mum is achieved by one of the vertices of the polytope TC(G) representing the feasible dual solutions and defined as follows: TC(G) = Ά x ʦ ޑ + V : Α v ʦ V t(v)x(v) Յ c(t) for all t ʦ T · A classification of the vertices of this polytope will therefore lead to a comprehensive set of lower bounds that can be obtained from fractional tile covers. For any specific con- strained graph, such a classification can be obtained by using vertex enumeration soft- ware, e.g., the package lrs, developed by Avis [2]. In [18], 1-cliques in graphs with constraints c 0 , c 1 were considered. In this case the channel assignment was found to be equivalent to the tile cover problem. Moreover, the fractional tile cover problem is equivalent to the integral tile cover problem for 1-cliques, leading to a family of lower bounds that can always be attained. None of the bounds was new. Two bounds were clique bounds of the type mentioned earlier. The third bound was first given by Gamst in [12], and can be stated as follows: S(G, w) Ն max{c 0 w(v) + ( c 1 – c 0 )w(C – v) – c 0 |C a clique of G, v ʦ C} (5.6) where is such that ( – 1)c 1 < c 0 Յ c 1 . The tile cover approach led to a number of new bounds for graphs with constraints c 0 , c 1 , c 2 . The bounds are derived from so-called nested cliques. A nested clique is a d 1 -clique that contains a d 2 -clique as a subset (d 2 < d 1 ). It is characterized by a node partition (Q, R), where Q is the d 2 -clique and R contains all remaining nodes. A triple (k, u, a) will denote the constraints k = c 0 , u = c d 2 , and a = c d 1 in a nested clique. Note that in a nested clique with node partition (Q, R) with constraints (k, u, a), every pair of nodes from Q has a con- straint of at least u, while the constraint between any pair of nodes in the nested clique is at least a. The following is a lower bound for a nested clique (Q, R) with parameters (k, a, u): S(G, w) Ն a Α vʦQ w(v) + u Α vʦR w(v) – u (5.7) This bound was first derived in [12] using ad-hoc methods. The same bound can also be derived using edge covers. Using tile covers, a number of new bounds for nested cliques with parameters (k, u,1) are obtained in [22]. The following is a generalization of bound (5.6). (The notation w Qmax and w Rmax is used to denote the maximum weight of any node in Q and R, respectively.) S(G, w) Ն (k – ␦ )w Qmax + ␦ Α vʦQ w(v) + ⑀ Α vʦR w(v) – k (5.8) where = , ␦ = ( + 1)u – k k ᎏ u 5.2 LOWER BOUNDS 103 and ⑀ = min Ά , · Bound (1.3), obtained from the total weight on a clique, was extended, leading to S(G, w) Ն u Α vʦQ w(v) + w Rmax + Α vʦR,vv Rmax w(v) – k (5.9) A bound of (2u – 1)w Qmax + ⌺ vʦR w(v) – for nested cliques where Q consists of one node was obtained in [34]. This bound is generalized in [22] to all nested cliques: S(G, w) Ն (2u – 1)w Qmax + Α vʦQ,vv Qmax w(v) + Α vʦR w(v) – k (5.10) where = u – max Ά , · Finally, we mention the following two tile cover bounds from [22] for nested cliques with parameters (k, u, a): S(G, w)) Ն (3u – k + 2 ␦ ) Α vʦQ w(v) + (k – 2 ␦ )w Rmax + ␦ Α vʦR w(v) – k (5.11) where ␦ = 3a – k, and S(G, w) Ն u Α vʦV w(v) + w Rmax + Α vʦR,vv Rmax w(v) – k (5.12) In [40], a bounding technique based on network flow is described. Since no explicit formulas are given, it is hard to compare these bounds with the ones given in this section. However, in an example the authors of [40] obtain an explicit lower bound that can be im- proved upon using edge covers [1] or tile cover bounds [22]. 5.3 ALGORITHMS In this section, an overview is given of algorithms for channel assignment with general constraints. Some of these algorithms are adaptations of graph multicoloring algorithms as described in the previous chapter and others are based on graph labeling. An overview of the best-known performance ratios of algorithms for different types of graphs and con- straints is presented in Table 5.1. 3a – u ᎏ 2 ␦ – 1 ᎏ – 1 u – 1 ᎏ k – u ᎏ k – 1 2u + ␦ – ᎏᎏ k + 1 ␦ ᎏᎏ k – 2u + 1 104 CHANNEL ASSIGNMENT AND GRAPH LABELING Most of the work done has been for the case where only a cosite constraint c 0 and one edge constraint c 1 are given. As with multicoloring, a base coloring of a graph G with one color per node can be used to generate a coloring for a weighted channel assignment prob- lem having G as its underlying graph. Algorithm A (for graphs with chromatic number k) Let G = (V, E, c 0 , c 1 ) be a constrained graph, and w an arbitrary weight vector. Assume that a base coloring f : V Ǟ {0, 1, , k – 1} of the nodes of G is given. A SSIGNMENT : Let s = max{c 0 , kc 1 }. Each node v receives the channels f (v) + is, i = 0, 1, , w(v) – 1. Algorithm A has a performance ratio of max{1, kc 1 /c 0 }, and is therefore optimal if c 0 Ն kc 1 . It is a completely distributed algorithm, since every node can assign its own chan- nels independently of the rest of the network. The only information needed by a node to be able to compute its assignment is its base color. The base coloring used in Algorithm A can be seen as a graph labeling satisfying the constraint c 1 = 1. A modified version of Algorithm A, based on graph labelings, can be formulated as follows. 5.3 ALGORITHMS 105 TABLE 5.1 An overview of the performance ratios of the best known algorithms for different types of graphs. A * indicates that the performance ratio depends heavily on the constraints; see the text of Section 5.3 for details Constraints Performance ratio Reference Bipartite graphs c 0 , c 1 : c 0 Ն 2c 1 1 c 0 , c 1 : c 0 > 2c 1 1 [14] Paths c 0 , c 1 , c 2 max{1, (2c 1 + c 2 )/c 0 [42, 13] c 0 , c 1 , c 2 , c 3 * [39] Bidimensional grid c 0 , c 1 , c 2 max{1, (2c 1 + 3c 2 )/c 0 } [13, 39] c 0 , c 1 , c 2 , c 3 max{1, 5c 1 /c 0 , 10c 2 /c 0 } [3] Odd cycles (length n) c 0 , c 1 : c 0 Ն (2nc 1 )/(n – 1) 1 [21] c 0 , c 1 : 2c 1 Յ c 0 < (2nc 1 )/(n – 1) 1 + 1/(4n – 3) [21] c 0 , c 1 : c 1 Յ c 0 < 2c 1 1 + 1/(n – 1) [21] c 0 , c 1 , c 2 max{1, 3c 1 /c 0 , 6c 2 /c 0 } [15] Hexagon graphs c 0 , c 1 : 9/4 c 1 Յ c 0 max{1, 3c 1 /c 0 }— c 0 , c 1 : 2c 1 < c 0 Յ 9/4 c 1 <4/3 + 1/100 [21] c 0 , c 1 : c 0 Յ 2c 1 4/3 [21] c 0 , c 1 , c 2 * [39] c 0 , c 1 , c 2 : c 1 Ն 2c 2 5/3 + c 1 /c 2 . [9] Algorithm A ЈЈ (based on graph labeling) Let G = (V, E, c 0 , c 1 , , c k ) be a constrained graph, and w an arbitrary weight vector. As- sume that a labeling f : V Ǟ ގ is given which satisfies the constraints c 1 , , c k and has cyclic span M. A SSIGNMENT : Let s = max{c 0 , M}. Each node v receives the channels f (v) + is, i = 0, 1, . . . , w(v) – 1. Algorithm AЈ has a performance ratio of max{1, M/c 0 } and is therefore optimal if c 0 Ն M. Like Algorithm A, it is a completely distributed algorithm, where the only local infor- mation needed at each node is the value of the labeling at that node. The method of repeating a basic channel assignment of one channel per node has exist- ed since the channel assignment problem first appeared in the literature. This method is referred to as fixed assignment (FA), as each node has a fixed set of channels available for its assignment (see for example [9, 35, 25, 28]). A type of labeling that gives regular, periodic graph labelings for lattices was defined in [39], and called labeling by arithmetic progression. Such a labeling is a linear, modular function of the coordinates of each node. Definition 5.3.1 A labeling f of a t-dimensional lattice is a labeling by arithmetic progression if there exist nonnegative integers a 1 , , a t and n such that for each node v with coordinates (m 1 , , m t ), f (v) = a 1 m 1 + . . . + a t m t mod n. The parameter n is called the cyclic span of the label- ing. Given integers c 1 , c 2 , , where c 1 Ն c 2 Ն , such a labeling satisfies the con- straints c 1 , c 2 , . . . if for all pairs of nodes u, v at graph distance i in the lattice, | f (u) – f (v)| Ն max{c i , n – c i }. A labeling by arithmetic progression is considered optimal for a given set of constraints if its cyclic span is as small as possible. Given f, a labeling by arithmetic progression, f (m 1 , m 2 ) denotes the value of the label- ing at the node with coordinates (m 1 , m 2 ). Labelings by arithmetic progression are easy to define and with Algorithm AЈ they can be used to find channel assignment algorithms. Moreover, their regularity may be helpful in designing borrowing methods that will give better channel assignments for nonuniform weights. 5.3.1 Bipartite Graphs For bipartite graphs with constraints c 0 and c 1 , Algorithm A gives optimal channel assign- ments if c 0 Ն 2c 1 . If c 0 < 2c 1 , bipartite graphs can be colored optimally using Algorithm B, given by Gerke [14]. Like Algorithm A, this algorithm uses base coloring of the nodes, but if a node has demand greater than any of its neighbors, it initially gets some channels that are 2c 1 apart (which allows interspersing the channels of its neighbors), while the lat- er channels are c 0 apart. Algorithm B (for bipartite graphs when c 1 Յ c 0 Յ 2c 1 ) Let G = (V, E, c 0 , c 1 ) be a constrained bipartite graph of n nodes, where c 1 Յ c 0 Յ 2c 1 , and w an arbitrary weight vector. Assume a base coloring f : V Ǟ {0,1} is given. For each node v, define p(v) = max{w(u) | uv ʦ E or u = v}. 106 CHANNEL ASSIGNMENT AND GRAPH LABELING A SSIGNMENT : Initially, each node v receives channels f (v)c 1 +2ic 1 , i = 0, 1, , p(v) – 1. If w(v) > p(v), then v receives the additional channels f (v)c 1 + 2p(v)c 1 + ic 0 , i = 0, , w(v) – p(v) – 1. The span of the assignment above is at most max (uv)ʦE {c 0 w(u) + (2c 1 – c 0 )w(v)}. It fol- lows from lower bound 5.6 that the algorithm is (asymptotically) optimal. In fact, [14] gives a more detailed version of the algorithm above that is optimal in the absolute sense. For higher constraints, the only results available are for graph labelings of specific bi- partite graphs. Van den Heuvel et al. [39] give labelings by arithmetic progression for sub- graphs as the line lattice (paths). Such labelings only have n (the cyclic span) and a 1 = a as parameters. If f is such a labeling, then a node v defined by the vector me will have value f (v) = ma mod n. The parameters of the labelings are displayed in the table below. These labelings are optimal in almost all cases. The exception is the case where there are three constraints c 1 , c 2 , and c 3 , and 2c 2 – c 3 Յ c 1 Յ ( 1 – 2 )c 2 + c 3 . For this case, a periodic labeling not based on arithmetic progressions is given in the same paper. Constraints na c 1 , c 2 2c 1 + c 2 c 1 c 1 , c 2 , c 3 : c 1 Ն c 2 + c 3 2c 1 + c 2 c 1 c 1 , c 2 , c 3 : c 2 + (1/3)c 3 Յ c 1 Յ c 2 + c 3 3c 2 + 2c 3 c 2 + c 3 c 1 , c 2 , c 3 : c 1 Յ c 2 + (1/3)c 3 3c 1 + c 3 c 1 For paths of size at least five, these labelings include the optimal graph labeling satis- fying constraints c 1 = 2, c 2 = 1 given by Yeh in [42], and the path labelings for general con- straints c 1 , c 2 by Georges and Mauro in [13]. Note that Algorithm AЈ, used with any of these labelings with cyclic span n, has a performance ratio of max{1, n/c 0 }. The near-optimal labeling for unit interval graphs given in [32] can be applied to paths with constraints c 1 , c 2 , , c 2r , where c 1 = c 2 = = c r = 2 and c r+1 = c 2r = 1, to give a labeling with cyclic span 2r + 1. Using this labeling in Algorithm AЈ leads to a perfor- mance ratio of max{1, (2r + 1)/c 0 }. Van de Heuvel et al. [39] also give an optimal labeling by arithmetic progression for the square lattice and constraints c 1 , c 2 . The labeling given has cyclic span n = 2c 1 + 3c 2 and is defined by the parameters a 1 = c 1 , a 2 = c 1 + c 2 . The square lattice is the Cartesian product graph of two infinite paths, and similar labelings can also be derived from the re- sults on products of paths given in [13]. Bertossi et al. [3] give a labeling for constraints c 1 = 2, c 1 = c 3 = 1 of span 8 and cyclic span 10. This labeling can be transformed into a labeling for general c 1 , c 2 , c 3 as follows. Let c = max{c 1 /2, c 2 }, and let f be the labeling for c 1 , c 2 , c 3 = 2, 1, 1. Let f Ј (u) = cf (u). It is easy to check that f Ј is a labeling for c 1 , c 2 , c 3 of cyclic span 10c. Using this labeling with Algorithm AЈ gives a performance ratio of max{1, 5c 1 /c 0 , 10c 2 /c 0 }. The same authors give a labeling for bidimensional grids with constraints c 1 = 2, c 2 = 1, which is just a special case the labeling by arithmetic progression given above. The same authors also give labelings for graphs they call hexagonal grids, with con- straints c 1 , c 2 = 2, 1 and c 2 , c 1 , c 3 = 2, 1, 1. Hexagonal grids are not to be confused with hexagon graphs, which will be discussed in Section 5.3.3. In fact, hexagonal grids are 5.3 ALGORITHMS 107 subgraphs of the planar dual of the infinite triangular lattice. Hexagonal grids form a reg- ular arrangement of 6 cycles, and are bipartite. Labelings for the hypercube Q n were described and analyzed in [15, 24, 41]. Graph la- belings for trees with constraints c 1 , c 2 = 2, 1 were treated in [5] and [15]. These labelings are obtained using a greedy approach, which is described in Section 5.3.4. 5.3.2 Odd Cycles Channel assignment on odd cycles was first studied by Griggs and Yeh in [15]. The au- thors give a graph labeling for constraints c 1 , c 2 = 2, 1 of span 4 and cyclic span 6. The la- beling repeats the channels 0, 2, 4 along the cycle, with a small adaptation near the end if the length of the cycle is not divisible by 3. As described in the previous section, this la- beling can be used for general constraints c 1 , c 2 if all values assigned by the labeling are multiplied by max{c 2 , c 1 /2}. Using Algorithm AЈ, this leads to an algorithm with perfor- mance ratio max{1, 3c 1 /c 0 , 6c 2 /c 0 }. In [21], three basic algorithms for odd cycles are combined in different ways to give optimal or near-optimal algorithms for all possible choices of two constraints c 0 and c 1 . The first of the three algorithms in [21] is based on a graph labeling that satisfies one constraint c 1 . This labeling has cyclic span c R = 2nc 1 /(n – 1). It starts by assigning zero to the first node, and then adding c 1 (modulo c R ) to the previously assigned channel and as- signing this to the next node in the cycle. At a certain point, this switches to an alternating assignment. This labeling is then used repeatedly, as in Algorithm AЈ. Since this particular form of Algorithm AЈ will be used to describe the further results in this chapter, I will state it explicitly below. Algorithm C (for odd cycles) Let G = (V, E, c 0 , c 1 ) be a constrained cycle of n nodes, where n > 3 is odd, and w be an ar- bitrary weight vector. Fix s = max{c 0 , c R }. Let the nodes of the cycle be numbered {1, , n}, numbered in cyclic order, where node 1 is a node of maximum weight in the cy- cle. Let m > 1 be the smallest odd integer such that s Ն 2m/(m – 1)c 1 (it can be shown that such an integer must exist). A SSIGNMENT : To each node i, the algorithm assigns the channels b(i) + js, where j = 0, . . . , w(i) – 1, and the graph labeling b : V Ǟ [0, s – 1] is defined as follows: (i – 1)c 1 mod s when 1 Յ i Յ m, b(i) = Ά 0 when i > m and i is even, (m – 1)c 1 mod s when i > m and i is odd. Note that this algorithm can only be implemented in a centralized way, since every node must know all weights, in order to calculate m, and so determine its initial assign- ment value. The second algorithm is a straightforward adaptation of the optimal algorithm for mul- ticoloring an odd cycle, described in [29] and discussed in the previous chapter. The span used by this algorithm is /2s. 108 CHANNEL ASSIGNMENT AND GRAPH LABELING Algorithm D (for odd cycles) Let G = (V, E, c 0 , c 1 ) be a constrained cycle of n nodes, where n > 3 is odd, and w be an ar- bitrary weight vector. Fix s = max{c 0 , 2c 1 }, and = max{2⌺ vʦV w(v)/(n – 1), 2w max }. Let f be an optimal multicoloring of (G, w) using the colors {0, 1, , – 1}. Such an f exists since (G, w) Յ . A SSIGNMENT : For each node v, replace each color i in f (v) with the channel f i , where is if i Յ \2 – 1, f i = Ά c 1 + (i – – 2 )s otherwise. Algorithms C and D only give good assignments for weight vectors with specific prop- erties, but they can be combined to give near-optimal algorithms for any weight vector. How they are combined will depend on the relation between the parameters. First, note that Algorithm C is optimal if c 0 Ն c R = 2nc 1 /(n – 1). If 2c 1 Յ c 0 < c R , then Algorithms A, C, and D can be combined to give a linear time al- gorithm with performance ratio 1 + 1/(4n – 3), where n is the number of nodes in the cy- cle. The algorithm is described below. Given a weight vector w, compute ␦ = ⌺ vʦV w(v) – (n – 1)w max . If ␦ Յ 0, Algorithm D is used, with spectrum [0, c 0 w max ]. The span is at most c 0 w max , which is within a constant of lower bound (5.2), so the assignment is optimal. If instead ␦ > 0, Algorithm C is combined with either Algorithm A or D to derive an as- signment. Denote by f 1 the assignment computed by Algorithm C for (G, wЈ) where wЈ(v) = min{w(v), ␦ }. This assignment has span at most c R ␦ . Consider the remaining weight w ෆ after this assignment. Clearly w ෆ max = w max – ␦ . We will denote by f 2 the assignment for (G, w ෆ ), and compute it in two different ways depend- ing on a key property of w ෆ . If there is a node v with w ෆ (v) = 0 at this stage, we have a bipar- tite graph left. Then f 2 is the assignment computed by Algorithm A for (G, w ෆ ). This assign- ment has a span of at most c 0 w ෆ max . If all nodes have nonzero weight, then Algorithm D is used to compute f 2 , the assign- ment for (G, w ෆ ). It can be shown that in this case, = 2w ෆ max , so this assignment also has a span of at most c 0 /2 = c 0 w max . Thus, in either case, f 2 has span at most c 0 w ෆ max . The two assignments f 1 and f 2 are then combined by adding c R ␦ + c 0 to every channel in f 2 , and then merging the channel sets assigned by f 1 and f 2 at each node. This gives a final assignment of span at most (c R – c 0 ) ␦ + c 0 w max + c 0 . Using the lower bounds (5.2) and (5.3), it can be shown that the performance ratio of the algorithm is as claimed. If c 0 < 2c 1 , Algorithms B and C can be combined into a linear time approximation al- gorithm with performance ratio 1 + 1/(n – 1), where n is the number of nodes in the cycle. The combination algorithm is formed as follows. First, find the assignment f 1 computed by Algorithm C for (G, wЈ) where wЈ(v) = w min for every node v. Then, find the assignment f 2 computed by Algorithm B for (G, wЈЈ) where wЈЈ(v) = w(v) – w min . Finally, combine the two assignments by adding c R w min + c 0 to each channel of f 2 and then merging the channel sets assigned by f 1 and f 2 . Using bound 1.6, it can be shown that the algorithm has performance ratio 1 + 1/(n – 1) as claimed. 5.3 ALGORITHMS 109 In [13], optimal graph labelings for odd cycles with constraints c 1 , c 2 are given. If c 1 > 2c 2 , or c 1 Յ 2c 2 and n ϵ 0 mod 3, the span is 2c 1 , and the cyclic span is 3c 1 . Using Algo- rithm AЈ in combination with this labeling gives a performance ratio of max{1, 3c 1 /c 0 }. For the remaining case, the span is c 1 + 2c 2 and the cyclic span is c 1 + 3c 2 , leading to a performance ratio for Algorithm AЈ of max{1, (c 1 + 3c 2 )/c 0 }. In [3], Bertossi et al. give a graph labeling for cycles of length at least 4 with constraints c 1 , c 2 , c 3 = 2, 1, 1. The span of the labeling is 4, and its cyclic span is 6. Adapting this labeling to general parameters c 1 , c 2 , c 3 and using Algorithm AЈ gives a performance ratio of max{1, 3c 1 /c 0 , 6c 2 /c 0 }. 5.3.3 Hexagon Graphs The first labelings for hexagon graphs were labelings by arithmetic progression given by van den Heuvel et al. in [39]. The labelings, as defined by their parameters a 1 , a 2 , and n, are given in the table below. Parameters na 1 a 2 c 1 Ն 2c 2 3c 1 + 3c 2 2c 1 + c 2 c 1 (3/2)c 2 Յ c 1 Յ 2c 1 9c 2 5c 2 2c 2 c 1 Յ (3/2) c 2 4c 1 + 3c 2 2c 1 + c 2 c 1 It can be easily seen that hexagon graphs admit a regular coloring with three colors. Hence Algorithm A will be optimal for constraints c 0 , c 1 so that c 0 Ն 3c 1 . A channel as- signment algorithm for hexagon graphs with constraints c 0 , c 1 = 2, 1 with performance ra- tio 4/3 was given in [36]. In [21], further approximation algorithms for hexagon graphs and all values of con- straints c 0 , c 1 are given. All algorithms have performance ratio not much more than 4/3, which is the performance ratio of the best known multicoloring algorithm for hexagon graphs (see [28]). The results are obtained by combining a number of basic algorithms for hexagon graphs and bipartite graphs. The algorithm described below is similar to the one in [36]. Algorithm E (for 3-colorable graphs) Let G = (V, E, c 0 , c 1 ) be a constrained graph, and w be an arbitrary weight vector. Fix s = max{c 1 , c 0 /2} and T Ն 3w max , T a multiple of 6. Let f : V Ǟ {0, 1, 2} be a base coloring of G. Denote base colors 0, 1, 2 as red, blue and green, respectively. A set of red channels is given, consisting of a first set R 1 = [0, 2s, , (T/3 – 2)s] and a second set R 2 = [(T/3 + 1)s + c 0 , (T/3 + 3)s + c 0 , , (2T/3 – 1)s + c 0 ]. Blue channels consist of first set B 1 = [(T/3)s + c 0 , (T/3 + 2)s + c 0 , , (2T/3 – 2)s + c 0 ] and second set B 2 = [(2T/3 + 1)s + 2c 0 , (2T/3 + 3)s + 2c 0 , , (T – 1)s + 2c 0 ], and green channels consist of first set G 1 = [(2T/3)s + 2c 0 , (2T/3 + 2)s + 2c 0 , , (T – 2)s + 2c 0 ] and second set G 2 = [s, 3s, , (T/3 – 1)s]. A SSIGNMENT : Each node v is assigned w(v) channels from those of its color class, where the first set is exhausted before starting on the second set, and lowest numbered channels are always used first within each set. 110 CHANNEL ASSIGNMENT AND GRAPH LABELING Note that the spectrum is divided into three parts, each containing T/3 channels, with a separation of s between consecutive channels. The first part of the spectrum consists of al- ternating channels from R 1 and G 2 , the second part has alternating channels from B 1 and R 2 , and the third part has alternating channels from G 1 and B 2 . The span used by Algo- rithm E equals sT + 2c 0 = max{c 1 , c 0 /2}T + 2c 0 , where T is at least 3w max . To obtain the optimal algorithms for hexagon graphs and different values of the para- meters c 0 , c 1 , Algorithm E is modified and combined with Algorithms A and B. Algorithm A for hexagon graphs has a performance ratio of max{1, 3c 1 /c 0 }. As noted, when c 0 Ն 3c 1 the algorithm is optimal. When c 0 Ն (9/4)c 1 , the performance ratio of equals 3c 1 /c 0 , which is at most 4/3. For the case where 2c 1 < c 0 Յ (9/4)c 1 , a combination of Algorithms A for hexagon graphs and Algorithm E followed by a borrowing phase and an application of Algorithm B results in an algorithm with performance ratio less than 4/3 + 1/100 The algorithm is outlined below. Let D represent the maximum weight of any maximal clique (edge or triangle) in the graph. It follows from lower bound (1.3) that S(G, w) Ն c 1 D – c 1 . For ease of explanation, we assume that D is a multiple of 6. Phase 1: If D > 2w max , use Algorithm A for hexagon graphs on (G, wЈ) where wЈ(v) = min{w(v), D – 2w max }. If D Յ 2w max , skip this phase, and take wЈ(v) = 0 for all v. The span needed for this phase is no more than max{0, D – 2w max }3c 1 . Phase 2: Let T = min{2w max , 6w max – 2D}. Use Algorithm E on (G, wЈЈ), where wЈЈ(v) = min{w(v) – wЈ(v), T/3}, taking T as defined. The span of the assignment is min{2w max , (6w max – 2D)}c 0 /2 + 2c 0 . It follows from the description that after this phase, in every triangle there is at least one node that has received a number of channels equal to its de- mand. Phase 3: Any node that has still has unfulfilled demand tries to borrow channels assigned in Phase 2 from its neighbors according to the following rule: red nodes borrow only from blue neighbors, blue from green, and green from red. A red node v with w(v) > wЈ(v) + wЈЈ(v), where w B (v) is the maximum number of channels used during Phase 2 by any blue neighbor of v, receives an additional min{w(v) – wЈ(v) – wЈЈ(v), T/3 – w B (v), T/6} channels from the second blue channel set B 2 , starting from the highest channels in the set. A similar strategy is followed for blue and green nodes. It can be shown that the graph induced by the nodes that still have unfulfilled demand after this phase is bipartite. Phase 4: Let w ෆ denote the weight left on the nodes after the assignments of the first three phases. Use Algorithm A to find an assignment for (G, w ෆ ), which has a span of c 0 w ෆ max . The assignments of all four phases are then combined without conflicts, as in the theo- rems for odd cycles. The final assignment has span at most (2w max )c 0 /2 + c 0 (w max /3) + ⌰(1) = (4/3)c 0 w max + ⌰(1). It then follows from lower bounds (5.2) and (5.3) that the per- formance ratio equals 1 + 3(c 0 – 2c 1 )/c 0 + (9c 1 – 4c 0 )/3c 1 . When 2c 1 < c 0 Յ (9/4)c 1 , this is always less than 4/3 + 1/100. In particular, the maximum value is reached when c 0 /c 1 = 3/ ͙ 2 ෆ . When c 0 = 2c 1 or c 0 = 9c 1 /4, the performance ratio is exactly 4/3. When c 0 Յ 2c 1 , a linear time approximation algorithm with performance ratio 4/3 is obtained from an initial assignment by Algorithm E, followed by a borrowing phase and a 5.3 ALGORITHMS 111 phase where assigned channels are rearranged in the spectrum, and finally an application of Algorithm B. The algorithm follows. Let L = max{c 0 w(u) + (2c 1 – c 0 )(w(v) + w(r))|{u, v, r} a triangle} and let T be the smallest multiple of 6 larger than max{L, Dc 1 }/c 1 . It follows from lower bounds (5.6) and (5.3) that Tc 1 – ⌰(1) is a lower bound for the span of any assignment. Phase 1: Use Algorithm E on (G, wЈ) where wЈ(v) = min{w(v), T/3} and T is defined above. In this case s, the separation between channels, equals c 1 , so the span of the as- signment is Tc 1 . Phase 2: Any red node v of weight greater than T/3 borrows min{w(v) – T/3, T/3 – w B (v), T/6} channels, where w B (v) is the maximum weight of any blue neighbor of v. The channels are taken only from the second blue channel set, and start with the highest channels. Blue and green nodes borrow analogously, following the borrowing rules giv- en earlier (red Ǟ blue Ǟ green Ǟ red). Phase 3: Any red node v of weight more than T/3, whose blue neighbors have weight at most T/6, will squeeze their assigned channels from their second set as much as possi- ble. More precisely, the last T/6 – w B (v) channels assigned to v from R 2 are replaced by min{w(v) – T/3 – w B (v), 2c 1 /c 0 (T/6 – w B (v))} channels with separation c 0 which fill the part of the spectrum occupied by the last T/6 – w B (v) channels of R 2 . For example, let T = 24, c 0 = 3, and c 1 = 2. Suppose v is a red corner node with at least two green neigh- bors, where w(v) = 13 and let w B (v) = 1. In Phase 1, v received the channels 21, 25, 29, 33 from the set R 2 , whereas at least one blue neighbor of v received the channel 19 from B 1 and no other channels from B 1 or B 2 were used by any neighbor of v. Then in Phase 2, v borrows all four blue channels in B 2 , and in Phase 3, squeezes the part of the spec- trum [21, 33] of R 2 to get five channels. In particular, it uses the channels 21, 24, 27, 30, 33 instead of the four channels mentioned above. The reader can verify that in this ex- ample, cosite and intersite constraints are respected. Phase 4: Let w ෆ be the weight vector remaining after Phase 3. It can be shown that the graph induced by the nodes with positive remaining weight is bipartite. We use Algo- rithm B to find an assignment for (G, w ෆ ), which has a span of L Ј = max{c 0 w ෆ (u) + (2c 1 – c 0 )w ෆ (v)|(u, v) ʦ E}. The assignments of different phases are then combined without causing conflicts, in the same way as described before, to give a final assignment of span at most (4/3) Tc 1 + ⌰(1). From the definition of T, we have that Tc 1 – ⌰(1) is a lower bound, which gives the required performance ratio of 4/3. In [3], a labeling is given for hexagon graphs with constraints c 1 , c 2 , c 3 = 2, 1, 1. (Hexa- gon graphs are referred to as cellular grids in this paper.) The labeling has a span of 8, which is proven to be optimal, and a cyclic span of 9. Moreover, when examined it can be determined that this labeling is, in fact, a labeling by arithmetic progression, with parame- ters n = 9, a = 2, b = 6. It therefore follows from the results of van de Heuvel et al. that the labeling is optimal, since 9 is the optimal span even for constraints c 1 , c 2 = 2, 1. This la- 112 CHANNEL ASSIGNMENT AND GRAPH LABELING [...]... assignment and graph multicoloring, in I Stojmenovic (Ed.), Handbook of Wireless Networks and Mobile Computing, New York: Wiley, 2001 29 L Narayanan and S Shende, Static frequency assignment in cellular networks, in Proceedings of SIROCCO 97, pp 215–227 Carleton Scientific Press, 1977 To appear in Algorithmica 30 M G C Resende R A Murphey, P M Pardalos, Frequency assignment problems, in D.-Z Du and P M... (Eds.), Handbook of Combinatorics Kluwer Academic Publishers, 1999 31 A Raychaudhuri, Intersection assignments, T-colourings and powers of graphs, PhD thesis, Rutgers University, 1985 32 D Sakai, Labeling chordal graphs: Distance two condition, SIAM J Discrete Math., 7: 133 –140, 1994 33 D Smith and S Hurley, Bounds for the frequency assignment problem, Discr Math., 167/168: 571–582, 1997 34 C Sung and. .. 1990 Handbook of Wireless Networks and Mobile Computing, Edited by Ivan Stojmenovic ´ Copyright © 2002 John Wiley & Sons, Inc ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic) CHAPTER 6 Wireless Media Access Control ANDREW D MYERS and STEFANO BASAGNI Department of Computer Science, University of Texas at Dallas 6.1 INTRODUCTION The rapid technological advances and innovations of the past few... access services Given the slow reaction time of government bureaucracy and the high cost of licensing, wireless operators are typically forced to make due with limited bandwidth resources The aim of this chapter is to provide the reader with a comprehensive view of the role and details of the protocols that define and control access to the wireless channel, i.e., wireless media access protocols (MAC) protocols... characteristics of wireless systems and their impact on the design and implementation of MAC protocols (Section 6.2) Section 6 .3 explores the impact of the physical limitations specific to MAC protocol design Section 6.4 lists the set of MAC techniques that form the core of most MAC protocol designs Section 6.5 overviews channel access in cellular telephony networks and other centralized networks Section... algorithm for channel assignment under conchannel and adjacent channel interference constraint, IEEE Trans Veh Techn., 46 (3) , 1997 35 S W Halpern, Reuse partitioning in cellular systems, in Proc IEEE Conf on Veh Techn., pp 32 2 32 7 New York: IEEE, 19 83 36 S Ubéda and J Zerovnik, Upper bounds for the span in triangular lattice graphs: application to REFERENCES 37 38 39 40 41 42 117 frequency planning for cellular... 25 I Katzela and M Naghshineh, Channel assignment schemes for cellular mobile telecommunications: a comprehensive survey, IEEE Personal Communications, pp 10 31 , June 1996 26 R A Leese, Tiling methods for channel assignment in radio communication networks, Z Angewandte Mathematik und Mechanik, 76: 30 3 30 6, 1996 27 Colin McDiarmid and Bruce Reed, Channel assignment and weighted colouring, Networks, 1997... provide an overview of two of the most prevalent centralized wireless networks Cellular telephony is the most predominant form of wireless system in current operation Wireless ATM is generating a lot of interest for its ability to deliver broadband multimedia services across a wireless link Each system will be briefly highlighted and the MAC protocol will be examined 128 WIRELESS MEDIA ACCESS CONTROL... toward wireless broadband access to the Internet and multimedia content With predictions of near exponential growth in the number of wireless subscribers in the coming decades, pressure is mounting on government regulatory agencies to free up the RF spectrum to satisfy the growing bandwidth demands This is especially true with regard to the next generation (3G) cellular systems that integrate voice and. .. for hexagonal cellular networks with constraints, Technical Report G-2000-14, GERAD, HEC, Montreal, March 2000 10 D Fotakis, G Pantziou, G Pentaris, and P Spirakis, Frequency assignment in mobile and radio networks, in Proceedings of the Workshop on Networks in Distributed Computing, DIMACS Series AMS, 1998 11 D A Fotakis and P G Spirakis, A hamiltonian approach to the assignment of non-reusable frequencies, . c 3 : c 1 Ն c 2 + c 3 2c 1 + c 2 c 1 c 1 , c 2 , c 3 : c 2 + (1 /3) c 3 Յ c 1 Յ c 2 + c 3 3c 2 + 2c 3 c 2 + c 3 c 1 , c 2 , c 3 : c 1 Յ c 2 + (1 /3) c 3 3c 1 + c 3 c 1 For paths of size at least five,. assignment and weighted colouring, Networks, 1997. To appear. 28. L. Narayanan. Channel assignment and graph multicoloring, in I. Stojmenovic (Ed.), Handbook of Wireless Networks and Mobile Computing, . [(T /3) s + c 0 , (T /3 + 2)s + c 0 , , (2T /3 – 2)s + c 0 ] and second set B 2 = [(2T /3 + 1)s + 2c 0 , (2T /3 + 3) s + 2c 0 , , (T – 1)s + 2c 0 ], and green channels consist of first set G 1 = [(2T /3) s