Spectral Based Solutions for (Near) Optimum Channel/Frequency Allocation Zekeriya Uykan Electronics and Communications Eng. Dept. Dogus University Kadiköy, 34722 Istanbul TURKEY
[email protected] Abstract—Optimum channel/frequency allocation problem in a general mobile radio networks is well-known to be NP-complete. The optimum general solution for a mobile radio network for even 2-channel case is not known. In this paper, we focus on the case L = 2 k where L is the number of channels/frequencies and k is a positive integer. In order to minimize the sum of the network level interference, we show that a hybrid solution (first finding the maximum eigenvector centrally and then running the well-known distributed standard minimum-interference-channelallocation algorithm) outperforms both the spectral solution and the standard distributive algorithm. The proposed solution can be adopted by any cellular, cognitive, ad-hoc or mesh type radio networks. Keywords- mobile radio systems; optimum channel/frequency allocation; max cut, weighted graph partitioning; spectral clustering; min-interference-channel-allocation algorithm, GADIA.
I.
INTRODUCTION
Channel/frequancy allocation is an important and essential mechanism in order to mitigate the interference in a wireless cellular network. Therefore, it has been a focus of intensive research in both academia and industry in the last two decades especially. As various new types of wireless cellular and/or non-cellular networks like cognitive, ad-hoc or mesh-type radio systems are about to emerge, the optimum channel/frequency allocation obviously remains to be a hot research topic in both academia and industry in the coming “era of converging wireless networks/systems/ecosystems” already next decade. The channel/frequency allocation problem in wireless cellular system is known to be NP-complete (Nondeterministic Polynomial time - complete) (see e.g. [11], [10], among others). This means that no polynomial-time algorithms are available for this problem. The optimum general solution for any mobile network for even 2-channel case is not known. There is a vast literature work in the area of channel/frequency allocation/assignment in various wireless radio systems. For a survey, and further references see e.g. [12], [13], among others. One of the most widely used algorithms in practical systems is simply to assign the mobile/base to the channel where it experiences the minimum interference in a distributive fashion. Although this algorithm performs well in practice, it has, in general, local minima problem and therefore its solution does not give any performance guarantee. An asynchronous version of such an algorithm is called GADIA and is examined in [9] in detail. The author would like to thank Prof. Vahid Tarokh for the discussions on the topic during the author’s visit at Harvard University, MA, in 2008-09.
Channel assignment problem in wireless systems is also examined as a graph multi-coloring problem, which is wellknown to be NP-complete as well. For various algorithms used in graph multi-coloring and for further references, please see e.g. [3] and [10], among others. In this paper, we follow this path which formulates the average network/channel interference minimization problem as a graph partitioning problem. However, here, our main contribution is that turning the channel allocation problem into a maxCut problem, we propose and examine new spectral based solutions for that. One of the proposed solutions is based on the spectral maxCut approximation, while other is a hybrid algorithm which outperforms both the spectral and the standard minimuminterference-channel allocation algorithm. The rest of the paper is organized as follows: Section II gives the formulation of the problem. The proposed spectral based solutions are presented in section III. The computer simulations are shown in section IV, followed by the conclusions in section V. II.
FORMULATION
In this section, we formulize the total and average network interferences to be minimized by channel allocation. Let the number of co-channel transmitters be M for a given channel. Here, we name a transmitter as a Mobile Base (MB) to indicate that the base stations may be mobile as well in a network. Then the received Signal-to-Interference+Noise-Ratio (SINR) at receiver i can be mathematically sketched as (see. e.g. [7], [8]) θ i (k ) =
g ii pi ( k ) N
ϕ i + ∑ g ij p j ( k )
,
i = 1,2," , M
(1)
j =1
where θ i (k ) denotes the received SINR at receiver i at time k; pi (k ) is the transmit power of transmitter i at time k; g ij is link gain from transmitter j to receiver i (involving path loss, shadowing, etc), and ϕ i is the thermal noise at receiver i. The link gain g ij can be modeled as follows (see e.g. [7].) g ij =
sij cij d β ij
,
i, j = 1, 2,", M
(2)
where sij is the shadow fading term, d ijβ is propagation loss with pathloss exponent β , and
cij is for multipath fading [7].
For information about modeling of radio wave propagation, see e.g. [7], [6]. Since, in this paper, we focus on the optimum channel allocation problem, we assume that the powers of the transmitters are fixed. Let the total number of MBs is N and that of channels/frequencies be L (where N> L). So, we need to allocate the N MB’s to L channels/frequencies. As an attempt to maximize the average received SINR in the network, in this paper we aim to minimize the average interference per MB or total network interference. In other words, ⎧N ⎛ ⎞⎫ N N ⎜ ⎟⎪ ⎧N ⎫ ⎪ max ⎨∑ θ i ⎬ ≡ min ⎨∑ I i = ∑ ⎜ ϕ i + ∑ g ij ⎟ ⎬ i =1 ⎜ j =1 ⎩ i =1 ⎭ ⎟⎪ ⎪ i =1 j ≠i ⎝ ⎠⎭ ⎩
(3)
III.
SPECTRAL-BASED SOLUTIONS FOR OPTIMUM CHANNEL/FREQUENCY ALLOCATION
In this section, we examine the optimization problem defined in section II for the case L=2, because i) its general solution is not known (NP-complete), and ii) for L = 2 q , q ≥ 2 , our proposed algorithm will be based on iterating the same procedure 2 q − 1 times. In what follows, we propose two spectral based solutions: A. SpecPure Algorithm Let’s first define the following matrices:
. Without loss of
Definition: Denoting the received signal strength (RSS) at base i of the pilot signal from base j as rij , we define so-called
generality and for the sake of brevity, we assume that N is an integer multiple of L. Once allocation of N MB’s to L channels is performed, then total co-channel interference in the network, ntw denoted as I tot , is given by
received-signal-strength (RSS) based adjacency system matrix as follows R = rij N × N = g ij p j (8) where g ij is given by (2), and g ii = 0 and p j is the transmit
N
where i=1,2,…,N, and I = ϕ + i i
∑g
ij
j =1 ( j ≠i )
L
L
ntw I tot = ∑ IS = ∑ s∈1
s =1
NS
NS
∑ ∑g
ij
s = 1," ,L
,
j∈C S i∈C S (i≠ j )
(4)
where C S represents the set of MBs assigned to channel/frequency s; N S is the length of the set C S (i.e., the number of MBs in channel s), and g ij represents the corresponding link gains, I S is the sum of the interferences experienced by those MBs using the same channel s, and L N = N . Similarly, we define average co-channel
∑
S
s =1
ntw interference, I ave , as L
ntw I ave =∑ s∈1
L 1 1 IS = ∑ NS s =1 N S
NS
NS
∑ ∑g
ij
j∈CS i∈CS (i≠ j )
,
s = 1," ,L
ntw network interference I tot in (4), i.e.:
determine CS , ( s =1,", L )
ntw I tot
ntw Similarly, for the average co-channel interference, I ave , in eq. (5)
solution defined in section 1: i) is equal to the optimum maxCut solution of the proposed RSS-based adjacency matrix; and ii) can be approximated by the maximum eigenvector computation of the unnormalized Laplacian matrix after relaxation. Proof: The entrywise 1-norm of the system link-gain matrix G is equal to
min
determine CS , ( s =1,", L )
(7)
N
N
i =1
j =1
∑ ∑g
G1=
N1
G 1 =∑
N1
N1
N2
N2
ntw tot
be chosen in minimizing the total network interferences I or ntw in (5) and (6), respectively, and this is an NP-complete I ave problem.
N2
N2
N1
∑g +∑ ∑g +∑ ∑g +∑ ∑g ij
ij
i∈C1 j∈C2
ij
i∈C2 j∈C2
(10)
ij
i∈C2 j∈C1
Eq.(10) can be written as (11)
G 1 = constant = I1 + J 1 + I 2 + J 2 where I = ∑ S
NS
∑g
ij
is the total co-channel interference
i∈C S j∈C S
N1
for channel s, and J = J = ∑ 1 2 So, channel allocation procedure determines which g ij ’s to
(9)
ij
Then, considering the grouping of bases into two groups C1 and C2 , we write
NS
ntw I ave
]
Proposition 1: The optimum channel/frequency allocation
i∈C1 j∈C1
(6)
[
power of MB j. We assume that p j is constant, and received signal strength from MB i to j is equal to the signal strength from MB j to i due to the reciprocity assumption since signal would travel approximately the same route, i.e., rij = r ji .
(5)
where N S is the number of MBs in channel s. From (1)-(4), we formulate the channel allocation problem as determining the sets C S of mobiles (s=1, …, L) which minimizes the total
min
[ ]
N2
∑g
i∈C1 j∈C2
ij
=
N2
N1
∑ ∑g
ij
,
i∈C2 j∈C1
where s=1 and 2, represents the total interference which is eliminated once C1 and C 2 are determined by the channel allocation process. From graph theory point of view (e.g. [4], [2]), using (9)-(11), we can write
1. 2. 3. 4.
TABLE I SPECPURE CHANNEL/FREQUENCY ALLOCATION ALGORITM Every mobile base (MB) transmits a pilot signal. Every MB calculates received signal strength (RSS) from all other MBs. Sends the RSS information to the center MB. The center MB establishes the RSS based dissimilarity matrix o Repeat for n=1:L Determine the MB subset of N / 2 L to be spectrally clustered for L=2. Perform the clustering with respect to the maximum eigenvector of the corresponding N / 2 L × N / 2 L dimensional unnormalized Laplacian matrix in (14). Finally, allocate the MBs to the L channels according to the results obtained after performing step 4.
(
5.
(
)
vol (G ) = vol ( A) + cut ( A, B ) + vol ( B ) + cut ( B, A)
(12)
where vol ( A) = I1 , vol ( B) = I 2 , and cut ( A, B ) = cut ( B, A) . From eq.(11) and (12) , (13) min C1 ,C2 (I1 + I 2 ) ≡ max C1 ,C2 (2cut ( A, B) ) This is equal to well-known weighted maxCut problem in graph theory (e.g. [4], ). The unnormalized Laplacian matrix (e.g. [4], [5]) is given as
L = D−G where diagonal matrix
(14)
⎧ N ⎪ ∑ g ij , if m = n . D = [d mn ] = ⎨ j =1, ( j ≠i ) ⎪ 0, otherwise ⎩
It’s straightforward to see that the maxCut problem can be formulized as follows (see e.g. [4], [5])
⎧N N 2⎫ max x T Lx = max ⎨∑ ∑ g ij (xi − x j ) ⎬ , ⎩ i =1 j =1 ⎭ where xi , x j ∈ {− 0.5,+0.5}
{
}
(15)
Relaxing the optimization in (15) such that x ∈ ℜ N ×1 (instead of being discrete) gives the optimum solution as real numbers which is equal to the maximum eigenvector, denoted as e max , of the Laplacian matrix in (14). So, the relaxation result is x
opt
= sign (e max )
(16)
which completes the proof. In the light of the result of Proposition 1, we summarize our first proposed algorithm, called SpecPure, in Table I. Corollary: The solution in Proposition 1 which minimizes ntw the total co-channel interference in the network I tot in (4) ntw in (5) also minimizes the average co-channel interference I ave if it’s required that the total number of MBs should be evenly distributed over the channels. Proof: If N 1 = N 2 = ( N / 2) , from eq.(4), (5) and Proposition 1, it’s straightforward to obtain (17) ⎧2 ⎫ ntw G 1 − cut ( A, B ) ⎬ min I ave = min ⎨ ⎩N ⎭
{ }
which gives the Corollary.
(
)
)
B. Hybrid Solution (SpecHybrid algorithm): The proposed hybrid algorithm, which SpecHybrid, can be summarized as follows:
is
named
1.
Centralized phase: Apply the spectral based algorithm summarized in Table I and allocate the N MBs to the L channels accordingly.
2.
Distributed phase: Take the result of the centralized phase as initial condition, and apply the standard minimum- interference- channel allocation algorithm (e.g. an asynchronous version called GADIA in [9].):
x j (k ) = s ∋ s = arg{min (I1 , I 2 ," , I L )}
(18)
where s ∈ {1,2 ," ,L} shows the channel index, and k is time index. The algorithm continues until no more states change. IV.
SIMULATION RESULTS
We compare the performances of the proposed SpecPure and SpecHyrid algorithms with the GADIA. Semi-definite programming of [1] is used for moderate size networks for comparison reasons. DS-CDMA network is considered in all the examples. For link gains modeling, attenuation factor β = 3 , the log-normally distributed sij in (2) is generated according to the model in [14], and the lognormal variance is 4 dB. A. Example 1: In order to give an insight into some of the cases where the SpecPure outperforms the GADIA [9] which gets stuck into a local minima, some sample snapshots are presented in Fig.1, 2 and 3. The SpecPure finds the optimum solution in these cases.
(a)
(b)
Figure 1. A snapshot of the network with N=4. The center base locations are shown as stars in squares, and the circles and diamonds indicate the MB locations with their channel allocations by (a) GADIA [9] and (b) SpecPure.
V.
(a)
(b)
Figure 2. A snapshot of the network with N=25. The circles and diamonds show the channel allocations by (a) GADIA [9] and (b) SpecPure.
(a)
(b)
Figure 3. A snapshot of the network with N=40. The circles and diamonds indicate the channel allocations by (a) GADIA [9] and (b) SpecPure.
B. Example 2: In this example, we examine random location networks for moderate (N=6 to 18) and large size networks (N=30 to 150). The area of simulation scenario is 50N [m] by 50N [m]. The average results are obtained over 1000 random snapshots. In order to have an idea of how far a solution is from the global optimum, we use semidefinite programming (SDP). We define a normalized SINR gain (in dB) with respect to GADIA SINR i.e, Gain SpecHyb _ normlzd = SINR SpecHyb - SINR GADIA [dB], and Gain SDP _ normlzd = SINR SDP - SINR GADIA [dB]. Figure 4. presents the average SINR-gain per MB, The total network gain is then found by multiplying the results with N. For example, for N=100, from the figure, the total network gain is over 7 dB. As seen from the figure, the SpecHybrid clearly outperforms the GADIA. The SpecPure, on the other hand, did not give statistically reliable results for random locations scenarios with even moderate N, and therefore it’s not presented here. Example 1, on the other hand, gives an insight into some cases where the SpecPure may outperform the GADIA.
In this paper, defining an RSS based adjacency matrix, we show that i) the optimum channel/frequency allocation problem turns out to be a maxCut problem of the proposed RSS-based adjacency matrix; ii) This immediately implies that a relaxation of the original problem yields an approximate solution which is based on finding the maximum eigenvector of the graph Laplacian matrix; iii) The hybrid solution, named as SpecHybrid, (which first finds the maximum eigenvector centrally and then runs the well-known distributed standard min-interference-channel-allocation algorithm outperforms both spectral solution and the standard distributive algorithm. The proposed solutions can be adopted by any cellular, cognitive, ad-hoc or mesh type radio networks. Simulation results also show that i) the other proposed algorithm, SpecPure, may give the optimal solution either for only special cases like “perfectly” symmetrical center locations like squares-type, Manhattan-type, circle-type, etc scenarios, or for relatively small N, but it may easily give quite poor and statistically unreliable solutions for random locations; ii) it’s computationally efficient to use semi-definite programs for only middle-size wireless systems (e.g. N
