Efficient Use of Radio Spectrum in Wireless Networks with Channel Separation between Close Stations Alan A. Bertossi
Cristina M. Pinotti
Richard B. Tan
Department of Mathematics University of Trento 38050 Povo (Trento), Italy
IEI-CNR 56100 Pisa, Italy
Dept. of Computer Science University of Sciences & Arts of Oklahoma, USA
[email protected]
[email protected]
ABSTRACT
This paper investigates the problem of assigning channels to the stations of a wireless network so that interfering transmitters are assigned channels with a given separation and the number of channels used is minimized. Two versions of the channel assignment problem are considered which are equivalent to two speci c coloring problems { called L(2; 1) and L(2; 1; 1) { of the graph representing the network topology. In these problems, channels assigned to adjacent vertices must be at least 2 apart, while the same channel can be reused only at vertices whose distance is at least 3 or 4, respectively. Ecient channel assignment algorithms using the minimum number of channels are provided for speci c, but realistic, network topologies, including buses, rings, hexagonal grids, bidimensional grids, cellular grids, and complete binary trees.
Categories and Subject Descriptors
C.2 [Computer-Communication Networks]; C.2.1 [Network Architecture and Design]: Wireless Communication 1. INTRODUCTION
The tremendous growth of wireless networks requires an ef cient use of the scarce radio spectrum allocated to wireless communications. However, the main diculty against an ecient use of radio spectrum is given by interferences, caused by unconstrained simultaneous transmissions. Interferences can be eliminated (or at least reduced) by means of suitable channel assignment techniques, which partition the given radio spectrum into a set of disjoint channels that can be used simultaneously by the stations while maintaining acceptable radio signals. By taking advantage of physical characteristic of the radio environment, the same channel can be reused by two stations at the same time without interferences (co-channel stations), provided that the two staThis work has been supported by the \Provincia Autonoma di Trento (PAT)" and \MURST" under research grants.
[email protected]
tions are spaced suciently apart. The minimum distance at which co-channels can be reused with no interferences is called co-channel reuse distance . Unfortunately, the interference phenomena may be so strong that even di erent channels used at near stations may interfere if the channels are too close. Therefore, channels assigned to near stations must be separated by at least a gap which is inversely proportional to the distance between the two stations. In other words, a minimum channel separation i is required between channels assigned to stations at distance i, with i < , such that i decreases when i increases [8]. The purpose of channel assignment algorithms is to assign channels to transmitters in such a way that the co-channel reuse distance and the channel separation constraints are veri ed and the difference between the highest and lowest channels assigned is kept as small as possible. Formally, the channel assignment problem with separation (CAPS) can be modeled as an appropriate coloring problem on an undirected graph G = (V; E ) representing the network topology, whose vertices in V correspond to stations, and edges in E correspond to pairs of stations that can hear each other transmission. Let the set of channels C be a set of nonnegative integers, and let the distance d(u; v) between stations u and v of the network be the number of edges on a shortest path between the corresponding vertices u and v of G. Given the co-channel reuse distance and the channel separation vector (1 ; 2 ; : : : ; ?1 ) of nonnegative integers, an L(1 ; 2 ; : : : ; ?1 )-coloring of the graph G is a function f from the vertex set V to the set C of channels, or colors, such that jf (u) ? f (v)j i if d(u; v) = i, for i = 1; 2; : : : ; ? 1. Note that the same color can be used for vertices which are at least at distance , while colors which are at least i apart must be used for vertices at distance i, with i < . In particular, the co-channel reuse constraint can be stated in terms of the channel separation constraint by xing all the i to 1, for 1 i ? 1. A k-L(1 ; 2 ; : : : ; ?1 ) -coloring is an L(1 ; 2 ; : : : ; ?1 )coloring of the graph G such that C = f0; : : : ; kg. An optimal L(1 ; 2 ; : : : ; ?1 ) -coloring is one k-L(1 ; 2 ; : : : ; ?1 )coloring using the smallest number k. The largest color used by an optimal L(1 ; 2 ; : : : ; ?1 )-coloring is denoted by (G). Note that, since the set C contains 0, the overall number of colors involved by an optimal coloring f is in fact (G) + 1 (although, due to the channel separation constraint, some colors in f1; : : : ; (G) ? 1g might not be actu-
ally assigned to any vertex). Finally, given a network G, the co-channel reuse distance and the channel separation vector (1 ; 2 ; : : : ; ?1 ), the CAPS problem does correspond to nding an optimal L(1 ; 2 ; : : : ; ?1 )-coloring of G. The CAPS problem has been shown to be NP -hard for general graphs, and therefore it is computationally intractable. If the channel separation vector (1 ; 2 ; : : : ; ?1 ) is equal to (1; 1; : : : ; 1), there is only the co-channel reuse constraint, but no channel separation constraint [9]. In such a case, the channel assignment problem has been widely studied in the past, e.g. see the paper by Chlamtac and Pinter [6]. In particular, the intractability of optimal L(1; 1; : : : ; 1)-coloring, for any positive integer , has been proved by McCormick [11]. For = 3, Battiti, Bertossi and Bonuccelli [1] found optimal L(1; 1)-colorings for rings, hexagonal, cellular, and bidimensional grids, as well as ecient heuristics for geometric graphs. Optimal L(1; 1; : : : ; 1)-colorings, for any positive integer , have been proposed by Bertossi and Pinotti [3] for rings, complete trees, and bidimensional grids. When the channel separation constraint is present, the problem has been studied only for = 3 and (1 ; 2 ) = (2; 1). The intractability of the L(2; 1)-coloring has been shown by Griggs and Yeh [7] along with bounds on the number of channels for buses, rings and hypercubes. Later, bounds for this problem on chordal graphs and arbitrary trees have been found, respectively, by Sakai [12], by Chang and Kuo [5]. Approximated solutions for outerplanar, permutation and split graphs are presented by Bodlaender et al. [4]. As a related case, when = 3 and (1 ; 2 ) = (0; 1), the L(0; 1)coloring problem models that of avoiding only the so-called hidden interferences, due to stations which are outside the hearing range of each other and transmit to the same receiving station. Optimal L(0; 1)-colorings have been provided by Makansi [10] for buses and bidimensional grids. Bertossi and Bonuccelli [2] proved the intractability of optimal L(0; 1)-coloring, giving also optimal solutions for rings and complete binary trees. Finally, as another related case, observe that the classical minimum vertex coloring problem on undirected graphs arises when = 2 and 1 = 1. Thus, the minimum vertex coloring problem consists in nding an optimal L(1)-coloring. This paper further investigates the L(2; 1)-coloring problem, and starts to investigate also the L(2; 1; 1)-coloring problem. Since Griggs and Yeh intractability proof for L(2; 1)coloring can be easily extended also to the L(2; 1; 1)-coloring problem, optimal solutions for special networks are considered. In particular, solutions to the L(2; 1)-coloring and L(2; 1; 1)-coloring problems are provided for hexagonal, bidimensional and cellular grids. Moreover, optimal solutions to the L(2; 1; 1)-coloring problem are exhibited also for rings and complete binary trees. For all networks, except for binary trees, a channel can be assigned to any vertex in constant time, provided that the relative position of the vertex in the network is locally known. For binary trees, instead, a logarithmic time in the number of vertices is required. 2. PRELIMINARIES
The channel assignment problem on a network N with no channel separation constraint and co-channel reuse distance , namely the L(1; 1; : : : ; 1)-coloring problem, can be reduced to a classical coloring problem on an augmented graph
GN; obtained as follows. The vertex set of GN; is the same as the vertex set of N , while an edge [r; s] belongs to the edge set of GN; if and only if the distance d(r; s) between the vertices r and s in N satis es d(r; s) ? 1. Now, colors must be assigned to the vertices of GN; so that every pair of vertices connected by an edge is assigned a couple of di erent colors and the minimum number of colors is used. Hence, the role of maximum clique in GN; is apparent for deriving lower bounds on the minimum number of channels for the L(1; 1; : : : ; 1)-coloring problem on N . A clique K for GN; is a subset of vertices of GN; such that for each pair of vertices in K there is an edge. By well-known graph theoretical results, a clique of size n in the augmented graph GN; implies that at least n di erent colors are needed to color GN; . In other words, the size of the largest clique in GN; is a lower bound for the number of channels required to solve the channel assignment problem without channel separation constraint. Clearly, in the presence of both the channel separation and the co-channel reuse constraints, at least as many channels are required as in the presence of the co-channel reuse constraint only. Formally, a lower bound for the L(1; 1; : : : ; 1)-coloring problem is also a lower bound for the L(1 ; 1; : : : ; 1)-coloring problem, with 1 1. In particular, lower bounds for L(1; 1) and L(1; 1; 1) hold also for L(2; 1) and L(2; 1; 1). Let the complement graph G = (V; E) of a graph G = (V; E ) be the graph having the same vertex set V as G and having the edge set E obtained by swapping edges with non-edges in E . Recall that a Hamilton path is a path that traverses each vertex of a graph exactly once. Lemma 1. Consider the L(1 ; 1; : : : ; 1)-coloring problem, with 1 2, on a graph G = (V; E ) such that d(u; v) < for every pair of vertices u and v in V . Then, (G) = jV j ? 1 if and only if G has a Hamilton path. Proof To satisfy the channel separation constraint, two vertices of G may get two consecutive colors if and only if they are not adjacent, that is, if and only if they are adjacent in G. If (G) = jV j ? 1 then there is an ordering v0 ; v1 ; : : : ; vjV j?1 of the vertices such that f (vi ) = i, for i = 0; 1; : : : ; jV j ? 1. Therefore, v0 ; v1 ; : : : ; vjV j?1 is a Hamilton path for G. Conversely, if G has a Hamilton path v0 ; v1 ; : : : ; vjV j?1 , then the optimal L(1 ; 1; : : : ; 1)-coloring is f (vi ) = i, for i = 0; 1; : : : ; jV j ? 1. 2 Consider the star graph S which consists of a center vertex c with degree , and ray vertices of degree 1. Lemma 2. Let the center c of S be already colored. Then, the largest color required for a k-L(2; 1)-coloring of S is at least: + 1 if f (c) = 0 or f (c) = + 1; k= + 2 if 0 < f (c) < + 1;
Proof By Lemma 1, k + 1 since S has no Hamilton path. If f (c) = 0 or f (c) = +1, either color 1 or violates the channel separation constraint and cannot be used at the ray vertices. Similarly, if 0 < f (c) < + 1, both colors f (c) ? 1 and f (c) + 1 cannot be used at the ray vertices. Therefore, one or two extra colors are required with respect to the star size, depending on the center color f (c). 2
0
3
0
3
0
2
5
2
5
2
4
1
4
1
4
0
3
0
3
0
2
5
2
5
2
4
1
4
1
4
Figure 1: Optimal coloring obtained by the Hexagonal-5-L(2; 1)-coloring algorithm. In this paper, optimal channel assignment algorithms are presented for the general case in which the network size is suciently large. 3. HEXAGONAL GRIDS
A hexagonal grid H of size r c has r rows and c columns, indexed respectively from 0 to r ? 1 (from top to bottom) and from 0 to c ? 1 (from left to right), with r 3 and c 2. A generic vertex u of H will be denoted by u = (i; j ), where i is its row index and j is its column index. Each vertex has degree 3, except for some vertices on the borders. In particular, each vertex (i; j ), which does not belong to the grid borders, is adjacent to the following 3 vertices:
1.
8 > > > < > > > :
(i; j + 1) if (i is even and j is even) or (i is odd and j is odd) or (i; j ? 1) if (i is even and j is odd) or (i is odd and j is even)
2. (i ? 1; j ) 3. (i + 1; j ) Lemma 3. For r 3 and c 3 there is a k-L(2; 1)coloring of a hexagonal grid H of size r c only if k 5.
Proof It follows immediately from Lemma 2 since there is at least one vertex of degree 3 that cannot be colored either 0 or 4. Hence, (H ) 5. 2 An optimal coloring for a hexagonal grid of size 6 5 is illustrated in Figure 1. Below, an optimal 5-L(2; 1)-coloring is given to color all the vertices of any hexagonal grid H of size r c, with r 3 and c 3.
Algorithm Hexagonal-5-L(2; 1)-coloring (H; r; c); If r 3 and c 3, then assign to each vertex u = (i; j ) the color f (u) = (2i + 3j ) mod 6
Theorem 1. The Hexagonal-5-L(2; 1)-coloring algorithm is optimal for any hexagonal grid H of size r c, with r 3 and c 3. 2
(0,0)
(0,1)
(0,0)
(0,1)
(1,0)
(1,1)
(1,0)
(1,1)
(2,0)
(2,1)
(2,0)
(2,1) HS
HS
Figure 2: The subgraph HS with the dummy edges (dashed), and its complement HS . Lemma 4. For r 3 and c 3, or r 5 and c = 2, there is a k-L(2; 1; 1)-coloring of a hexagonal grid H only if k 6.
Proof Consider the augmented graph GH;4 = (V; E 0 ) and the subset S = f(0; 0); (0; 1); (1; 0); (1; 1); (2; 0); (2; 1)g. Since all the 6 vertices in S are mutually at distance at most 3 in H , they form a clique in GH;4 . Therefore, (H ) 5. In the case where r 3 and c 3, consider the subgraph HS
induced by S and also the vertex (3; 0). To satisfy the cochannel reuse constraint using exactly 6 colors, vertex (3; 0) must get the same color as vertex (0; 1). Moreover, due to the channel separation constraint, the colors assigned to vertices (2; 0) and (3; 0) must have a gap of at least 1 = 2. Hence, also the colors of (2; 0) and (0; 1) must have a gap of at least 1 = 2. This is equivalent to add in HS a dummy edge between vertices (2; 0) and (0; 1). The same reasoning can be repeated for the pairs of vertices (1; 1) and (1; 0), and (2; 1) and (0; 0), as illustrated in Figure 2. Now, consider the complement HS of HS , depicted also in Figure 2. There is no Hamilton path in HS since it consists of two connected components. It follows from Lemma 1 that (H ) 6. By a similar argument, the same lower bound can be proved when r 5 and c = 2. 2 Below, an optimal 6-L(2; 1; 1)-coloring is given to color all the vertices of a hexagonal grid H of size r c, with r 3 and c 3.
Algorithm Hexagonal-6-L(2; 1; 1)-coloring (H; r; c); If r 3 and c 3 or r 5 and c = 2, then assign to each vertex u = (i; j ) the color f (u) given as follows: 8 > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > :
0 if or 4 if or 6 if or 2 if or 1 if or 5 if or
(i 0 mod 6 and j is even) (i 3 mod 6 and j is odd) (i 0 mod 6 and j is odd) (i 3 mod 6 and j is even) (i 1 mod 6 and j is even) (i 4 mod 6 and j is odd) (i 1 mod 6 and j is odd) (i 4 mod 6 and j is even) (i 2 mod 6 and j is even) (i 5 mod 6 and j is odd) (i 2 mod 6 and j is odd) (i 5 mod 6 and j is even)
An optimal L(2; 1; 1)-coloring for a hexagonal grid is illustrated in Figure 3.
0
4
0
4
0
6
2
6
2
6
1
5
1
5
1
4
0
4
0
4
2
6
2
6
2
5
1
5
1
5
(i+6,j)
(i-3,j-1)
Proof Consider a generic vertex u = (i; j ) of H . By con-
struction, the channel separation constraint is immediately veri ed. Indeed, for any vertex v adjacent to u such that v = (i; j 1), f (v) = f (u) 4 holds. Moreover, as shown in Figure 3, any pair u; v of adjacent vertices on the same column can be colored only as follows f (u) = 0 and f (v) = 6, f (u) = 6 and f (v) = 1, f (u) = 1 and f (v) = 4, f (u) = 4 and f (v) = 2, f (u) = 2 and f (v) = 5, f (u) = 5 and f (v) = 0. Therefore, the gap between the colors assigned to each pair of adjacent vertices is at least 2. Now, in order to prove that the co-channel reuse constraint is veri ed, let us show that two vertices assigned to the same color are at distance greater than 3. First of all, note that each row of H is colored with two colors, and any 3 consecutive rows of H use di erent colors. The i-th and (i +3)-th rows of H , for any i, are colored with the same two colors, except that the order of such two colors along the row is swapped. That is, vertices (i; j ) and (i; j + 1) are colored, respectively, as vertices (i + 3; j + 1) and (i + 3; j ). Hence, two vertices in rows i and (i + 3) get the same color only if their distance is at least 4. Moreover, the i-th and (i + 6)-th rows of H , for any i, are colored the same. Hence, the same color can be reused on the same column only in two vertices at distance 6. Finally, all the even (resp., odd) columns are colored the same. Although the vertices (i; j ) and (i; j +2) get the same color, their distance is 4 since there are no two consecutive horizontal edges. Summarizing, a given color is reused on H according to the pattern shown in Figure 4. The coloring optimality follows from Lemma 4. 2 4. BIDIMENSIONAL GRIDS
A bidimensional grid B of size r c, with both r 2 and c 2, can be obtained from a hexagonal grid H of the same size connecting all the pairs of consecutive nodes lying on the same row. Thus, a generic vertex u = (i; j ) of B , which does not belong on the borders, has degree 4. In particular, vertex u = (i; j ) is adjacent to the vertices (i ? 1; j ); (i; j + 1); (i + 1; j ) and (i; j ? 1). Lemma 5. The optimal L(2; 1)-coloring of a bidimensional grid B of size r c, with r 3 and c 3, has (B ) = 6. Proof The lower bound for k follows immediately since there is at least a vertex of B with degree 4 that cannot
(i,j)
(i,j-2)
Figure 3: Optimal coloring obtained by the Hexagonal-6-L(2; 1; 1)-coloring algorithm. Theorem 2. Algorithm Hexagonal-6-L(2; 1; 1)-coloring is optimal for hexagonal grids of size r c, with r 3 and c 3.
(i-3,j+1)
(i,j+2)
(i+3,j+1)
(i+3,j-1)
(i-6,j)
Figure 4: The vertices closest to vertex u = (i; j ) where the color f (u) is reused. be colored either 0 or 5. Hence, from Lemma 2, (B ) = 6. An optimal solution is given by the Grid-6-L(2; 1)-coloring algorithm and illustrated in Figure 5. 2
Algorithm Grid-6-L(2; 1)-coloring (B; r; c); If r 3 and c 3, assign to each vertex u = (i; j ) the color f (u) = (2i + 4j ) mod 7.
Lemma 6. There is a k-L(2; 1; 1)-coloring of a bidimensional grid B , with r 5 and c 4 or r 4 and c 5, only if k 8.
Proof For a bidimensional grid B = (V; E ) of size r c, with r 5 and c 4, consider the augmented graph GB;4 = (V; E 0 ). For any pair of vertices on the same column u = (i; j ) and v = (i + 3; j ), with 0 i r ? 4 and 1 j c ? 2, let Su;v be the subset of vertices f(i; j ); (i + 1; j ); (i + 2; j ); (i + 3; j ); (i + 1; j ? 1); (i + 2; j ? 1); (i + 1; j + 1); (i + 2; j + 1)g at pairwise distance no more than 3. Similarly, for any pair of vertices on the same row u = (i; j ) and w = 0 (i; j + 3), with 1 i r ? 2 and 0 j c ? 4, let Su;w be the subset of vertices f(i; j ); (i; j + 1); (i; j + 2); (i; j + 3); (i + 1; j + 1); (i + 1; j + 2); (i ? 1; j + 1); (i ? 1; j + 2)g 0
2
4
6
1
3
5
4
6
1
3
5
0
2
1
3
5
0
2
4
6
5
0
2
4
6
1
3
2
4
6
1
3
5
0
Figure 5: Optimal coloring obtained by the Grid-6-
L(2; 1)-coloring algorithm.
u h
u
a
g
p
t
s
b
f
z
w
y
c
e
v
d
u
p
t
s
p
t
s
z
w
y
z
w
y
v
v
M
M
Figure 8: Subgrid M with dummy edges (dashed), and its complement M . Figure 6: The subsets of vertices Lu;v (all) and Su;v (white). at pairwise distance no more than 3. Since both Su;v and 0 induce a clique in GB;4 , at least 8 colors are needed to Su;w satisfy the co-channel reuse constraint. However, as proved in the following, 8 colors are not enough to color the set Lu;v of vertices depicted in Figure 6, which consists of Su;v along with all the vertices of B at horizontal distance exactly 1 from the vertices on the border of Su;v . Indeed, to color vertices a = (i; j + 1) and b = (i + 1; j + 2), consider the vertex p = (i + 1; j ? 1), the subsets of vertices 0 , and the bidimensional subgrid M induced by Su;v and Sp;b Su;v . Once Su;v has been0 assigned to all di erent colors, the two vertices b and a of Sp;b must be assigned to the two colors used for the two vertices z = (i + 2; j ? 1) and v = (i + 3; j ) of Su;v , if only 8 colors are to be used. Due to the channel separation constraint, the colors assigned to vertices a and b must be at least 2 apart from the color assigned to the vertex s = (i + 1; j + 1). This is equivalent to add to M two dummy edges: one between vertices s and z, and the other between vertices s and v, as shown in Figure 7.
Now repeating the same argument for the three pairs of vertices c = (i + 2; j + 2) and d = (i + 3; j + 1), e = (i + 3; j ? 1) and f = (i +2; j ? 2), and g = (i +1; j ? 2) and h = (i; j ? 1), other dummy edges must be added (see Figure 8), namely those between p and y, u and y, u and z, s and z, p and y, and p and v. By the previous discussion, either vertex h or g is colored as vertex v. Analogously, either vertex f or e is colored as vertex u. Examining the set of vertices fv; e; f; g; h; ug, it is easy to be convinced that whatever is the color assignment adopted for such vertices, the colors f (u) and f (v) must be assigned to two adjacent vertices among fv; e; f; g; h; ug. Namely, f (u) and f (v) must appear in one of the following pair of vertices: u and h, or v and e, or f and g. Thus, one further dummy edge between vertices u and v must be added to M , as shown in Figure 8. Finally, let us build M , the complement of M . Since M consists of two connected components (see also Figure 8), M does not contain a Hamilton path. Hence, by Lemma 1, there is no 7-L(2; 1; 1)-coloring for bidimensional grids of size r c, with r 5 and c 4. Thus, (B ) 8. The proof when r 4 and c 5 is analogous. 2 The algorithm for optimally L(2; 1; 1)-coloring a grid, with r 5 and c 4 or r 4 and c 5, is given below.
Algorithm Grid-8-L(2; 1; 1)-coloring (G; r; c); If r 5 and c 4 or r 4 and c 5, then assign to u
a
p
t
s
z
w
y
each vertex u = (i; j ) the color
’ b Sp,b
v Su,v
0 along Figure 7: The subsets of vertices Su;v and Sp;b with the bidimensional subgrid M with two dummy edges (dashed).
8 > > > > > > > > >
> > > > > > > > :
0 1 2 3 5 6 7 8
if (i + j ) 0 mod 4; i is even,j is even if (i + j ) 0 mod 4; i is odd,j is odd if (i + j ) 2 mod 4; i is even,j is even if (i + j ) 2 mod 4; i is odd,j is odd if (i + j ) 3 mod 4; i is odd,j is even if (i + j ) 3 mod 4; i is even,j is odd if (i + j ) 1 mod 4; i is even,j is odd if (i + j ) 1 mod 4; i is odd,j is even
An example of optimal coloring for a bidimensional grid of size 5 5 is illustrated in Figure 9. Theorem 3. The Grid-8-L(2; 1; 1)-coloring algorithm is optimal for bidimensional grid B of size r c, with r 5 and c 4 or r 4 and c 5.
0
7
2
6
0
0
2
4
6
8
1
3
5
7
0
8
3
5
1
8
3
5
7
0
2
4
6
8
1
3
2
6
0
7
2
6
8
1
3
5
7
0
2
4
6
5
1
8
3
5
0
2
4
6
8
1
3
5
7
0
7
2
6
5
0
2
4
6
8
0
3
7
0
Figure 9: Optimal coloring obtained by the Grid-8L(2; 1; 1)-coloring algorithm. Proof In order to prove that the channel separation constraint is veri ed, one notes that two consecutive colors cannot be assigned to two adjacent vertices. For example, consider the pair of colors 2 and 3. A vertex u = (i; j ) gets color 2 if and only if both i and j are even, and i + j 2 mod 4, while a vertex v = (h; k) gets color 3 if and only if both h and k are odd, and h + k 2 mod 4. Therefore, the distance between the vertices u and v is at least 2. An analogous argument can be repeated for any pair of consecutive colors c and c + 1, with 0 c 8. To show that the co-channel reuse constraint holds, one notes that two vertices u = (i; j ) and v = (h; k) are assigned to the same color if and only if their distance d(u; v) = 4, and both ji ? hj and jj ? kj are even. The optimality follows from Lemma 6. 2 5. CELLULAR GRIDS
A cellular grid C of size r c, with r 2 and c 2, is obtained from a bidimensional grid B of the same size augmenting the set of edges with left-to-right diagonal connections. Speci cally, each vertex u = (i; j ) of C is also connected to the vertices v = (i ? 1; j ? 1) and z = (i +1; j +1). Hence, each vertex has degree 6, except for the vertices on the borders. Lemma 7. The optimal L(2; 1)-coloring for a cellular grid C of size r c, with r 5 and c 3 or r 3 and c 5, or r 4 and c 4, has (C ) = 8. 2
Algorithm Cellular-8-L(2; 1)-coloring (C ); If r 4 and c 4, or r = 3 and c 5, or r 5
and c = 3 then assign to each vertex u = (i; j ) the color f (u) = (3i + 2j ) mod 9.
Figure 10 illustrates the optimal coloring obtained by means of the Cellular-8-L(2; 1)-coloring algorithm described above. Lemma 8. There is a k-L(2; 1; 1)-coloring of a cellular grid C of size r c, with r 4 and c 4, only if k 11.
Proof Given the cellular grid0 C = (V; E ), consider the aug-
mented graph GC;4 = (V; E ) and the subgraph D of C illustrated in Figure 11. All the 12 vertices of D form a clique in GC;4 . Hence, they must be assigned to all di erent colors, and (C ) 11. 2
1
3
Figure 10: A cellular grid C of size 5 10 colored by the Cellular-8-L(2; 1)-coloring algorithm. Figure 11 shows how to color the subgraph D in such a way that the channel separation constraint is veri ed for every two adjacent vertices. Moreover, Figure 12 shows a complete coloring of a cellular grid C obtained by replicating the coloring for the subgraph D. Note that the channel separation constraint is veri ed not only for the vertices belonging to each copy of D, but also for the vertices belonging to the borders of two contiguous copies of D. Formally, the coloring of a cellular grid can be described as follows.
Algorithm Cellular-11-L(2; 1; 1)-coloring (C ); If r 4 and c 4, then assign to each vertex u = (i; j ) the color 8 > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > :
0 1 2 3 4 5 6 7 8 9 10 11
if (i + j ) 2 mod 6; i even; j even if (i + j ) 0 mod 6; i even; j even if (i + j ) 4 mod 6; i even; j even if (i + j ) 1 mod 6; i odd; j even if (i + j ) 3 mod 6; i odd; j even if (i + j ) 5 mod 6; i odd; j even if (i + j ) 5 mod 6; i even; j odd if (i + j ) 2 mod 6; i odd; j odd if (i + j ) 4 mod 6; i odd; j odd if (i + j ) 1 mod 6; i even; j odd if (i + j ) 3 mod 6; i even; j odd if (i + j ) 0 mod 6; i odd; j odd
Theorem 4. The Cellular-11-L(2; 1; 1)-coloring algorithm is optimal for cellular grids of size r c, with r 4 and c 4. 1
9
0
3
7
4
8
10
2
6
5
11
Figure 11: The subgraph D of C whose vertices form a clique in GC;4 , and an optimal 11-L(2; 1; 1)-coloring for it.
1
9
0
10
2
3
7
4
8
5
0
10
2
6
4
8
5
11
2
6
1
9
5
11
3
7
6
9
0
10
7
4
8
0
10
2
6
5
11
11
1
9
3
7
4
8
0
10
2
6
4
8
5
11
1
Lemma 9. There is a k-L(2; 1; 1)-coloring for a ring R of size n 12 only if k 4.
Proof Since any optimal L(2; 1; 1)-coloring uses at least as many colors as an optimal L(2; 1)-coloring, (R) 4, as proved in [7] for the L(2; 1)-coloring problem on rings. 3
9
0
4 1
1 3
7
1
9
0
10
2
6
1
9
0
10
3
7
4
8
5
11
3
7
4
8
0
10
2
6
10
2
6
3 4 0
2 2
0 1
9
0
3 4
8
2
6
5
11
11
5 1 3
3
7
4
8 6
1
9
11
3
7
9
0
10
2
7
4
8
5
5
11
Figure 12: Optimal 11-L(2; 1; 1)-coloring for a cellular grid C . Proof In order to prove that the channel separation con-
straint is veri ed, it is useful to introduce the Manhattan distance m(u; v) between any two vertices u and v, where m(u; v) is the length of a shortest path between u and v including only horizontal and vertical edges, thus excluding diagonal edges. Now, any two consecutive colors are considered and it will be proved that such colors cannot be assigned to two adjacent vertices. For example, consider the pair of colors 2 and 3. A vertex u = (i; j ) gets the color 2 if and only if both i and j are even, and i + j 4 mod 6, while a vertex v = (h; k) is colored 3 if and only if h is odd, k is even, and h + k 1 mod 6. The vertices u and v might belong to the same column, but to di erent rows. In this case, their distance is at least 3. In the case that they do not belong to the same column, they have Manhattan distance m(u; v) = 3. Hence, the vertex v which is closest to u and assigned to color 3 is v = (i +1; j +2), as illustrated in Figure 12. Keeping track of the diagonal edges, the actual distance d(u; v) is 2, and therefore the channel separation constraint is still veri ed. An analogous argument can be repeated for any pair of consecutive colors c and c + 1, with 0 c 10. To show that the co-channel reuse constraint holds, one notes that two vertices u = (i; j ) and v = (h; k) get the same color if and only if their Manhattan distance m(u; v) = 6, and both ji ? hj and jj ? kj are even. Due to the diagonal edges, the actual distance d(u; v) is at least 4. Indeed, the actual distance d(u; v) could be 3 when m(u; v) = 6, but in this case ji ? hj and jj ? kj cannot be both even. The optimality follows from the lower bound shown in Lemma 8. 2 6. RINGS
A ring R of size n is a sequence of n vertices, indexed consecutively from 0 to n ? 1, such that vertex i is connected to both vertices (i ? 1) mod n and (i + 1) mod n.
2
1
4
Figure 13: Optimal coloring obtained by the Ring4-L(2; 1; 1)-coloring algorithm for a ring of size 14. Algorithm Ring-4-L(2; 1; 1)-coloring (R; n); ? if n 12, let = 4 b n4 c ? (n mod 4) , and assign to each vertex i of R the color 8 > > > > > > > > > > >
> > > > > > > > > > :
0 if i 0 mod 4 and i < ; or (i ? ) 0 mod 5 and i 1 if i 2 mod 4 and i < ; or (i ? ) 3 mod 5 and i 2 if (i ? ) 1 mod 5 and i 3 if i 3 mod 4 and i < ; or (i ? ) 4 mod 5 and i 4 if i 1 mod 4 and i < ; or (i ? ) 2 mod 5 and i
Theorem 5. The Ring-4-L(2; 1; 1)-coloring algorithm is optimal for rings of size n 12.
Proof Note that the ring is colored repeating b n4 c? (n mod 4) times the color sequence 0; 4; 1; 3 of length 4, followed by n mod 4 times the color sequence 0; 2; 4; 1; 3 of length 5 (see Figure 13, where n = 14, 14 mod 4 = 2 and thus b 144 c ? 2 = 1). Therefore since two vertices i and j get the same color only if i and j are at distance at least 4 or 5, and since two adjacent colors have a gap of at least 2, both the co-channel reuse constraints and the channel separation are readily veri ed. The optimality follows from Lemma 9. 2 7. COMPLETE BINARY TREES
A complete binary tree T of size n is a rooted tree in which all the leaves are at the same level and each internal vertex has exactly 2 children. The level of vertex u 2 T is de ned as the number of edges on the path from u to the root, which is at level 0. The height h of T , that is the maximum level of its vertices, is h = blog2 nc. Denoted the root of T as r = (0; 0), the (j + 1)-th vertex at level i, u = (i; j ),
Turn
0 1 2
0 0 0 2
1 1 2 1
Shift
0 1 2
0 1 1 -1
1 3 -1 3
0 1 2 3 4 5 0 6 1 4 2 5
Color
Table 1: The functions Turn, Shift, Color used in the Tree-6-L(2; 1; 1)-coloring algorithm. with 0 < i < h and 0 j 2i ? 1, is connected to its father v = (i ? 1; b 2j c), and to z = (i +1; 2j ) and w = (i +1; 2j +1), respectively, its left and right child. Lemma 10. There is a k-L(2; 1; 1)-coloring for a complete binary tree T of size n 31 only if k 6. Proof When n 31, given a complete binary tree0 T = (V; E ), consider the augmented graph GT;4 = (V; E ), the vertex u = (i + 3; 8j ), with 0 i h ? 3, 0 j 2i ? 1, and the subset Su of vertices at reciprocal distance no more than 3, where Su = f(i +3; 8j ); (i +3; 8j +1); (i +2; 4j ); (i + 2; 4j + 1); (i + 1; 2j ); (i; j )g. Since Su forms a clique in GT;4 , there is a k-L(2; 1; 1)-coloring for a complete binary tree T only if k 5. However, one more color is necessary to satisfy the channel separation constraint as noted below. First of all, consider Su and observe that, to color Su with 6 colors, jf (p) ? f (l)j 3 must hold for the father-child pair p = (i +1; 2j ) and l = (i +2; 4j ). Now, let v = (i +3; 8j +2), and consider Sv = f(i + 3; 8j + 2); (i + 3; 8j + 3); (i + 2; 4j + 1); (i + 2; 4j ); (i + 1; 2j ); (i; j )g. Again, for the father-child pair p = (i + 1; 2j ) and r = (i + 2; 4j + 1), jf (p) ? f (r)j 3 must result in order to color Sv with 6 colors. Repeating the same reasoning for Sw and Sz , where w = (i + 3; 8j + 4) and z = (i + 3; 8j + 6), implies that the colors assigned to any father-child pair at levels i +1 and i +2 must be at least 3 apart. Since the same property holds for any father-child pair at levels i + 2 and i + 3, for any vertex v at level i + 2 all the three vertices adjacent v must get a color which is at least 3 apart from that of v. Observing that there must be at least one vertex at level i + 2 whose color is di erent from 0 and 5, (T ) 6 follows. 2 In order to describe the Tree-6-L(2; 1; 1)-coloring for complete binary trees, consider for any vertex u = (i; j ) the binary representation of j using i bits. Note that the kth bit of such representation is 1 (resp., 0) if and only if the k-th vertex on the unique path from the root to u is a right (resp., left) child. Moreover, let the functions Turn: f0; 1; 2g f0; 1g ! f0; 1; 2g, Shift: f0; 1; 2g f0; 1g ! f?1; 1; 3g, and Color: f0; : : : ; 5g ! f0; : : : ; 6g be de ned as shown in Table 1. The tree coloring is derived from the 6-L(2; 1; 1)-coloring of an in nite bidimensional grid B with 2 columns, by mapping the entire tree T on B . For the sake of simplicity, the in nite grid can be represented by a 62 grid with toroidal columns. As an example, a mapping for a tree of size n = 31 is shown in Figure 14. Once B has been colored, the root r = (0; 0) of T is mapped to a vertex of B colored with 0. The left child a (resp., right child b) of r is mapped to the vertex (i +1; 1) (resp., (i ? 1; 1)) which is below (resp., above) (i; 1) on B . Then, the left and right children c and d of the left child a of the root are mapped, respectively, to the vertices (i ? 2; 1) and (i ? 1; 0). Similarly, the left and right children e
and f of b are mapped, respectively, to the vertices (i + 2; 1) and (i + 1; 0). Again, the left and right children of the tree vertex c, mapped into the grid vertex (i + 2; 1), are mapped to the vertices (i +3; 1) and (i +2; 0), respectively, and so on. In particular, the left and right children o and p of the tree vertex f , mapped into the grid vertex (i ? 1; 0), are mapped to the vertices (i ? 2; 0) and (i ? 1; 1), respectively. The above reasoning can be then summarized as follows. Given a tree vertex v, mapped to the grid vertex u = (i; j ), its right child is mapped to the grid vertex (i; (j + 1) mod 2), except when the father of v has already been mapped to such a vertex. In such a case, the right child of u is mapped to the grid vertex above u, that is (i ? 1; j ). Similarly, given the tree vertex v, mapped to the grid vertex u = (i; j ), its left child is mapped to the grid vertex (i ? 1; j ) except when the father of v has already been mapped to such a vertex. When this happens, the left child of u is mapped to the grid vertex below u, that is to (i +1; j ). Hence, the left and right children of a vertex v are mapped to di erent grid vertices depending on where their father has been mapped. Iterating this argument, it follows that the left and right children are mapped to di erent grid vertices depending on how many right vertices, or turns, there are on the path from the root to v. In particular, the number of turns W on the path are counted by the function Turn as follows:
if a right vertex is encountered and there was 0 or 1
turn already counted, the number of turns increases by 1, while if there were 2 turns the number of turns decreases by 1; if a left vertex is encountered, the number of turns remains the same except when there was already 1 turn counted; in this case, the number of turns returns to 0.
Intuitively, for a given level k of the path, W [k] = 0 means that the mapping of the tree path is proceeding southwards 0
4
6
2
r
g k
p l
1 f
4 o j
2
a
e
b
5 c
b
e
d
f
0 g
r
h
i
j
k
l
o
p
6
d
a
5
1
c
h i
Figure 14: Mapping of a complete binary tree into an in nite bidimensional grid.
Network G bus ring complete binary tree hexagonal grid bidimensional grid cellular grid References
L(1) 2 2 or 3 2 2 2 3
folklore
L(0; 1) L(1; 1) L(2; 1) L(1; 1; 1) 2 3 5 4 2 or 3 3 or 4 5 4 or 5 3 4 5 6 3 4 6 6 4 5 7 8 6 7 9 12 [2, 10] [1, 3] [5, 7], this paper [3]
L(2; 1; 1) 5 5 7 7 9 12
this paper
Table 2: Minimum number (G) + 1 of channels used for a suciently large network G. on the grid, W [k] = 2 means that it is proceeding northwards, while W [k] = 1 means that it is crossing a horizontal edge. A tree vertex, once mapped to a grid vertex, inherits the same color. By the optimal L(2; 1; 1)-coloring of a grid when c = 2, the grid B is colored in a cyclic way by repeating on the two columns the color pattern sequence 0; 6; 2; 4; 1; 5, which is speci ed by the function Color. The algorithm Tree-6-L(2; 1; 1)-coloring, instead of computing the grid vertex to which each tree vertex is mapped, directly maps each grid edge to a shift on the color pattern sequence, thus obtaining the color of a tree vertex as the sum of the shifts along the tree path. In particular, crossing a vertical edge upwards corresponds to one left cyclic shift on the pattern sequence, crossing a vertical edge downwards corresponds to one right cyclic shift, and crossing on a horizontal edge corresponds to a cyclic shift of three positions. In conclusion, given the walking direction W , the function shift maps any tree edge into a speci c grid edge, and returns the cyclic shift on the color pattern sequence. Summarizing, the algorithm for coloring a single tree vertex u = (i; j ) is the following:
Algorithm Tree-6-L(2; 1; 1)-coloring (T; n; u = (i; j )); If n 31 then { If u = (0; 0) then f (u) = 0 { If u =6 (0; 0) then Consider bin[1::i], the binary
representation of j using i bits, and the arrays Shift[0::2; 0::1], Turn[0::2; 0::1], and Color[0::5]; Consider the vector W [1::i] and ll it as follows: W [0] = 1, W [k] = Turn [W [k ? 1]; bin[k]], for k = 1; : : : ; i; Consider the vector S[1::i] and ll it as follows: S [k] = Shift [W [k ? 1]; bin[k]], for k =P 1; : : : ; i; s = ik=1 S [k] mod 6; Assign to vertex u the color f (u) = Color[s].
In Table 3, it is shown how to ll the vectors W and S in order to color the vertex u = (6; 43). Note that computing f (u) for a single tree vertex takes O(log n) time. However,
it is easy to see that the entire tree can be colored in O(n) time. Theorem 6. The Tree-6-L(2; 1; 1)-coloring algorithm is optimal for complete binary trees of size n 31.
Proof Since the 6-L(2; 1; 1)-coloring of the grid satis es the
channel separation constraint, and adjacent tree vertices are mapped to adjacent grid vertices, the Tree-6-L(2; 1; 1)coloring algorithm also veri es the channel separation constraint. To show that the co-channel reuse constraint is veri ed too, observe that any two consecutive tree edges are mapped to two di erent grid edges. Namely, no tree path traverses twice the same grid edge consecutively (in opposite directions). For example, consider a horizontal grid edge. It can be crossed when a right vertex is encountered on the tree path and the number of turns is either 0 (i.e., W [k] = 0) or 2 (i.e., W [k] = 2). However, in the next move, since the number of turns has become 1 (i.e., W [k + 1] = 1), both the left and right vertices are mapped to vertical grid edges, and the horizontal grid edge is not used. In conclusion, on any grid path that corresponds to a tree path, the minimum distance between two vertices colored the same is 4. The optimality follows immediately from Lemma 10. 2 8. CONCLUSIONS
Table 2 reports the results for optimal L(1 ; 2 ; : : : ; ?1 )coloring, known up to now in the literature. Speci cally, such a table indicates, for the most common regular networks, the minimum number of channels required by a suf ciently large network. In all the cases, there are ecient algorithms to assign channels to vertices. The channel assigned to any vertex can be computed locally provided that the relative position of the vertex in the network is known. Such a computation can be performed in constant time for all the networks, except for binary trees which require logarithmic time in the number of vertices. In particular, the bit W S
0 1 1 1 2 -1
2 0 2 -1
3 1 1 3
4 0 0 1
5 1 1 3
6 1 2 -1
Table 3: The vectors bit, W, and S computed by the Tree-6-L(2; 1; 1)-coloring algorithm to color vertex u = (6; 43). Since ?P6k=1 S [k] mod 6 = 0, the color assigned to u is f (u) = Color(4) = 2.
present paper dealt with the L(2; 1)- and L(2; 1; 1)-coloring problems which consider a channel separation of at least 2 between adjacent vertices, as required in many real wireless environments. The proposed results extend those already known in the literature for the L(0; 1)-, L(1; 1)- and L(1; 1; 1)-coloring problems. For the sake of completeness, Table 2 reports also the minimum number of colors required by the L(1)-coloring problem, which reduces to the classical minimum vertex coloring of a graph. From a theoretical point of view, it remains as an open question to solve the general L(1 ; 2 ; : : : ; ?1 )-coloring problem, with 1 2 : : : ?1 .
9. REFERENCES
[1] R. Battiti, A.A. Bertossi, and M.A. Bonuccelli, \Assigning Codes in Wireless Networks: Bounds and Scaling Properties", Wireless Networks, Vol. 5, 1999, pp. 195-209. [2] A.A. Bertossi and M.A. Bonuccelli, \Code Assignment for Hidden Terminal Interference Avoidance in Multihop Packet Radio Networks", IEEE/ACM Transactions on Networking, Vol. 3, 1995, pp. 441-449. [3] A.A. Bertossi and M.C. Pinotti, \Mappings for Con ict-Free Access of Paths in Bidimensional Arrays, Circular Lists, and Complete Trees", Tec. Rep. UTM-563, Depatment of Mathematics, University of Trento, 1999. [4] H.L. Bodlaender, T. Kloks, R.B. Tan and J. van Leeuwen, \Approximation -Coloring on Graphs", to appear in Proceedings STACS 2000. [5] G. J. Chang and D. Kuo, \The L(2; 1)-Labeling Problem on Graphs", SIAM Journal on Discrete Mathematics, Vol. 9, 1996, pp. 309-316. [6] I. Chlamtac and S.S. Pinter, \Distributed nodes Organizations Algorithm for Channel Access in a Multihop Dynamic Radio Network", IEEE Transactions on Computers, Vol. 36, 1987, pp. 728-737. [7] J. R. Griggs and R.K. Yeh, \Labelling Graphs with a Condition at Distance 2", SIAM Journal on Discrete Mathematics, Vol. 5, 1992, pp. 586-595. [8] W.K. Hale, \Frequency Assignment: Theory and Application", Proceedings of the IEEE, Vol. 68, 1980, pp. 1497-1514. [9] I. Katzela and M. Naghshineh, \Channel Assignment Schemes for Cellular Mobile Telecommunication Systems: A Comprehensive Survey", IEEE Personal Communications, June 1996, pp. 10-31. [10] T. Makansi, \Transmitted Oriented Code Assignment for Multihop Packet Radio", IEEE Transactions on Communications, Vol. 35, 1987, pp. 1379-1382.
[11] S.T. McCormick, \Optimal Approximation of Sparse Hessians and its Equivalence to a Graph Coloring Problem", Mathematical Programming, Vol. 26, 1983, pp. 153{171. [12] D. Sakai, \Labeling Cordal Graphs: Distance Two Condition", SIAM Journal on Discrete Mathematics, Vol. 7, 1994, pp. 133-140.
