Handbook of Wireless Networks and Mobile Computing phần 3 doc

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

mum is achieved by one of the vertices of the polytope TC(G) representing the feasible

dual solutions and defined as follows:

TC(G) = 冦x僆 ⺡+: v冱僆V t(v)x(v)  c(t) for all t 僆 T冧

A classification of the vertices of this polytope will therefore lead to a comprehensiveset 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 c0, c1were considered In this case thechannel assignment was found to be equivalent to the tile cover problem Moreover, thefractional 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 wasnew Two bounds were clique bounds of the type mentioned earlier The third bound wasfirst given by Gamst in [12], and can be stated as follows:

S(G, w)  max{c0w(v) + (c1– c0)w(C – v) – c0|C a clique of G, v 僆 C} (5.6)where is such that (– 1)c1< c0c1

The tile cover approach led to a number of new bounds for graphs with constraints c0,

c1, c2 The bounds are derived from so-called nested cliques A nested clique is a d1-clique

that contains a d2-clique as a subset (d2< d1) It is characterized by a node partition (Q, R), where Q is the d2-clique and R contains all remaining nodes A triple (k, u, a) will denote the constraints k = c0, u = c d2 , and a = c d1in 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 straint of at least u, while the constraint between any pair of nodes in the nested clique is at least a

con-The following is a lower bound for a nested clique (Q, R) with parameters (k, a, u):

S(G, w)  a v冱僆Q w(v) + u v冱僆R w(v) – u (5.7)This bound was first derived in [12] using ad-hoc methods The same bound can also bederived 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)  (k –  )wQmax+ v冱僆Q w(v) + v冱僆R w(v) – k (5.8)where

=   k , = (+ 1)u – k u

Bound (1.3), obtained from the total weight on a clique, was extended, leading to

S(G, w)  u冢v冱僆Q w(v) + w Rmax冣+ v 冱 w(v) – k (5.9)

A bound of (2u – 1)wQmax+ v僆R 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)  (2u – 1)wQmax+ v 冱 w(v) + v冱僆R w(v) – k (5.10)where

Finally, we mention the following two tile cover bounds from [22] for nested cliques

with parameters (k, u, a):

S(G, w))  (3u – k + 2 ) v冱僆Q w(v) + (k – 2 )wRmax+ v冱僆R w(v) – k (5.11)where = 3a – k, and

S(G, w)  u冢v冱僆V w(v) + w Rmax冣+ v 冱 w(v) – k (5.12)

In [40], a bounding technique based on network flow is described Since no explicitformulas 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 generalconstraints 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 straints is presented in Table 5.1

con-3a – u

2

– 1

Most of the work done has been for the case where only a cosite constraint c0and one

edge constraint c1are 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, c0, c1) be a constrained graph, and w an arbitrary weight vector Assume that a base coloring f : V 씮 {0, 1, , k – 1} of the nodes of G is given.

ASSIGNMENT: Let s = max{c0, kc1} Each node v receives the channels f (v) + is, i = 0, 1, , w( v) – 1.

Algorithm A has a performance ratio of max{1, kc1/c0}, and is therefore optimal if c0

 kc1 It is a completely distributed algorithm, since every node can assign its own nels independently of the rest of the network The only information needed by a node to beable to compute its assignment is its base color

chan-The base coloring used in Algorithm A can be seen as a graph labeling satisfying the

constraint c1= 1 A modified version of Algorithm A, based on graph labelings, can beformulated as follows

TABLE 5.1 An overview of the performance ratios of the best known algorithms for differenttypes of graphs A * indicates that the performance ratio depends heavily on the constraints; see thetext of Section 5.3 for details

Algorithm A

(based on graph labeling)

Let G = (V, E, c0, c1, , ck) be a constrained graph, and w an arbitrary weight vector sume that a labeling f : V 씮 ⺞ is given which satisfies the constraints c1, , ckand has

As-cyclic span M.

ASSIGNMENT: Let s = max{c0, M} Each node v receives the channels f (v) + is, i = 0, 1, , w(v) – 1.

Algorithm A
has a performance ratio of max{1, M/c0} and is therefore optimal if c0

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 appearexist-ed in the literature This method isreferred to as fixed assignment (FA), as each node has a fixed set of channels available forits 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

label-straints c1, c2, if for all pairs of nodes u, v at graph distance i in the lattice, | f (u) – f (v)|

 max{ci , n – c i} A labeling by arithmetic progression is considered optimal for a givenset of constraints if its cyclic span is as small as possible

Given f, a labeling by arithmetic progression, f (m1, m2) denotes the value of the

label-ing at the node with coordinates (m1, m2) Labelings by arithmetic progression are easy todefine and with Algorithm A
they can be used to find channel assignment algorithms.Moreover, their regularity may be helpful in designing borrowing methods that will givebetter channel assignments for nonuniform weights

5.3.1 Bipartite Graphs

For bipartite graphs with constraints c0and c1, Algorithm A gives optimal channel

assign-ments if c0 2c1 If c0< 2c1, 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 2c1apart (which allows interspersing the channels of its neighbors), while the

lat-er channels are c0apart

Algorithm B (for bipartite graphs when c1 c0 2c1)

Let G = (V, E, c0, c1) be a constrained bipartite graph of n nodes, where c1 c0 2c1, and

w an arbitrary weight vector Assume a base coloring f : V씮 {0,1} is given

For each node v, define p(v) = max{w(u) | uv 僆 E or u = v}.

106 CHANNEL ASSIGNMENT AND GRAPH LABELING

ASSIGNMENT: Initially, each node v receives channels f (v)c1+2ic1, i = 0, 1, , p( v) – 1.

If w( v) > p(v), then v receives the additional channels f (v)c1+ 2p( v)c1+ ic0, i = 0, , w(v) – p(v) – 1.

The span of the assignment above is at most max(u 僆E {c0w(u) + (2c1– c0)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 a1= 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 c1, c2, and c3, and 2c2– c3 c1 (1)c2+ c3 For this case, a periodic labelingnot based on arithmetic progressions is given in the same paper

For paths of size at least five, these labelings include the optimal graph labeling

satis-fying constraints c1= 2, c2= 1 given by Yeh in [42], and the path labelings for general

con-straints c1, c2by Georges and Mauro in [13] Note that Algorithm A
, used with any of

these labelings with cyclic span n, has a performance ratio of max{1, n/c0}

The near-optimal labeling for unit interval graphs given in [32] can be applied to paths

with constraints c1, c2, , c 2r , where c1= c2= = c r = 2 and c r+1 = c 2r= 1, to give

a labeling with cyclic span 2r + 1 Using this labeling in Algorithm A
leads to a mance ratio of max{1, (2r + 1)/c0}

perfor-Van de Heuvel et al [39] also give an optimal labeling by arithmetic progression for

the square lattice and constraints c1, c2 The labeling given has cyclic span n = 2c1+ 3c2and is defined by the parameters a1= c1, a2= c1+ c2 The square lattice is the Cartesianproduct 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 c1= 2, c1= c3= 1 of span 8 and cyclic

span 10 This labeling can be transformed into a labeling for general c1, c2, c3as follows

Let c = max{c1/2, c2}, and let f be the labeling for c1, c2, c3= 2, 1, 1 Let f
(u) = cf (u) It is easy to check that f
is a labeling for c1, c2, c3of cyclic span 10c Using this labeling with

Algorithm A
gives a performance ratio of max{1, 5c1/c0, 10c2/c0} The same authors give

a labeling for bidimensional grids with constraints c1= 2, c2= 1, which is just a specialcase the labeling by arithmetic progression given above

The same authors also give labelings for graphs they call hexagonal grids, with straints c1, c2= 2, 1 and c2, c1, c3= 2, 1, 1 Hexagonal grids are not to be confused withhexagon graphs, which will be discussed in Section 5.3.3 In fact, hexagonal grids are

con-subgraphs of the planar dual of the infinite triangular lattice Hexagonal grids form a ular arrangement of 6 cycles, and are bipartite

reg-Labelings for the hypercube Q nwere described and analyzed in [15, 24, 41] Graph

la-belings for trees with constraints c1, c2= 2, 1 were treated in [5] and [15] These labelingsare 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 c1, c2= 2, 1 of span 4 and cyclic span 6 The beling repeats the channels 0, 2, 4 along the cycle, with a small adaptation near the end ifthe length of the cycle is not divisible by 3 As described in the previous section, this la-

la-beling can be used for general constraints c1, c2if all values assigned by the labeling are

multiplied by max{c2, c1/2} Using Algorithm A
, this leads to an algorithm with

perfor-mance ratio max{1, 3c1/c0, 6c2/c0}

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 c0and c1 The first of the three algorithms in [21] is based on a graph labeling that satisfies one

constraint c1 This labeling has cyclic span c R = 2nc1/(n – 1) It starts by assigning zero to the first node, and then adding c1(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 alternatingassignment This labeling is then used repeatedly, as in Algorithm A
Since this particularform of Algorithm A
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, c0, c1) be a constrained cycle of n nodes, where n > 3 is odd, and w be an bitrary weight vector Fix s = max{c0, c R} Let the nodes of the cycle be numbered {1,

ar- ar- ar- , n}, numbered in cyclic order, where node 1 is a node of maximum weight in the cle Let m > 1 be the smallest odd integer such that s  2m/(m – 1)c1(it can be shown thatsuch an integer must exist)

cy-ASSIGNMENT: To each node i, the algorithm assigns the channels b(i) + js, where j = 0, , w(i) – 1, and the graph labeling b : V 씮 [0, s – 1] is defined as follows:

(i – 1)c1mod s when 1  i  m,

b(i) = 冦0 when i > m and i is even, (m – 1)c1mod 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 ticoloring an odd cycle, described in [29] and discussed in the previous chapter The spanused by this algorithm is /2s

mul-108 CHANNEL ASSIGNMENT AND GRAPH LABELING

Algorithm D (for odd cycles)

Let G = (V, E, c0, c1) be a constrained cycle of n nodes, where n > 3 is odd, and w be an bitrary weight vector Fix s = max{c0, 2c1}, and  = max{2v僆V w(v)/(n – 1), 2wmax}

ar-Let f be an optimal multicoloring of (G, w) using the colors {0, 1, , – 1} Such an

prop-that Algorithm C is optimal if c0 cR = 2nc1/(n – 1)

If 2c1 c0< 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 = v僆V w(v) – (n – 1)wmax If  0, Algorithm D

is used, with spectrum [0, c0wmax] The span is at most c0wmax, 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 f1the assignment computed by Algorithm C for (G, w
) where w
(v)

= min{w( v), } This assignment has span at most cR

Consider the remaining weight w 苶 after this assignment Clearly w苶max= wmax– We

will denote by f2the assignment for (G, w苶), and compute it in two different ways

depend-ing on a key property of w 苶 If there is a node v with w 苶(v) = 0 at this stage, we have a tite graph left Then f2is the assignment computed by Algorithm A for (G, w苶) This assign-

bipar-ment has a span of at most c0w苶max

If all nodes have nonzero weight, then Algorithm D is used to compute f2, the

assign-ment for (G, w苶) It can be shown that in this case, = 2w苶max, so this assignment also has a

span of at most c0/2 = c0wmax Thus, in either case, f2has span at most c0w苶max

The two assignments f1and f2are then combined by adding c R + c0to every channel in

f2, and then merging the channel sets assigned by f1and f2at each node This gives a final

assignment of span at most (c R – c0) + c0wmax+ c0 Using the lower bounds (5.2) and(5.3), it can be shown that the performance ratio of the algorithm is as claimed

If c0< 2c1, 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 f1computed by Algorithm C for (G, w
) where w
(v) = wmin

for every node v Then, find the assignment f2 computed by Algorithm B for (G, w

) where w

(v) = w(v) – wmin Finally, combine the two assignments by adding c R wmin+ c0to

each channel of f2and then merging the channel sets assigned by f1and f2

Using bound 1.6, it can be shown that the algorithm has performance ratio 1 + 1/(n – 1)

as claimed

In [13], optimal graph labelings for odd cycles with constraints c1, c2are given If c1>

2c2, or c1 2c2and n ⬅ 0 mod 3, the span is 2c1, and the cyclic span is 3c1 Using rithm A
in combination with this labeling gives a performance ratio of max{1, 3c1/c0}

Algo-For the remaining case, the span is c1+ 2c2and the cyclic span is c1+ 3c2, leading to aperformance ratio for Algorithm A
of max{1, (c1+ 3c2)/c0} In [3], Bertossi et al give a

graph labeling for cycles of length at least 4 with constraints c1, c2, c3= 2, 1, 1 The span

of the labeling is 4, and its cyclic span is 6 Adapting this labeling to general parameters

c1, c2, c3and using Algorithm A
gives a performance ratio of max{1, 3c1/c0, 6c2/c0}

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 a1, a2, and n,

are given in the table below

It can be easily seen that hexagon graphs admit a regular coloring with three colors

Hence Algorithm A will be optimal for constraints c0, c1so that c0 3c1 A channel

as-signment algorithm for hexagon graphs with constraints c0, c1= 2, 1 with performance tio 4/3 was given in [36]

ra-In [21], further approximation algorithms for hexagon graphs and all values of

con-straints c0, c1are given All algorithms have performance ratio not much more than 4/3,which is the performance ratio of the best known multicoloring algorithm for hexagongraphs (see [28]) The results are obtained by combining a number of basic algorithms forhexagon 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, c0, c1) be a constrained graph, and w be an arbitrary weight vector Fix s = max{c1, c0/2} and T  3wmax, T a multiple of 6 Let f : V씮 {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 R1= [0, 2s, , (T/3 – 2)s] and a second set R2= [(T/3 + 1)s + c0, (T/3 + 3)s + c0, , (2T/3 – 1)s + c0] Blue channels

consist of first set B1= [(T/3)s + c0, (T/3 + 2)s + c0, , (2T/3 – 2)s + c0] and second set

B2= [(2T/3 + 1)s + 2c0, (2T/3 + 3)s + 2c0, , (T – 1)s + 2c0], and green channels consist

of first set G1= [(2T/3)s + 2c0, (2T/3 + 2)s + 2c0, , (T – 2)s + 2c0] and second set G2=

[s, 3s, , (T/3 – 1)s]

ASSIGNMENT: 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 channelsare 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 R1and G2, the second part has alternating channels from B1and

R2, and the third part has alternating channels from G1and B2 The span used by

Algo-rithm E equals sT + 2c0= max{c1, c0/2}T + 2c0, where T is at least 3wmax

To obtain the optimal algorithms for hexagon graphs and different values of the

para-meters c0, c1, Algorithm E is modified and combined with Algorithms A and B

Algorithm A for hexagon graphs has a performance ratio of max{1, 3c1/c0} As noted,

when c0  3c1 the algorithm is optimal When c0  (9/4)c1, the performance ratio of

equals 3c1/c0, which is at most 4/3 For the case where 2c1< c0 (9/4)c1, 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)  c1D – c1 For ease of explanation,

we assume that D is a multiple of 6

Phase 1: If D > 2wmax, use Algorithm A for hexagon graphs on (G, w
) where w
(v) = min{w( v), D – 2wmax} If D  2wmax, skip this phase, and take w
(v) = 0 for all v The span needed for this phase is no more than max{0, D – 2wmax}3c1

Phase 2: Let T = min{2wmax, 6wmax– 2D} Use Algorithm E on (G, w

), where w

(v) = min{w( v) – w
(v), T/3}, taking T as defined The span of the assignment is min{2wmax,

(6wmax– 2D)}c0/2 + 2c0 It follows from the description that after this phase, in everytriangle 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 onlyfrom blue neighbors, blue from green, and green from red A red node v with w(v) >

w
(v) + w

(v), where wB(v) is the maximum number of channels used during Phase 2

by any blue neighbor of v, receives an additional min{w(v) – w
(v) – w

(v), T/3 –

w B(v), T/6} channels from the second blue channel set B2, starting from the highestchannels in the set A similar strategy is followed for blue and green nodes It can beshown that the graph induced by the nodes that still have unfulfilled demand after thisphase is bipartite

Phase 4: Let w苶 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 苶), which has a span of c0w苶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 (2wmax)c0/2 + c0(wmax/3) +

(1) = (4/3)c0wmax+ (1) It then follows from lower bounds (5.2) and (5.3) that the

per-formance ratio equals 1 + 3(c0– 2c1)/c0+ (9c1– 4c0)/3c1 When 2c1< c0 (9/4)c1, this is

always less than 4/3 + 1/100 In particular, the maximum value is reached when c0/c1=3/兹2苶 When c0= 2c1or c0= 9c1/4, the performance ratio is exactly 4/3

When c0 2c1, a linear time approximation algorithm with performance ratio 4/3 isobtained from an initial assignment by Algorithm E, followed by a borrowing phase and a

phase where assigned channels are rearranged in the spectrum, and finally an application

of Algorithm B The algorithm follows

Let

L = max{c0w(u) + (2c1– c0)(w( v) + w(r))|{u, v, r} a triangle}

and let T be the smallest multiple of 6 larger than max{L, Dc1}/c1 It follows from lower

bounds (5.6) and (5.3) that Tc1– (1) is a lower bound for the span of any assignment

Phase 1: Use Algorithm E on (G, w
) where w
(v) = min{w(v), T/3} and T is defined above In this case s, the separation between channels, equals c1, so the span of the as-

en earlier (red 씮 blue 씮 green 씮 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 R2are replaced by

min{w( v) – T/3 – w B(v), 2c1/c0(T/6 – w B(v))} channels with separation c0which fill the

part of the spectrum occupied by the last T/6 – w B(v) channels of R2 For example, let T

= 24, c0= 3, and c1= 2 Suppose v is a red corner node with at least two green bors, where w( v) = 13 and let w B( v) = 1 In Phase 1, v received the channels 21, 25, 29,

neigh-33 from the set R2, whereas at least one blue neighbor of v received the channel 19 from

B1and no other channels from B1or B2were used by any neighbor of v Then in Phase

2, v borrows all four blue channels in B2, and in Phase 3, squeezes the part of the

spec-trum [21, 33] of R2to 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 ample, cosite and intersite constraints are respected

ex-Phase 4: Let w苶 be the weight vector remaining after Phase 3 It can be shown that thegraph induced by the nodes with positive remaining weight is bipartite We use Algo-

rithm B to find an assignment for (G, w 苶), which has a span of L
= max{c0w 苶(u) + (2c1–

c0)w 苶(v)|(u, v) 僆 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) Tc1+

(1) From the definition of T, we have that Tc1– (1) is a lower bound, which gives therequired performance ratio of 4/3

In [3], a labeling is given for hexagon graphs with constraints c1, c2, c3= 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 bedetermined 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 c1, c2= 2, 1 This la-

112 CHANNEL ASSIGNMENT AND GRAPH LABELING

beling can be used with Algorithm A
to give a performance ratio of max{1, 9 (c1/2)/c0,

9c2/c0}

Algorithm A
is based on a uniform repitition of an assignment of one channel pernode, and will therefore work best when the distribution of weights in the network is fair-

ly uniform To accommodate for nonuniform weights, Fitzpatrick et al [9] give an

algo-rithm for hexagon graphs with parameters c0, c1, c2, where c0= c1and c1 2c2, whichcombines an assignment phase based on a labeling by arithmetic progression with twoborrowing phases, in which nodes with high demand borrow unused channels from theirneighbors

The labeling f that is the basis of the algorithm is defined by the parameters a = c1, b = 3c1+ c2, and n = 5c1+ 3c2 It can be verified that f indeed satisfies the constraints c1and

c2 It is also the case that c2 f (i, j)  n – c2even for nodes (i, j) at graph distance 3 of (0, 0) So, any channel assignment derived from f has the property that the nodes at graph dis- tance 3 also have separation at least c2 (This implies that the given labeling satisfies the

constraints c1, c2, c3; in fact, when c1, c2= 2, 1, the labeling is the same as the one given in[3].)

More precisely, v can calculate T(v), where

T(v) = max冦u冱僆C w(u) | C a clique, d(u, v)  1 for all u 僆C冧

The algorithm then proceeds in three phases, as described below

Phase 1 Node v receives channels f (v) + in, 0  i < min{w(v), T(v)/3}.

Phase 2 If v has weight higher than T(v)/3, then v will borrow any unused channels from its neighbor x = (i + 1, j).

Phase 3 If v still has unfulfilled demand after the last phase, then v borrows the ing channels from its neighbor y = (i + 1, j – 1).

remain-The algorithm can be implemented in a distributed manner Every node v = (i, j) knows its value under f, f (i, j), and is able to identify its neighbors and their position with respect

to itself, and receive information about their weight Specifically, v is able to identify the neighbors (i + 1, j) and (i + 1, j – 1), and to calculate the maximum weight on a clique

among its neighbors

Using lower bound 5.1, applied to a 2-clique of the graph, it can be shown that the

per-formance ratio of this algorithm equals 5/3 + c1/c2

work involving greedy labelings has been done for constraints c1, c2= 2, 1 In this section

we will assume that the constraints are these, unless otherwise noted

Any labeling for the given constraints will have span at least  + 1, as can be deducedfrom examining a node of maximum degree and its neighbors It can be deduced from

Brooks’ theorem (see [8]), that each graph G with maximum degree  has a labeling with

span at most 2+ 2

Griggs and Yeh [15] observe that trees have a labeling of span at most  + 2 Nodes arelabeled so that nodes closer to the root come first Each unlabeled node then has at mostone labeled neighbor, and at most  – 1 labeled nodes at distance 2 from it The authorsconjecture that it is NP-hard to decide whether a particular tree has minimum span  + 1

or  + 2 This conjecture was proven false by Chang and Kuo [5]

Sakai [32] uses a perfect elimination ordering to show that chordal graphs have a

label-ing of span at most ( + 3)2/4 A perfect elimination ordering v1, 2, , v nof the nodes

has the property that for all i, 1  i n, the neighbors of viin the subgraph induced by v1,

v2, , v i–1form a clique A similar approach was later used by Bodlaender et al [4] toobtain upper bounds on labelings of graphs with fixed tree width

Planar graphs are of special interest in the context of channel assignment, since a graphrepresenting adjacency relations between cells will necessarily be planar In [38], van denHeuvel and McGuinness use methods such as used in the proof of the four color theorem

to prove that all planar graphs with constraints c1, c2 admit a graph labeling of span at

most (4c1– 2) + 10c2+ 38c1– 23

5.4 CONCLUSIONS AND OPEN PROBLEMS

I have given an overview of channel assignment algorithms that take channel spacing straints into consideration I have also reviewed the lower bounds and lower boundingtechniques available for this version of the channel assignment problem Many of the al-gorithms described are based on graph labeling, hence an overview of relevant results ongraph labeling is included in this exposition

con-All the algorithms reviewed in this chapter have proven performance ratios Very little

is known about the best possible performance ratio that can be achieved A worthwhile deavor would be to find lower bounds on the performance ratio of any channel assignmentalgorithm for specific graphs and/or specific constraint parameters

en-Other types of constraints may arise in cellular networks Many cellular systems

oper-ate under intermodulation constraints, which forbid the use of frequency gaps that are

multiples of each other Channel assignment under intermodulation constraints is related

to graceful labeling of graphs Another type of constraint forbids the use of certain

chan-nels in certain cells Such constraints may be external, resulting from interference withother systems, or internal, when an existing assignment must be updated to accomodate

growing demand This problem is related to list coloring

In practice, the most commonly encountered channel separation constraints are cositeconstraints and intersite constraints of value 1 or 2 This situation corresponds to a con-

strained graph with parameters c0, c1, , ck, where c1= = cj = 2 and cj+1 = = ck=

1 Much work on graph labelings focusses on constraints 1 and 2, most specifically,

con-straints c1, c2= 2, 1 and c1, c2, c3= 2, 1, 1 As shown above, graph labelings can be

repeat-ed to accomodate demands of more than one channel per node It would be useful to see if

114 CHANNEL ASSIGNMENT AND GRAPH LABELING

there are any better ways to use these graph labelings, possibly via borrowing techniques,

to accomodate high, nonuniform demand

2 D Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm,

May 1998 ftp://mutt.cs.mcgill ca/pub/doc/avis/Av98a.ps.gz

3 A A Bertossi, C M Pinotti, and R B Tan, Efficient use of radio spectrum in wireless

net-works with channel separation between close stations, in Proceedings of DialM 2000, August

2000

4 H L Bodlaender, T Kloks, R B Tan, and J van Leeuwen, Approximations for -coloring of

graphs, in H Reichel and S Tison (Eds.), STACS 2000, Proceedings 17th Annual Symposium on Theoretical Aspects of Computer Science, volume 1770 of Lecture Notes in Computer Science,

Optimiza-8 R Diestel, Graph Theory, 2nd ed New York: Springer-Verlag, 2000

9 S Fitzpatrick, J Janssen, and R Nowakowski, Distributive online channel assignment forhexagonal 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

Se-ries AMS, 1998

11 D A Fotakis and P G Spirakis, A hamiltonian approach to the assignment of non-reusable

fre-quencies, in Foundations of Software Technology and Theoretical Computer Science—FST TCS’98, volume LNCS 1530, pp 18–29, 1998

12 A Gamst, Some lower bounds for a class of frequency assignment problems, IEEE Trans Veh Technol., 35(1): 8–14, 1986

13 J P Georges and D W Mauro, Generalized vertex labelings with a condition at distance two,

16 J Janssen and K Kilakos, Polyhedral analysis of channel assignment problems: (I) Tours, nical Report CDAM-96-17, London School of Economics, LSE, London, 1996

Tech-17 J Janssen and K Kilakos, A polyhedral analysis of channel assignment problems based on

tours, in Proceedings of the 1997 IEEE International Conference on Communications New

York: IEEE 1997 Extended abstract

18 J Janssen and K Kilakos, Polyhedral analysis of channel assignment problems: (II) Tilings,Manuscript, 1997

19 J Janssen and K Kilakos, An optimal solution to the “Philadelphia” channel assignment

prob-lem, IEEE Transactions on Vehicular Technology, 48(3): 1012–1014, May 1999

20 J Janssen and K Kilakos, Tile covers, closed tours and the radio spectrum, in B Sansó and P

Soriano (Eds.), Telecommunications Network Planning, Kluwer, 1999

21 J Janssen and L Narayanan, Channel assignment algorithms for cellular networks with

con-straints, Theoretical Comp Sc A, 1999 to appear, extended abstract published in the

proceed-ings of ISAAC’99

22 J C M Janssen and T E Wentzell, Lower bounds from tile covers for the channel assignmentproblem, Technical Report G-2000-09, GERAD, HEC, Montreal, March 2000

23 D S Johnson, L A McGeoch, and E E Rothberg, Asymptotic experimental analysis for the

Held-Karp traveling salesman bound, in Proceedings of the 7th Annual ACM-SIAM Symposium

on Discrete Algorithms, 1996 To appear

24 K Jonas, Graph Coloring Analogues with a Condition at Distance Two: L(2, 1)-Labelings and List -Labelings PhD thesis, Dept of Math., University of South Carolina, Columbia, SC,

1993

25 I Katzela and M Naghshineh, Channel assignment schemes for cellular mobile

telecommuni-cations: 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 wandte Mathematik und Mechanik, 76: 303–306, 1996

Ange-27 Colin McDiarmid and Bruce Reed, Channel 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, 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 Pardalos (Eds.), Handbook of Combinatorics Kluwer Academic Publishers, 1999

31 A Raychaudhuri, Intersection assignments, T-colourings and powers of graphs, PhD thesis,

34 C Sung and W Wong, Sequential packing 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.

322–327 New York: IEEE, 1983

36 S Ubéda and J Zerovnik, Upper bounds for the span in triangular lattice graphs: application to

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frequency planning for cellular network Technical Report 97–28, Laboratoire de l’Informatique

du Parallélisme, ENS, Lyon, France, September 1997

37 J van den Heuvel, Radio channel assignment on 2-dimensional lattices Technical Report CDAM-98-05, Centre for Discrete and Applicable Mathematics, LSE, 1998

LSE-38 J van den Heuvel and S McGuinness, Colouring the square of a planar graph Technical ReportLSE-CDAM-99-06, Centre for Discrete and Applicable mathematics, LSE, http://www.cdam.lse.ac.uk/Reports, 1999

39 J van den Heuvel, Robert Leese, and Mark Shepherd, Graph labelling and radio channel

assign-ment, Journal of Graph Theory, 29(4), 1998

40 Dong wan Tcha, Yong Joo Chung, and Taek jin Choi, A new lower bound for the frequency

as-signment problem, ACM/IEEE Trans Networking, 5(1): 34–39, 1997

41 M A Whittlesey, J P Georges, and D W Mauro, On the lambda-coloring of Q nand related

graphs, SIAM J Discr Math., 8: 499–506, 1995

42 R K Yeh, Labeling graphs with a condition at distance 2 PhD thesis, Department of

Mathe-matics, University of South Carolina, Columbia, SC, 1990

CHAPTER 6

Wireless Media Access Control

ANDREW D MYERS and STEFANO BASAGNI

Department of Computer Science, University of Texas at Dallas

portabil-With predictions of near exponential growth in the number of wireless subscribers inthe coming decades, pressure is mounting on government regulatory agencies to free upthe RF spectrum to satisfy the growing bandwidth demands This is especially true withregard to the next generation (3G) cellular systems that integrate voice and high-speeddata access services Given the slow reaction time of government bureaucracy and thehigh cost of licensing, wireless operators are typically forced to make due with limitedbandwidth resources

The aim of this chapter is to provide the reader with a comprehensive view of the role anddetails of the protocols that define and control access to the wireless channel, i.e., wirelessmedia access protocols (MAC) protocols We start by highlighting the distinguishing char-acteristics of wireless systems and their impact on the design and implementation of MACprotocols (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 ofmost MAC protocol designs Section 6.5 overviews channel access in cellular telephonynetworks and other centralized networks Section 6.6 focuses on MAC solutions for ad hocnetworks, namely, network architectures with decentralized control characterized by themobility of possibly all the nodes A brief summary concludes the chapter

6.2 GENERAL CONCEPTS

In the broadest terms, a wireless network consists of nodes that communicate by ing “packets” via radio waves These packets can take two forms A unicast packet con-

exchang-119

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)

tains information that is addressed to a specific node, whereas a multicast packet utes the information to a group of nodes The MAC protocol simply determines when anode is allowed to transmit its packets, and typically controls all access to the physical lay-

distrib-er Figure 6.1 depicts the relative position of the MAC protocol within a simplified col stack

proto-The specific functions associated with a MAC protocol vary according to the systemrequirements and application For example, wireless broadband networks carry datastreams with stringent quality of service (QoS) requirements This requires a complexMAC protocol that can adaptively manage the bandwidth resources in order to meet thesedemands Design and complexity are also affected by the network architecture, communi-cation model, and duplexing mechanism employed These three elements are examined inthe rest of the section

6.2.1 Network Architecture

The architecture determines how the structure of the network is realized and where thenetwork intelligence resides A centralized network architecture features a specializednode, i.e., the base station, that coordinates and controls all transmissions within its cover-age area, or cell Cell boundaries are defined by the ability of nodes to receive transmis-sions from the base station To increase network coverage, several base stations are inter-connected by land lines that eventually tie into an existing network, such as the publicswitched telephone network (PTSN) or a local area network (LAN) Thus, each base sta-tion also plays the role of an intermediary between the wired and wireless domains Figure6.2 illustrates a simple two-cell centralized network

User Application User Application

Communication from a base station to a node takes place on a downlink channel, andthe opposite occurs on an uplink channel Only the base station has access to a downlinkchannel, whereas the nodes share the uplink channels In most cases, at least one of theseuplink channels is specifically assigned to collect control information from the nodes Thebase station grants access to the uplink channels in response to service requests received

on the control channel Thus, the nodes simply follow the instructions of the base station.The concentration of intelligence at the base station leads to a greatly simplified nodedesign that is both compact and energy efficient The centralized control also simplifiesQoS support and bandwidth management since the base station can collect the require-ments and prioritize channel access accordingly Moreover, multicast packet transmission

is greatly simplified since each node maintains a single link to the base station On theother hand, the deployment of a centralized wireless network is a difficult and slowprocess The installation of new base stations requires precise placement and system con-figuration along with the added cost of installing new landlines to tie them into the exist-ing system The centralized system also presents a single point of failure, i.e., no base sta-tion equals no service

The primary characteristic of an ad hoc network architecture is the absence of any defined structure Service coverage and network connectivity are defined solely by nodeproximity and the prevailing RF propagation characteristics Ad hoc nodes communicatedirectly with one another in a peer-to-peer fashion To facilitate communication betweendistant nodes, each ad hoc node also acts as a router, storing and forwarding packets onbehalf of other nodes The result is a generalized wireless network that can be rapidly de-ployed and dynamically reconfigured to provide on-demand networking solutions An adhoc architecture is also more robust in that the failure of one node is less likely to disruptnetwork services Figure 6.3 illustrates a simple ad hoc network

pre-Although a generic architecture certainly has its advantages, it also introduces severalnew challenges All network control, including channel access, must be distributed Each

ad hoc node must be aware of what is happening in its environment and cooperate withother nodes in order to realize critical network services Considering that most ad hoc sys-tems are fully mobile, i.e., each node moves independently, the level of protocol sophisti-cation and node complexity is high Moreover, each ad hoc node must maintain a signifi-

6.2 GENERAL CONCEPTS 121

Wireless LinkBase Station

cant amount of state information to record crucial information such as the current networktopology.

Given its distributed nature, channel access in an ad hoc network is achieved throughthe close cooperation between competing nodes Some form of distributed negotiation isneeded in order to efficiently allocate channel resources among the active nodes Theamount of overhead, both in terms of time and bandwidth resources, associated with thisnegotiation will be a critical factor of the overall system performance

6.2.2 Communication Model

The communication model refers to the overall level of synchronization present in thewireless system and also determines when channel access can occur There are differentdegrees of synchronization possible; however, there are only two basic communicationmodels The synchronous communication model features a slotted channel consisting ofdiscrete time intervals (slots) that have the same duration With few exceptions, these slotsare then grouped into a larger time frame that is cyclically repeated All nodes are thensynchronized according to this time frame and communication occurs within the slotboundaries

The uniformity and regularity of the synchronous model simplifies the provision ofquality of service (QoS) requirements Packet jitter, delay, and bandwidth allotment can all

be controlled through careful time slot management This characteristic establishes the chronous communication model as an ideal choice for wireless systems that support voiceand multimedia applications However, the complexity of the synchronization process de-pends on the type of architecture used In a centralized system, a base station can broadcast

syn-a besyn-acon signsyn-al to indicsyn-ate the beginning of syn-a time frsyn-ame All nodes within the cell simplylisten for these beacons to synchronize themselves with the base station The same is nottrue of an ad hoc system that must rely on more sophisticated clock synchronization mech-anisms, such as the timing signals present in the global positioning system (GPS)

The asynchronous communication model is much less restrictive, with communicationtaking place in an on-demand fashion There are no time slots and thus no need for anyglobal synchronization Although this certainly reduces node complexity and simplifiescommunication, it also complicates QoS provisioning and bandwidth management Thus,

an asynchronous model is typically chosen for applications that have limited QoS

require-Node

Wireless Link

Figure 6.3 Ad hoc network architecture

ments, such as file transfers and sensor networks The reduced interdependence betweennodes also makes it applicable to ad hoc network architectures.

at the same time, which dramatically increases the rate at which feedback can be obtained.However, FDD systems require more complex hardware and frequency management

6.3 WIRELESS ISSUES

The combination of network architecture, communication model, and duplexing nism define the general framework within which a MAC protocol is realized Decisionsmade here will define how the entire system operates and the level of interaction betweenindividual nodes They will also limit what services can be offered and delineate MACprotocol design However, the unique characteristics of wireless communication must also

mecha-be taken into consideration In this section, we explore these physical constraints and cuss their impact on protocol design and performance

dis-Radio waves propagate through an unguided medium that has no absolute or able boundaries and is vulnerable to external interference Thus, wireless links typicallyexperience high bit error rates and exhibit asymmetric channel qualities Techniques such

observ-as channel coding, bit interleaving, frequency/space diversity, and equalization increobserv-asethe survivability of information transmitted across a wireless link An excellent discussion

on these topics can be found in Chapter 9 of [1] However, the presence of asymmetrymeans that cooperation between nodes may be severely limited

The signal strength of a radio transmission rapidly attenuates as it progresses awayfrom the transmitter This means that the ability to detect and receive transmissions is de-pendent on the distance between the transmitter and receiver Only nodes that lie within aspecific radius (the transmission range) of a transmitting node can detect the signal (carri-er) on the channel This location-dependent carrier sensing can give rise to so-called hid-den and exposed nodes that can detrimentally affect channel efficiency A hidden node isone that is within range of a receiver but not the transmitter, whereas the contrary holdstrue for an exposed node Hidden nodes increase the probability of collision at a receiver,whereas exposed nodes may be denied channel access unnecessarily, thereby underutiliz-ing the bandwidth resources

Performance is also affected by the signal propagation delay, i.e., the amount of timeneeded for the transmission to reach the receiver Protocols that rely on carrier sensing areespecially sensitive to the propagation delay With a significant propagation delay, a nodemay initially detect no active transmissions when, in fact, the signal has simply failed to

6.3 WIRELESS ISSUES 123

reach it in time Under these conditions, collisions are much more likely to occur and tem performance suffers In addition, wireless systems that use a synchronous communica-tions model must increase the size of each time slot to accommodate propagation delay.This added overhead reduces the amount of bandwidth available for information transmis-sion.

sys-Even when a reliable wireless link is established, there are a number of additional ware constraints that must also be considered The design of most radio transceivers only al-low half-duplex communication on a single frequency When a wireless node is activelytransmitting, a large fraction of the signal energy will leak into the receive path The powerlevel of the transmitted signal is much higher than any received signal on the same frequen-

hard-cy, and the transmitting node will simply receive its own transmission Thus, traditional lision detection protocols, such as Ethernet, cannot be used in a wireless environment.This half-duplex communication model elevates the role of duplexing in a wirelesssystem However, protocols that utilize TDD must also consider the time needed toswitch between transmission and reception modes, i.e., the hardware switching time.This switching can add significant overhead, especially for high-speed systems that op-erate at peak capacity [2] Protocols that use handshaking are particularly vulnerable tothis phenomenon For example, consider the case when a source node sends a packet andthen receives feedback from a destination node In this instance, a turnaround time of 10

col-␮s and transmission rate of 10 Mbps will result in an overhead of 100 bits of lost nel capacity The effect is more significant for protocols that use multiple rounds of mes-sage exchanges to ensure successful packet reception, and is further amplified whentraffic loads are high

chan-6.4 FUNDAMENTAL MAC PROTOCOLS

Despite the great diversity of wireless systems, there are a number of well-known MACprotocols whose use is universal Some are adapted from the wired domain and others areunique to the wireless one Most of the current MAC protocols use some subset of the fol-lowing techniques

6.4.1 Frequency Division Multiple Access (FDMA)

FDMA divides the entire channel bandwidth into M equal subchannels that are

sufficient-ly separated (via guard bands) to prevent cochannel interference (see Figure 6.4) Ignoringthe small amount of frequency lost to the guard bands, the capacity of each subchannel is

C/M, where C is the capacity associated with the entire channel bandwidth Each source

node can then be assigned one (or more) of these subchannels for its own exclusive use

To receive packets from a particular source node, a destination node must be listening on

the proper subchannel The main advantage of FDMA is the ability to accommodate M

si-multaneous packet transmissions (one on each subchannel) without collision However,this comes at the price of increased packet transmission times, resulting in longer packet

delays For example, the transmission time of a packet that is L bits long is M · L/C This is

M times longer than if the packet was transmitted using the entire channel bandwidth The

exclusive nature of the channel assignment can also result in underutilized bandwidth sources when a source node momentarily lacks packets to transmit.

re-6.4.2 Time Division Multiple Access (TDMA)

TDMA divides the entire channel bandwidth into M equal time slots that are then

orga-nized into a synchronous frame (see Figure 6.5) Conceptually, each slot represents one

channel that has a capacity equal to C/M, where C is again the capacity of the entire

chan-nel bandwidth Each node can then be assigned one (or more) time slots for its own sive use Consequently, packet transmission in a TDMA system occurs in a serial fashion,

exclu-6.4 FUNDAMENTAL MAC PROTOCOLS 125

Figure 6.4 Frequency division multiple access

Figure 6.5 Time division multiple access

2

Time

with each node taking turns accessing the channel Since each node has access to the

en-tire channel bandwidth in each time slot, the time needed to transmit a L bit packet is then L/C When we consider the case where each node is assigned only one slot per frame, however, there is a delay of (M – 1) slots between successive packets from the same node.

Once again, channel resources may be underutilized when a node has no packet(s) totransmit in its slot(s) On the other hand, time slots are more easily managed, allowing thepossibility of dynamically adjusting the number of assigned slots and minimizing theamount of wasted resources

6.4.3 Code Division Multiple Access (CDMA)

While FDMA and TDMA isolate transmissions into distinct frequencies or time instants,CDMA allow transmissions to occupy the channel at the same time without interference.Collisions are avoided through the use of special coding techniques that allow the infor-mation to be retrieved from the combined signal As long as two nodes have sufficientlydifferent (orthogonal) codes, their transmissions will not interfere with one another.CDMA works by effectively spreading the information bits across an artificially broad-ened channel This increases the frequency diversity of each transmission, making it lesssusceptible to fading and reducing the level of interference that might affect other systemsoperating in the same spectrum It also simplifies system design and deployment since allnodes share a common frequency band However, CDMA systems require more sophisti-cated and costly hardware, and are typically more difficult to manage

There are two types of spread spectrum modulation used in CDMA systems Direct quence spread spectrum (DSSS) modulation modifies the original message by multiplying

se-it wse-ith another faster rate signal, known as a pseudonoise (PN) sequence This naturally creases the bit rate of the original signal and the amount of bandwidth that it occupies Theamount of increase is called the spreading factor Upon reception of a DSSS modulated sig-nal, a node multiplies the received signal by the PN sequence of the proper node This in-creases the amplitude of the signal by the spreading factor relative to any interfering signals,which are diminished and treated as background noise Thus, the spreading factor is used toraise the desired signal from the interference This is known as the processing gain.Nevertheless, the processing gain may not be sufficient if the original information signalreceived is much weaker than the interfering signals Thus, strict power control mechanismsare needed for systems with large coverage areas, such as a cellular telephony networks.Frequency hopping spread spectrum (FHSS) modulation periodically shifts the trans-mission frequency according to a specified hopping sequence The amount of time spent

in-at each frequency is referred to as the dwell time Thus, FHSS modulin-ation occurs in twophases In the first phase, the original message modulates the carrier and generates a nar-rowband signal Then the frequency of the carrier is modified according to the hopping se-quence and dwell time

6.4.4 ALOHA Protocols

In contrast to the elegant solutions introduced so far, the ALOHA protocols attempt toshare the channel bandwidth in a more brute force manner The original ALOHA protocol

was developed as part of the ALOHANET project at the University of Hawaii [3].Strangely enough, the main feature of ALOHA is the lack of channel access control.When a node has a packet to transmit, it is allowed to do so immediately Collisions arecommon in such a system, and some form of feedback mechanism, such as automatic re-peat request (ARQ), is needed to ensure packet delivery When a node discovers that itspacket was not delivered successfully, it simply schedules the packet for retransmission.Naturally, the channel utilization of ALOHA is quite poor due to packet vulnerability.The results presented in [4] demonstrate that the use of a synchronous communicationmodel can dramatically improve protocol performance This slotted ALOHA forces eachnode to wait until the beginning of a slot before transmitting its packet This reduces theperiod during which a packet is vulnerable to collision, and effectively doubles the chan-

nel utilization of ALOHA A variation of slotted ALOHA, known as p-persistent slotted ALOHA, uses a persistence parameter p, 0 < p < 1, to determine the probability that a

node transmits a packet in a slot Decreasing the persistence parameter reduces the ber of collisions, but increases delay at the same time

num-6.4.5 Carrier Sense Multiple Access (CSMA) Protocols

There are a number of MAC protocols that utilize carrier sensing to avoid collisions withongoing transmissions These protocols first listen to determine whether there is activity

on the channel An idle channel prompts a packet transmission and a busy channel presses it The most common CSMA protocols are presented and formally analyzed in [5].While the channel is busy, persistent CSMA continuously listens to determine whenthe activity ceases When the channel returns to an idle state, the protocol immediatelytransmits a packet Collisions will occur when multiple nodes are waiting for an idle chan-nel Nonpersistent CSMA reduces the likelihood of such collisions by introducing ran-domization Each time a busy channel is detected, a source node simply waits a randomamount of time before testing the channel again This process is repeated with an expo-nentially increasing random interval until the channel is found idle

sup-The p-persistent CSMA protocol represents a compromise between persistent and

non-persistent CSMA In this case, the channel is considered to be slotted but time is not chronized The length of each slot is equal to the maximum propagation delay, and carriersensing occurs at the beginning of each slot If the channel is idle, the node transmits a

syn-packet with probability p, 0 < p < 1 This procedure continues until either the syn-packet is

sent, or the channel becomes busy A busy channel forces a source node to wait a randomamount of time before starting the procedure again

6.5 CENTRALIZED MAC PROTOCOLS

In this section, we provide an overview of two of the most prevalent centralized wirelessnetworks Cellular telephony is the most predominant form of wireless system in currentoperation Wireless ATM is generating a lot of interest for its ability to deliver broadbandmultimedia services across a wireless link Each system will be briefly highlighted and theMAC protocol will be examined

6.5 CENTRALIZED MAC PROTOCOLS 127

6.5.1 Cellular Telephony

The advanced mobile phone system (AMPS) is an FDMA-based cellular system [6] Thesystem features 832 full-duplex channels that are grouped into control and data channels.Each cell has a full-duplex control channel dedicated to system management, paging,and call setup There are also 45–50 data channels that can be used for voice, fax, or data.The base station grants access to a data channel in response to a call setup request sent onthe control channel A data channel remains assigned to a specific node until it is relin-quished or the node moves outside the current cell Access to the control channel is deter-mined using a CSMA-based MAC protocol The base station periodically broadcasts thestatus of the control channel, and a node transmits its setup request (possibly in contentionwith other nodes) when the control channel is idle Collisions among setup requests are re-solved using randomized retransmissions

The IS-136 cellular system is a digital version of the AMPS system [7] As such, it ates within the same spectrum using the same frequency spacing of the original AMPS sys-tem Each data channel is then slotted and a time frame of six slots is used This allows thesystem to support multiple users within a single AMPS data channel An assignment of oneslot per frame can support a total of six users transmitting at a rate of 8.1 kb/s Higher datarates can be achieved by successively doubling the number of assigned slots up to a maxi-mum of 48.6 kb/s Channel access remains relatively unchanged from the original AMPSsystem

oper-The IS-95 cellular system is a CDMA-based wireless network in which all the base tions share a common frequency band with individual transmissions being distinguished

sta-by their PN sequences [8] Strict power control ensures that all transmitted signals reachthe base station with the same power level This allows a more equitable sharing of thesystem power resources while minimizing systemwide cochannel interference However,the equalized power levels make it difficult to determine when a node is about to leave onecell and enter another A node must communicate with multiple base stations simultane-ously, allowing it to measure the relative signal quality of each base station Handover isthen made to the base station with the best signal characteristics This type of system re-quires complex and costly hardware both within the base stations and nodes

Cdma2000 is the third generation (3G) version of the IS-95 cellular system Cdma2000

is backward compatible with the current system, allowing legacy users to be

accommodat-ed in future 3G systems Many other proposaccommodat-ed 3G cellular systems have also adoptaccommodat-ed aCDMA interface This includes the 3G version of GSM known as the universal mobiletelecommunications services (UMTS) [9]

6.5.2 Wireless ATM

Asynchronous transfer mode (ATM) is a high-performance connection-oriented switchingand multiplexing technology that uses fixed-sized packets to transport a wide range of in-tegrated services over a single network These include voice, video, and multimedia ser-vices that have different QoS requirements The ability to provide specific QoS services isone of the hallmarks of ATM Wireless ATM is designed to extend these integrated ser-vices to the mobile user

Similar to cellular systems, wireless ATM nodes send requests to the base station forservice The specific QoS requirements of an application are included in these requestmessages The base station then collects these requirements and allocates the uplink anddownlink channels accordingly Thus wireless ATM MAC protocols typically follow athree-phase model In the first phase, a request message is sent on a random access controlchannel, usually using a slotted ALOHA protocol The second phase involves the base sta-tion scheduling uplink and downlink transmissions according to the QoS requirements ofthe current traffic mix Preference is given to delay-sensitive data, such as voice packets,whereas datagram services must make due with any remaining capacity The third phaseinvolves the transmission of packets according to the schedule created in phase two.The PRMA/DA [10] and DSA++ [11] protocols are two examples of this three-phaseMAC design using FDD, whereas MASCARA [12] and DTDMA [13] use TDD Each ofthese protocols are respectively illustrated in Figures 6.6 through 6.9 and Table 6.1 sum-marizes their relative characteristics.

6.5 CENTRALIZED MAC PROTOCOLS 129

Request

slots

CBR reserved slots

VBR reserved slots

Data reserved slots Fixed Time Frame

6.6 AD HOC MAC PROTOCOLS

Ad hoc networks do not have the benefit of predefined base stations to coordinate channelaccess, thus invalidating many of the assumptions held by centralized MAC designs Inthis section, we focus our attention on MAC protocols that are specifically designed for adhoc networks

A possible taxonomy of ad hoc MAC protocols includes three broad protocol gories that differ in their channel access strategy: contention protocols, allocation proto-cols, and a combination of the two (hybrid protocols)

cate-Contention protocols use direct competition to determine channel access rights, and solve collisions through randomized retransmissions The ALOHA and CSMA protocols

re-Contention slots

Uplink period Downlink period Frame structure

Reserved slots Broadcast

Fixed frame length

Figure 6.9 DTDMA protocol

TABLE 6.1 Wireless ATM MAC protocol relative characteristics

introduced in Sections 6.4.4 and 6.4.5 are prime examples With the exception of slottedALOHA, most contention protocols employ an asynchronous communication model Col-lision avoidance is also a key design element that is realized through some form of controlsignaling.

The contention protocols are simple and tend to perform well at low traffic loads, i.e.,when there are few collision, leading to high channel utilization and low packet delay.However, protocol performance tends to degrade as the traffic loads are increased and thenumber of collisions rise At very high traffic loads, a contention protocol can become un-stable as the channel utilization drops This can result in exponentially growing packet de-lay and network service breakdown since few, if any, packets can be successfully ex-changed

Allocation protocols employ a synchronous communication model and use a ing algorithm that generates a mapping of time slots to nodes This mapping results in atransmission schedule that determines in which particular slots a node is allowed to accessthe channel Most allocation protocols create collision-free transmission schedules, thusthe schedule length (measured in slots) forms the basis of protocol performance The timeslots can either be allocated statically or dynamically, leading to a fixed and variableschedule length

schedul-The allocation protocols tend to perform well at moderate to heavy traffic loads as allslots are likely to be utilized These protocols also remain stable even when the trafficloads are extremely high This is due to the fact that most allocation protocols ensure thateach node has collision-free access to at least one time slot per frame On the other hand,these protocols are disadvantaged at low traffic loads due to the artificial delay induced bythe slotted channel This results in significantly higher packet delays with respect to thecontention protocols

Hybrid protocols can be loosely described as any combination of two or more cols However, in this section, the definition of the term hybrid will be constrained to in-clude only those protocols that combine elements of contention- and allocation-basedchannel access schemes in such a way as to maintain their individual advantages whileavoiding their drawbacks Thus, the performance of a hybrid protocol should approximate

proto-a contention protocol when trproto-affic is light, proto-and proto-an proto-allocproto-ation protocol during periods ofhigh load

6.6.1 Contention Protocols

Contention protocols can be further classified according to the type of collision avoidancemechanism employed The ALOHA protocols make up the category of protocols that fea-ture no collision avoidance mechanism, i.e., they simply react to collision via randomizedretransmissions Most contention protocols, however, use some form of collision avoid-ance mechanism

The busy-tone multiple access (BTMA) protocol [14] divides the entire bandwidth intotwo separate channels The main data channel is used for the transmission of packets, andoccupies the majority of the bandwidth The control channel is used for the transmission

of a special busy-tone signal that indicates the presence of activity on the data channel.These signals are not bandwidth-intensive, thus the control channel is relatively small

6.6 AD HOC MAC PROTOCOLS 131

The BTMA protocol operates as follows When a source node has a packet to transmit,

it first listens for the busy-tone signal on the control channel If the control channel is idle,i.e., no busy-tone is detected, then the node may begin transmitting its packet Otherwise,the node reschedules the packet for transmission at some later time Any node that detectsactivity on the data channel immediately begins transmitting the busy-tone on the controlchannel This continues until the activity on the data channel ceases

In this way, BTMA prevents all nodes that are two hops away from an active sourcenode from accessing the data channel This significantly lowers the level of hidden nodeinterference, and therefore reduces the probability of collision However, the number ofexposed nodes is dramatically increased and this may result in a severely underutilizeddata channel

The receiver-initiated busy-tone multiple access (RI-BTMA) protocol [15] attempts tominimize the number of exposed nodes by having only the destination(s) transmit thebusy-tone Rather than immediately transmitting the busy-tone upon detection of an activedata channel, a node monitors the incoming data transmission to determine whether it is adestination This determination takes a significant amount of time, especially in a noisyenvironment with corrupted information During this time, the initial transmission re-mains vulnerable to collision This can be particularly troublesome in high-speed systemswhere the packet transmission time may be short

The wireless collision detect (WCD) protocol [2] essentially combines the BTMA andRI-BTMA protocols by using two distinct busy-tone signals on the control channel WCDacts like BTMA when activity is first detected on the main channel, i.e., it transmits a col-lision detect (CD) signal on the BTC RI-BTMA behavior takes over once a node deter-mines it is a destination In this case, a destination stops transmitting the CD signal andbegins transmitting a feedback-tone (FT) signal In this way, WCD minimizes the exposednodes while still protecting the transmission from hidden node interference

These busy-tone protocols feature simple designs that require only a minimal increase

in hardware complexity Because of its unique characteristics, the WCD protocol is theoverall performance leader, followed by RI-BTMA and BTMA, respectively [2] Further-more, the performance of busy-tone protocols are less sensitive to the hardware switchingtime since it is assumed that a node can transmit and receive on the data and control chan-nels simultaneously However, wireless systems that have a limited amount of RF spec-trum may not be able to realize a separate control and data channel In such cases, colli-sion avoidance using in-band signaling is necessary

The multiple access with collision avoidance (MACA) protocol [16] uses a shaking dialogue to alleviate hidden node interference and minimize the number of ex-posed nodes This handshake consists of a request-to-send (RTS) control packet that issent from a source node to its destination The destination replies with a clear-to-send(CTS) control packet, thus completing the handshake A CTS response allows the sourcenode to transmit its packet The absence of a CTS forces a node to reschedule the pack-

hand-et for transmission at some later time Figure 6.10 illustrates the operation of the MACAprotocol

Consider the case where node B wishes to send a packet to node A Node B first mits an RTS, which reaches nodes A, C, and D (Figure 6.10a) Node A then responds by

trans-sending a CTS, which reaches nodes B and C, thus completing the handshake (Figure 6.10b) At this point, B is free to send its packet (Figure 6.10c).

Notice that a hidden node is likely to overhear the CTS packet sent by a destinationnode, whereas an exposed node is not Thus, by including the time needed to receive aCTS and packet in the respective RTS and CTS packets, we reduce the likelihood of hid-den node interference and the number of exposed nodes simultaneously

The MACAW protocol [17] enhances MACA by including carrier sensing to avoid lisions among RTS packets, and a positive acknowledgement (ACK) to aid in the rapid re-covery of lost packets To protect the ACK from collision, a source node transmits a datasending (DS) control packet to alert exposed nodes of its impending arrival Improve-ments are also made to the collision resolution algorithm to ensure a more equitable shar-ing of the channel resources

col-The MACA with piggyback reservations (MACA/PR) protocol [18] enhances MACA

by incorporating channel reservations This allows the system to support QoS sensitive plications Each node maintains a reservation table (RT) that is used to record the channelreservations made by neighboring nodes A source node makes a reservation by first com-pleting a RTS/CTS exchange It then sends the first real-time packet, whose header con-tains the time interval specifying the interval in which the next one will be sent The desti-nation responds with an ACK carrying the equivalent time interval Other nodes withinrange note this reservation in their RT and remain silent during the subsequent time inter-vals Thus, the source node can send subsequent real-time packets without contention Toensure proper bookkeeping, the nodes periodically exchange their RTs

ap-The MACA by invitation (MACA-BI) protocol [19] reverses the handshaking logue of MACA In this case, the destination node initiates packet transmission by send-ing a request-to-receive (RTR) control packet to the source node The source node re-sponds to this poll with a packet transmission Thus, each node must somehow predictwhen neighbors have packets for it This means that each node must maintain a list of itsneighbors along with their traffic characteristics In order to prevent collision, the nodesmust also synchronize their polling mechanisms by sharing this information with theirneighbors

dia-These MACA-based contention protocols minimize collisions by reducing the negativeeffect of hidden and exposed nodes through simple handshaking dialogues However, theexchange of multiple control packets for each data packet magnifies the impact of signalpropagation delay and hardware switching time To some extent, the MACA/PR and

6.6 AD HOC MAC PROTOCOLS 133

C

D

E A B

MACA/BI protocols alleviate these problems by reducing the amount of handshaking, yetthe amount of state information maintained at each node can be substantial.

6.6.2 Allocation Protocols

There are two distinct classes of allocation protocols that differ in the way the sion schedules are computed Static allocation protocols use a centralized scheduling al-gorithm that statically assigns a fixed transmission schedule to each node prior to its oper-ation This type of scheduling is similar to the assignment of MAC addresses for Ethernetinterface cards Dynamic allocation protocols uses a distributed scheduling algorithm thatcomputes transmission schedules in an on-demand fashion

transmis-Since the transmission schedules are assigned beforehand, the scheduling algorithm of astatic allocation protocol requires global system parameters as input The classic TDMAprotocol builds its schedules according to the maximum number of nodes in the network

For a network of N nodes, the protocol uses a frame length of N slots and assigns each node

one unique time slot Since each node has exclusive access to one slot per frame, there is nothreat of collision for any packet type (i.e., unicast or multicast) Moreover, the channel ac-cess delay is bounded by the frame length Because of the equivalence between system sizeand frame length, classic TDMA performs poorly in large-scale networks

The time spread multiple access (TSMA) protocol [20] relaxes some of the strict quirements of classic TDMA to achieve better performance while still providing boundedaccess delay The TSMA scheduling algorithm assigns each node multiple slots in a singleframe, and permits a limited amount of collisions to occur These two relaxations allowTSMA to obtain transmission schedules whose lengths scale logarithmically with respect

re-to the number of nodes Furthermore, TSMA guarantees the existence of a collision-freetransmission slot to each neighbor within a single frame

The source of this “magic” is the scheduling algorithm that makes use of the matical properties of finite fields An excellent introduction to finite fields can be found

mathe-in [21] The schedulmathe-ing algorithm is briefly outlmathe-ined as follows For a network of N nodes, the parameters q (of the form q = p m , where p is a prime and m an integer) and integer k are chosen such that q k+1 ⱖ N and q ⱖkDmax+ 1, where Dmaxis the maximum node de-

gree Each node can then be assigned a unique polynomial f over the Galois field GF(q) Using this polynomial, a unique TSMA transmission schedule is computed where bit i = 1

if (i mod q) = f (i/q), otherwise i = 0.

As shown in [20], that this TSMA scheduling algorithm provides each node with atransmission schedule with guaranteed access in each time frame The maximum length ofthis schedule is bounded by

ᎏᎏlog2Dmax

a network of N = 1000 nodes For TSMA protocols a ⍀(log n) lower bound has been proved for L in [22] We notice that there is still a gap between the TSMA upper bound

and the mentioned logarithmic lower bound Therefore, there is still room for ment (more likely on the lower-bound side) TSMA-like protocols have also been de-ployed as a basis for implementing broadcast (i.e., one-to-all communication) in ad hocnetworks Upper and lower bound for deterministic and distributed TSMA-based broad-cast can be found in [23, 24] and [25], respectively

improve-With mobile ad hoc networks, nodes may be activated and deactivated without ing, and unrestricted mobility yields a variable network topology Consequently, globalparameters, such as node population and maximum degree, are typically unavailable ordifficult to predict For this reason, protocols that use only local parameters have been de-veloped A local parameter refers to information that is specific to a limited region of the

warn-network, such as the number of nodes within x hops of a reference node (referred to as an x-hop neighborhood) A dynamic allocation protocol then uses these local parameters to

deterministically assign transmission slots to nodes Because local parameters are likely tovary over time, the scheduling algorithm operates in a distributed fashion and is periodi-cally executed to adapt to network variations

Dynamic allocation protocols typically operate in two phases Phase one consists of aset of reservation slots in which the nodes contend for access to the subsequent transmis-sion slots This is similar to many of the wireless ATM protocols studied in Section 6.5.Lacking a coordinating base station, contention in this phase requires the cooperation ofeach individual node to determine and verify the outcome Successful contention in phaseone grants a node access to one or more transmission slots of phase two, in which packetsare sent

A great number of dynamic allocation protocols have been proposed The protocols[26–29] are just a few excellent examples of this two-phase design They use a contentionmechanism that is based on classic TDMA Essentially, the nodes take turns contendingfor slot reservations, with the earliest node succeeding This results in a high degree of un-fairness that is equalized by means of a reordering policy Although these protocols createtransmission schedules that are specific to the local network topology, they still requireglobal parameters

In contrast, the five-phase reservation protocol (FPRP) [29] is designed to be

arbitrari-ly scalable, i.e., independent of the global network size FPRP uses a complex frame ture that consists of two subframe types, namely reservation frames and information

struc-frames As illustrated in Fig 6.11, a reservation frame precedes a sequence of k

informa-tion frames Each reservainforma-tion frame consists of ᐉ reservainforma-tion slots that correspond to the ᐉinformation slots of each information frame Thus, if a node wants to reserve a specific in-formation slot, it contends in the corresponding reservation slot At the end of the reserva-

6.6 AD HOC MAC PROTOCOLS 135

TABLE 6.2 Frame lengths of classic TDMA versus TSMA

spec-trum [21, 33 ] of R2to get five channels In particular, it uses the channels 21, 24, 27, 30 ,

33 instead of the...

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. .. and

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