal at each receiver is the strongest fundamental limitations of the binary constraint model, besignal that is available at that receiver, and all other signals cause all of the system requirements are still expressed using are regarded as interference. only binary constraints. The solution we propose here to overcome the modelling di- The problem we are then faced with is to assign frequencies culties we identify is to introduce higher-order constraints to to the given transmitters in order to achieve some speci ed express the problem requirements. We demonstrate the eec- minimum acceptable value for the ratio of the carrier signal tiveness of higher-order constraints in the examples we con- strength to the combined strength of the interference signals, sider, and we discuss general ways in which such constraints at all positions in the region of interest. This ratio is generally referred to as the \carrier-to-interference ratio" (C-I). may be de ned for arbitrary problem instances.

c 1998 P.Jeavons, N.Dunkin

and J.Bater ECAI 98. 13th European Conference on Arti cial Intelligence Edited by Henri Prade Published in 1998 by John Wiley & Sons, Ltd.

The strength of the carrier and interference signals at any given position depends on the following factors.

 Signal propagation. The signal strength available from

a given transmitter at a given point is dependent on the power and direction of the transmitted signal, the distance from the transmitter, and the terrain in between, as well as unpredictable factors such as atmospheric conditions. In the simpli ed model used below we shall assume that the signal strength can be approximated as a simple function of distance (i.e. terrain variations and other non-regular features will be ignored).  Signal Frequency. As each receiver may be tuned to respond most strongly to particular frequencies, the eective signal strength from any transmitter depends on the frequency of the transmitted signal. In the simpli ed model used below we shall assume that the receiver is tuned to the frequency of the desired carrier signal, and the eective signal strength of all signals at different frequencies is reduced by a \ ltering" factor, which depends on the dierence in frequency from the frequency of the carrier signal, and on the design of the receiver.  External Noise. In any network some interference is caused by signals external to the network which cannot be altered. In the models developed here these external sources of interference will be ignored.

3 Examples 3.1 Example 1 - An in nite hexagonal lattice

Figure 1 shows a region of an in nite hexagonal grid of transmitter positions, where each transmitter is surrounded by a hexagonal \cell". Any receiver within the cell is assumed to be tuned to the transmitter at the centre of the cell, which is supplying the desired carrier signal for that receiver. This idealised form of frequency assignment problem has been extensively studied, in order to provide insight into more realistic networks, see for example [10, 11, 19, 21].

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With these assumptions, a coverage problem consists of:

   

A region - R. A set of transmitters - fT1 ; T2 ; : : : ; Tn g at xed positions. A set of available frequency channels - f0; 1; 2; : : : ; k ? 1g. A collection of rules governing signal propagation and receiver response.  A required minimum carrier-to-interference ratio - .

Figure 1.

Consider a case where the propagation rules and receiver characteristics are such that at most three of the transmitters in the second ring of cells around any transmitter Tx can be assigned the same frequency channel as Tx without the collective interference due to these transmitters becoming too great at any point in the cell around Tx . (In Figure 1 a typical transmitter Tx is identi ed, and the transmitters in the second ring of cells around Tx are indicated by shading.) Furthermore, assume that any number of transmitters which are further away from Tx than this second ring of cells can also be assigned this frequency channel without causing unacceptable interference in the cell around Tx . Because of the spatial uniformity in this problem, it follows that if the three transmitters in the second ring of cells around Tx which are assigned the same frequency channel as Tx are chosen to be either the transmitters T1 ; T3 ; T5 , or the transmitters T2 ; T4 ; T6 , as shown in Figure 1, then this pattern of assignments may be repeated across the in nite grid without causing excessive interference in any of the cells assigned this value. The result of repeating this assignment across the grid is shown in Figure 2. If we further assume that the frequency channels are suf ciently separated so that there is no interference between signals on dierent frequency channels, then this pattern of assignment can be applied again to a dierent set of transmitters, at a dierent frequency channel, giving the assignment

Achieving the required carrier-to-interference ratio at each point in the region imposes constraints on the possible assignments of frequencies to transmitters, and the whole problem may be considered as a constraint satisfaction problem (CSP). However, it is not immediately clear how the constraints arising from the problem speci cation may be best expressed as constraints in a CSP. What is commonly done is to associate a variable with each transmitter Ti , and then to impose constraints on the frequency channels assigned to pairs of transmitters, to obtain a binary CSP. These constraints typically take the form jfi ? fj j > m where fi and fj are the frequencies assigned to transmitters Ti and Tj respectively, and m  0 is an integer giving the required separation in terms of frequency channels. However, attempting to express the problem in this form ignores the fact that in any physical network the carrier-tointerference ratio at any point depends on the combined effect of signals received from all of the transmitters. In the following sections we demonstrate that in many cases of interest this fact makes it impossible to express the problem adequately using binary constraints.

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shown in Figure 3. This process can be repeated for two more frequency channel shifts as shown in Figure 4. 1

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of a range of possible frequency separations s, but in this particular example we have assumed that dierent frequency channels never cause excessive interference, and so ds = 0 for all s > 0. Hence, to generate binary constraints we simply have to choose a distance d0 , and then impose a constraint on any pair of transmitters which are closer together than d0 which ensures that they are assigned dierent frequency channels. However, this approach cannot capture the essential features of this particular example, because whatever value is chosen for d0 the binary constraints will not express the problem, as we shall now show. 1. If the re-use distance d0 is chosen to be smaller than, or equal to, the distance to the transmitters in the second ring, then solutions containing patterns such as the one shown in Figure 6 will be allowed. Any solution containing this pat-

Finally, the remaining transmitters can be assigned frequencies by repeating the same pattern (rotated) at a fth frequency channel, and then assigning a sixth frequency channel to the remaining transmitters, as shown in Figure 5. Hence we have obtained a complete solution to the problem using only 6 frequency channels. Regular problems of this type are generally modelled using binary constraints between pairs of transmitters, based on the notion of \re-use distances" [1, 12, 22]. For s = 0; 1; 2; : : :, the re-use distance, ds is the minimum distance from a transmitter Tx at which another transmitter may be assigned a frequency channel which is separated by s channels from the frequency channel used at Tx . In general, there is a dierent re-use distance, ds , for each

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3.2 Example 2 - A nite hexagonal grid

tern obviously breaks the condition stated above that there must be \no more than three at the same frequency channel in the second ring". The collective interference caused by the six transmitters at the same frequency channel would result in an unacceptable carrier-to-interference ratio at some positions, and hence a poor network performance.

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In the next example we consider a collection of 36 transmitters arranged in a hexagonal grid, as shown in Figure 7. Each transmitter is surrounded by a cell, within which the desired carrier signal at any receiver is the signal from that transmitter. The cells are determined by calculating the closest transmitter to each point in the region of interest.

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A solution satisfying loose binary constraints

Now consider the following propagation rules, based on values commonly used in the literature for analogue mobile telephone networks [21, 11]: 1. The signal strength at any point is given by 1 d4 where d is the distance from the point to the transmitter. 2. The strength of all signals on a frequency channel which is separated by s channels from the wanted signal is ltered at the receiver by a factor of

2. On the other hand, if the re-use distance d0 is chosen to be greater than the distance to the transmitters in the second ring, then only transmitters in the third ring around any Tx can use the same frequency channel. Results from [20] indicate that in that case at least 7 frequency channels would be required to construct a complete assignment, and hence the system would use an unnecessary number of channels. Of course, the particular solution found above (Figure 5) could be captured by specifying corresponding binary constraints on all pairs of transmitters. However, constraints obtained in that way would not be spatially symmetric, and it is not clear how such constraints could be chosen without rst nding a solution to the problem! Another point to note is that choosing binary constraints based on a particular solution in that way would rule out other valid solutions (such as rotations of the solution above). On the other hand, this example can easily be expressed precisely as a CSP with higher order constraints. One way in which this can be done is by imposing a constraint on the transmitters in the two rings of cells around each transmitter, together with that transmitter, giving constraints of arity 19. These constraints would allow any assignment in which at most three of the transmitters in the outer ring of cells were assigned the same frequency channel as the central transmitter. This example therefore provides a rst indication that higher order constraint representations can lead to better modelling of some frequency assignment problems.

ECAI98 Workshop on Non Binary Constraints

Transmitter locations and cells

?15(1 + log2 s)dB: 3. The total strength of the interference signals is obtained by adding the interfering signal from each transmitter. Finally, the required minimum carrier-to-interference ratio at all points in the region of interest is 15dB. To nd an assignment of frequency channels to all the transmitters which achieves this carrier-to-interference value everywhere we can simply conduct a search through all possible assignments, rejecting those which give unacceptable interference. The rst solution found by this method when there are 9 available channels is illustrated in Figure 8 Now we consider how this problem instance could be modelled using binary constraints. If the constraints are based on re-use distances, as described above, then we can deduce from the solution shown in Figure 8 that the re-use distances must be smaller than certain maximum values, otherwise this solution will be disallowed. Examining this solution reveals that the same frequency channel is used for some pairs of transmitters that are distance 2 apart (where distance 1 is the distance

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It is clear that this solution does not meet the requirements for a working system, and hence the constraints need to be tightened. However, this cannot be done by increasing the reuse distances without eliminating the valid solution shown in Figure 8, as noted above. This means that this problem instance cannot be modelled adequately with binary constraints based on re-use distances.

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We have shown that in order to capture the requirements of certain forms of frequency assignment problem it is sometimes inadequate to consider constraints on pairs of transmitters. Further experimental results supporting this conclusion for irregular transmitter arrangements are given in [4, 5]. This observation immediately raises the question of how to choose more expressive, higher-order constraints which do capture the problem requirements. This is the question which we are currently investigating. In view of the fact that the strength of radio signals decreases rapidly with distance from the transmitter, one obvious possibility is to consider constraints on groups of transmitters within a small local area. In fact, if we make the common assumption that signal strength decreases according to the function d1 , where d is the distance from the transmitter, then the combined interference from all transmitters beyond a cuto distance D is bounded by 1

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A frequency assignment achieving the desired carrier-to-interference ratio

between neighbouring pairs of transmitters). This means that the re-use distance d0 must be set to at most 2. Similarly, the re-use distance,pd1 , for channels at a separation of 1, must be set to at most 3=2, the re-use distance, d2 , for channels at a separation of 2 must be set to at most 1, and so on. However, when we set the re-use distances to these maximum values, calculate the corresponding binary constraints, and then solve the resulting binary CSP, the rst solution that we obtain is the solution shown in Figure 9. The dark areas in this Figure indicate areas where the calculated carrierto-interference ratio is below the required threshold value of 15dB.

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Hence, if > 2 and the distribution of the transmitters is uniform, the combined interference from all transmitters beyond distance D can be made arbitrarily small, by choosing a suciently large value for D. Figure 10 shows a collection of transmitters with a circular region of a xed radius centred on each transmitter, showing how the problem may be decomposed into overlapping local regions. The number of transmitters within these circles varies from 2 to 6. Hence, if we calculate appropriate constraints for each of these groups of transmitters, then the resulting constraint satisfaction problem will have constraints with arities which range from 2 to 6. One way to calculate constraints on these local groups of transmitters is to exhaustively enumerate all of the assignments which give an acceptable signal-to-noise ratio at all points within the cell served by the central transmitter. This set of assignments forms a relation which can be imposed on the frequency channel values assigned to these transmitters. The assignments in the constraint relation can be stored as a list of tuples of frequency channel values for all the transmitters in the region, or else (more eciently) as a list of tuples of osets from the frequency channel assigned to the central transmitter [3]. If there are too many possibilities to store them all explicitly, then it is possible to represent the constraint relation more compactly by storing only the minimal lists of osets, since increasing the frequency separation of any transmitter from the central transmitter can only reduce the interference and hence improve the carrier-to-interference ratio in the central cell. These representation methods are currently being investigated.

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Figure 10.

[6] L.A. Berry, `Potential contribution of optimum frequency assignment to ecient use of the spectrum', in Proceedings of the IEEE International Symposium on Electromagnetic Compatibility, pp. 409{412, (1990). [7] J.F. Boyce, H.D. Dimitropoulos, G. vom Scheidt, and J.G. Taylor, `GENET and Tabu search combinatorial optimisation problems', in World Congress on Neural Networks (WCNN95), Washington DC, (17-21 July 1995). [8] G.R. Bradbeer, `Graph colouring for frequency assignment of combat radio', Technical Report DRA/CIS(CIS1)/P132/D/06/002, Defence Evaluation and Research Agency, (December 1996). [9] M. Carlsson and M. Grindal, `Automatic frequency assignment for cellular telephones using contraint satisfaction techniques', in Proceedings of ICLP'93, ed., D.S. Warren, pp. 647{ 665, Budapest, (1993). MIT Press. [10] A. Gamst, `Homogeneous distribution of frequencies in a regular hexagonal cell system', IEEE Transactions on Vehicular Technology, VT-31(3), 132{144, (August 1982). [11] R. Gower and R. Leese, `The sensitivity of channel assignment to constraint speci cation', in EMC97 Symposium, pp. 131{ 136, Zurich, (1997). [12] W.K. Hale, `Frequency assignment: Theory and applications', Proceedings of the IEEE, 68(12), 1487{1514, (December 1980). [13] A. Kapsalis, V. Rayward-Smith, and G.D. Smith, `Using genetic algorithms to solve the radio link frequency assignment problem', in Proceedings of the 2nd international conference on arti cial neural networks and genetic algorithms, pp. 37{ 40, (1995). [14] A. Knalmann and A. Quellmalz, `Solving the frequency assignment problem with simulated annealing', in Proceedings of the Ninth International Conference on Electromagnetic Compatibility, pp. 233{240, (1994). [15] R. Leese, `Tiling methods for channel assignment in radio communication networks', in Proceedings of the 3rd International Conference on Industrial and Applied Mathematics, (1995). [16] E. Malesinska, `List colouring and optimization criteria for a channel assignment problem', Technical Report 458/1995, Technische Universit}at Berlin, (1995). [17] G. Ottosson and M. Carlsson, `Using global constraints for frequency allocation', Technical Report TR-97-07, ASTEC, (1997). [18] D.H. Smith, S. Hurley, and S.U. Thiel, `Improving heuristics for the frequency assignment problem', European Journal of Operations Research, (To Appear). [19] A.C. Stocker, `Cochannel interference and its avoidance in close-spacedsystems', IEEE Transactions on Vehicular Technology, VT-31(3), 145{150, (August 1982). [20] J. van den Heuvel, R.A. Leese, and M.A. Shepherd, `Graph labelling and radio channel assignment', Technical Report CDAM-9623, London School of Economics, (1996). (Available from www.maths.ox.ac.uk/users/gowerr/preprints.html). [21] S-W. Wang and S.S. Rappaport, `Signal-to-interference calculations for balanced channel assignment patterns in cellular communications systems', IEEE Transactions on Communications, 37(10), 1077{1087, (1989). [22] D.J. Withers, Radio Spectrum Management, Peter Peregrinus, Stevenage, United Kingdom, 1991. [23] J. Z erovnik, `Experiments with a randomized algorithm for a frequency assignment problem', Technical Report 97-27, Ecole Normale Superieure de Lyon, (September 1997).

Local areas centred on each transmitter

5 Conclusion

We have shown a number of dierent examples of the frequency assignment problem which cannot be adequately modelled using binary constraints between pairs of transmitters. This work suggests that to model this broad class of problems eectively we need to consider higher-order constraints over larger groups of transmitters. We have brie y described how an arbitrary frequency assignment problem can be modelled by decomposing into small groups of mutually constraining transmitters, and specifying higher order constraints over those groups of transmitters which directly capture the problem requirements. We believe that modelling frequency assignment problems in this more exible and powerful way can lead to solutions which combine high quality system performance with minimal spectrum usage.

REFERENCES

[1] J-F. Arnaud, `Frequency planning for Broadcast Services in Europe', Proceedings of the IEEE, 68(12), 1515{1521, (December 1980). [2] D.M. Balston, `The pan-European cellular technology', in Personal & Mobile Radio Systems, ed., Macario, R.C.V., chapter 14, 290{319, Peter Peregrinus, Stevenage, United Kingdom, (1991). [3] J.E. Bater, `Heuristic generation of arbitrary arity constraints for frequency assignment'. Working Paper, 1998. [4] J.E. Bater, P.G. Jeavons, and D.A. Cohen, `Are there optimal reuse distance constraints in FAPs with random Tx placements?', Technical Report CSD-TR-98-01, Royal Holloway College, University of London, (February 1998). [5] J.E. Bater, P.G. Jeavons, D.A. Cohen, and N.W. Dunkin, `Are there eective frequency separation constraints for FAPs with irregular Tx placements?'. Submitted to NATO Symposion on Frequency Assignment, Sharing and Conservation, March 1998.

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