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based on a greedy approach, the user in the worst condition usually suffers. In any event, these are complex schemes and simpler schemes are needed to nish the allocation within the coherence time. To cope with these challenges, in the following a simple, ef cient and fair subcarrier allocation scheme is introduced with iterative improvement [53]. The scheme is composed of two modules, referred to as scheduling and improvement modules. In the scheduling section, bits and subcarriers are distributed to the users and passed to the improvement module where the allocation is improved iteratively by bit swapping and subcarrier swapping algorithms. The fair scheduling algorithm starts the allocation procedure with the highest level of modulation scheme. In this way, it tries to nd the best subcarrier of a user to allocate the highest number of bits. In Koutsopoulos and Tassiulas [55] the strategy is described by an analogy: The best strategy to ll a case with stone, pebble and sand is as follows. First lling the case with the stones and then lling the gap left from the stones with pebbles and in the same way, lling the gap left from pebbles with sand. Since lling in opposite direction may leave the stones or pebbles outside . With this strategy more bits can be allocated and the scheme becomes immune to uneven QoS requirements. The fair scheduling algorithm (FSA) runs greedy release algorithm (GRA) if there are nonallocated subcarriers after the lowest modulation turn and the rate requirement is not satis ed. GRA decrements one bit of a subcarrier to gain power reduction, which is used to assign higher number of bits to the users on the whole. FSA is described as follows.
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FS algorithm (1) Set c = M, Select a k, and PT = 0;
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c (2) Find n = arg minn Pk,n ;
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(3) Set Rk = Rk c and k,n = 1, update PT , shift to the next k; (4) If PT > Pmax , step out and set c = c 1, GOTO STEP 2. (5) If k, Rk < c, set c = c 1, GOTO STEP 2. (6) If {c == 1}, STEP 2. (7) Finish. The greedy releasing algorithm tends to ll the unallocated subcarriers. It releases one of the bits of the most expensive subcarrier to gain power reduction in order to drive the process. GRA works in the opposite direction to BLA. GRA is described as follows.
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K k=1 N n=1
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k,n < N , PT > Pmax , run greedy release and GOTO
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GR algorithm c (1) Find {k, n, ck,n } = arg maxk,n,c Pk,n k,n c; (2) Set ck,n = ck,n 1, PT = PT (3) Set c = ck,n 1; (4) Finish. Pk,n (ck,n );
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The horizontal swapping algorithm (HSA) aims to smooth the bit distribution of a user. When the subcarriers are distributed, the bit weight per subcarrier can be adjusted to reduce power. One bit of a subcarrier may be shifted to the other subcarrier of the same user if there is a power reduction gain. Therefore, variation of the power allocation per subcarrier is reduced and a smoother transmission is performed. HSA is described as follows. HS algorithm (1) Set PC = ; c (1a) Find {k, n, ck,n } = arg maxk,n,c (Pk,n k,n ) < PC c; (2) De ne n Sk , where { k,n == 1} for n i ; (3) Set
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c (4) Set PC = Pk,n ; (4a) If n > 0, set PT = PT n (4b) Set ck,n = ck,n 1, ck,n = ck,n + 1 GOTO Step (1a) c (5) If {PC = mink,n,c (Pk,n k,n )}, nish.
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The vertical swapping algorithm (VSA) does vertical swapping for every pair of users. In each iteration, users try to swap their subcarriers such that the power allocation is reduced. There are different types of vertical swapping. For instance, in triple swapping, user i gives its subcarrier to user j and in the same way user j to user k and user k to user i. In Koutsopoulos ad Tassiulas [55], pairwise swapping is modi ed to cope with the adaptive modulation case. In this case, there is more than one class where each class is de ned with its modulation (i.e number of bits loaded to a subcarrier) and swapping is only within the class. Each pair of user swap their subcarriers that belong to the same class if there is a power reduction. In this way, adjustment of subcarrier is done across users, to try to approximate the optimal solution. VSA is described as follows. VS algorithm (1) pair of user {i, j}; c c (1a) Find Pi, j (n) = Pi,n P j,n and (1b) Find (1c) Set (1d) Add (2) Select (3) If (4) If
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Pi, j = max Pi, j (n), n Si ; = max P j,i (n), n S j ;
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c c P j, i(n) = P j,n Pi,n n P j,i n, n Pi, j = n Pi, j + n P j,i ; n, n Pi, j to the { } list;
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= max(i,n),( j, n ) ; GOTO STEP (1a);
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