Formally introduced in [9] for use in network modeling, and ied in conjunction with broadcast scheduling in [28], unit disk graphs [9, 3] model... An example of a unit disk graph model o
Trang 1CHAPTER 16
Broadcast Scheduling for TDMA in
Wireless Multihop Networks
The chapter is organized as follows In the next section we provide background and minology on broadcast scheduling and related topics Section 16.3 examines the computa-tional complexity of broadcast scheduling Sections 16.4 and 16.5 study approximation al-gorithms in centralized and distributed domains Section 16.6 briefly outlines somerelated results Finally, Section 16.7 summarizes the chapter and outlines prominent openproblems
ter-16.2 WHAT IS BROADCAST SCHEDULING?
Background and terminology associated with wireless multihop networks, the modeling
of such networks, and related concepts are provided in this section
16.2.1 Basic Concepts of Wireless Multihop Networks
We define a wireless multihop network as a network of stations that communicate witheach other via wireless links using radio signals All of the stations share a common chan-nel Each station in the network acts both as a host and as a switching unit It is required
347
Copyright © 2002 John Wiley & Sons, Inc ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic)
Trang 2that the transmission of a station be received collision-free by all of its one-hop (i.e., rect) neighbors This cannot occur if a station transmits and receives simultaneously or if astation simultaneously receives from more than one station A collision caused by trans-mitting and receiving at the same time is called a primary conflict A collision caused bysimultaneously receiving from two stations is called a secondary conflict We note that as
di-a prdi-acticdi-al mdi-atter, some existing multihop networks mdi-ay violdi-ate one or more of the di-aboveassumptions
A wireless multihop network can be modeled by a directed graph G = (V, A), where V is
a set of nodes denoting stations in the network and A is a set of directed edges between nodes, such that for any two distinct nodes u and v, edge (u, v) 僆 A if and only if v can re- ceive u’s transmission As is common throughout the literature, we assume that (u, v) 僆 A
if and only if (v, u) 僆 A That is, links are bidirectional, in which case it is common to use
an undirected graph G = (V, E) Throughout this chapter we use undirected graphs to
model wireless multihop networks
Sharing a common channel introduces the question of how the channel is accessed.Channel access mechanisms for wireless multihop networks fall into two general cate-gories: random access (e.g., ALOHA) and fixed access Broadcast scheduling, the focus
of this chapter, is a fixed access technique that preallocates the common channel by way
of TDMA so that collisions do not occur
16.2.2 Defining Broadcast Scheduling
The task of a broadcast scheduling algorithm is to produce and/or maintain an infiniteschedule of TDMA slots such that each station is periodically assigned a slot for transmis-sion and all transmissions are received collision-free In this framework, most broadcastscheduling algorithms operate by producing a finite length nominal schedule in whicheach station is assigned exactly one slot for transmission, and then indefinitely repeatingthat nominal schedule Except where noted otherwise, throughout this chapter the termbroadcast schedule refers to a nominal schedule
16.2.3 Graph Concepts and Terminology
In modeling wireless multihop networks by undirected graphs, many variants are possible
in regard to network topology Among the possibilities, the three most relevant to thischapter are:
1 Arbitrary graphs Such graphs can model any physical situation, including for ample, geographically close neighbors that cannot communicate directly due to in-terference (e.g., a mountain) on a direct line between the stations
ex-2 Planar graphs A graph is planar if and only if it can be drawn in the plane such that
no two edges intersect except at common endpoints Planar graphs are among themost widely studied classes of graphs
3 Unit disk graphs Formally introduced in [9] for use in network modeling, and ied in conjunction with broadcast scheduling in [28], unit disk graphs [9, 3] model
Trang 3stud-the situation in which all stations utilize a uniform transmission range R, and stud-there
is no interference Thus, the transmission of a station v will be received by all tions within a Euclidean distance R of v In these graphs, there is an edge between nodes u and v if and only if the Euclidean distance between stations u and v does not exceed R An example of a unit disk graph model of a wireless multihop net-
sta-work and a nominal schedule are shown in Figure 16.1 In that figure, there is a link
between a pair of stations if and only if the circles of radius R/2 centered at the pair
of stations intersect, including being tangent The slots assigned to the stations arethe numbers inside the brackets
Regardless of the graph model utilized, if there is an edge between nodes u and v, then u is
a one-hop neighbor/neighbor of v and likewise v is a neighbor of u The degree of a node
is the number of neighbors The degree of a network is the maximum degree of thenodes in the network The distance-2 neighbors of a node include all of its one-hop neigh-bors and the one-hop neighbors of its one-hop neighbors The two-hop neighbors of anode are those nodes that are distance-2 neighbors, but are not one-hop neighbors The
unit subset of node u consists of u and its distance-2 neighbors The distance-2 degree D(u) of u is the number of distance-2 neighbors of u, and the distance-2 degree D of a net-
work, is the maximum distance-2 degree of the nodes in the network
Relevant to broadcast scheduling is distance-2 coloring [22, 13] of a graph G = (V, E), where the problem is to produce an assignment of colors C : V씮 1, 2, such that notwo nodes are assigned the same color if they are distance-2 neighbors An optimal color-ing is a coloring utilizing a minimum number of colors A distance-2 coloring algorithm issaid to color nodes in a greedy fashion (i.e., greedily) if when coloring a node, the color
Figure 16.1 Broadcast scheduling modeled by a unit disk graph.
Trang 4assigned is the smallest number color that can be assigned without resulting in conflicts.Here, the constraint number of a node is the number of different colors assigned to thenode’s distance-2 neighbors
In the context of broadcast scheduling, determining a nominal schedule is directly stracted to distance-2 coloring, whereby slots that are assigned to stations are translatedinto colors that are assigned to nodes In this chapter, we will interchangeably use theterms network and graph, station and node, and slot and color
ab-16.2.4 Varieties of Broadcast Scheduling Algorithms
There are two main varieties of broadcast scheduling algorithms—centralized and uted
distrib-Centralized Algorithms
Centralized algorithms are executed at a central site and the results are then transmitted tothe other stations in the network This requires that the central site have complete informa-tion about the network This is a strong assumption that is not easy to justify for wirelessmultihop networks with mobile stations The study of centralized algorithms, however,provides an excellent starting point for both the theory of broadcast scheduling and the de-velopment of more practical algorithms Further, for some stationary wireless networks, it
is reasonable to run a centralized algorithm at the net management center and then ute schedules to stations
distrib-In the centralized algorithm context, there are two types of algorithms corresponding tohow the input is provided:
1 Off-line algorithms The network topology is provided to the central site in its
en-tirety The algorithm computes the schedule for the entire network once and for all
2 Adaptive algorithms: With off-line algorithms, if the network topology changes,
then the algorithm is rerun for the entire network However, as wireless networksare evolving towards thousands of stations spread over a broad geographical areaand operating in an unpredictable dynamic environment, the use of off-line schedul-ing algorithms is not realistic In practice, it is absolutely unaffordable to halt com-munication whenever there is a change in the network, so as to produce a newschedule “from scratch.” In such circumstances, adaptive algorithms are requiredThat is, given a broadcast schedule for the network, if the network changes (by thejoining or leaving of a station), then the schedule should be appropriately updated tocorrespond to the modified network Thus, an adaptive algorithm for broadcastscheduling is one that, given a wireless multihop network, a broadcast schedule forthat network, and a change in the network (i.e., either a station joining or leaving thenetwork), produces a broadcast schedule for the new network The twin objectives
of adaptive algorithms are much faster execution (than an off-line algorithm thatcomputes a completely new schedule) and the production of a provably high-qualityschedule We note that many other network changes can be modeled by joining orleaving or a combination of the two (e.g., the moving of a station from one location
to another)
Trang 5Distributed Algorithms
Although centralized algorithms provide an excellent foundation, algorithms in which thecomputation is distributed among the nodes of the network are essential for use in prac-tice In these distributed algorithms, network nodes have only local information and par-ticipate in the computation by exchanging messages Distributed algorithms are important
in order to respond quickly to changes in network topology Further, the decentralizationresults in decreased vulnerability to node failures We distinguish between two kinds ofdistributed algorithms:
1 Token passing algorithms A token is passed around the network When a station
holds the token, it computes its portion of the algorithm [26, 2] There is no centralsite, although a limited amount of global information about the network may bepassed with the token Token passing algorithms, while distributing the computa-tion, execute the algorithm in an essentially sequential fashion
2 Fully distributed algorithms No global information is required (other than the
glob-al slot synchronization associated with TDMA), either in individuglob-al or centrglob-al sites.Rather, a station executes the algorithm itself after collecting information from sta-tions in its local vicinity Multiple stations can simultaneously run the algorithm, aslong as they are not geographically too close, and stations in nonlocal portions ofthe network can transmit normally even while other stations are joining or leavingthe network Fully distributed algorithms are essentially parallel, and are typicallyable to scale as the network expands
16.3 THE COMPLEXITY OF BROADCAST SCHEDULING
In this section the computational complexity of broadcast scheduling is studied
16.3.1 Computing Optimal Schedules
As noted in the previous section, determining a minimum nominal schedule in a wirelessmultihop network is equivalent to finding a distance-2 graph coloring that uses a mini-mum number of colors The NP-completeness of distance-2 graph coloring is well estab-lished [22, 6, 26, 5] The strongest of these [6] shows that distance-2 graph coloring re-mains NP-complete even if the question is whether or not four colors will suffice Theyutilize a reduction from standard graph coloring Thus:
Theorem 1 Given an arbitrary graph and an integer kⱖ 4, determining if there exists a
broadcast schedule of length not exceeding k, is NP-complete
By way of contrast, in [25] it is shown that when k is three, the problem can be solved in
polynomial time Given the NP-completeness of the basic problem, we are left with thepossible approaches of utilizing approximation algorithms to determine approximatelyminimal solutions, or considering the complexity on restricted classes of graphs Most ofthe remainder of this chapter is devoted to the former In regard to the latter, we note:
Trang 6Theorem 2 [25] Given a planar graph, determining if there exists a broadcast schedule
of length not exceeding seven, is NP-complete
16.3.2 What About Approximations?
From the NP-completeness results cited above, finding minimum length broadcastschedules is generally not possible Thus, it is necessarily the case that we focus on al-gorithms that produce schedules that are approximately minimal For such an approxi-mation algorithm, its approximation ratio ␣[8, 10], is the worst case ratio of the length
of a nominal schedule produced by the algorithm to the length of an optimal nominalschedule Such an algorithm is said to produce ␣-approximate solutions In the context
of adaptive algorithms, the analagous concept is that of a competitive ratio Here, the tio is the length of the current nominal schedule produced by the algorithm to an opti-mal off-line nominal schedule Additional information on such ratios and alternativesmay be found in [8, 10]
ra-What quality of approximation ratio might be possible for broadcast scheduling? Mostoften, the goal in designing approximation algorithms is to seek an approximation ratiothat does not exceed a fixed constant (i.e., a constant ratio approximation algorithm) Onewould hope, as in bin packing and geometric traveling salesperson [10], that approxima-tion ratios of two or less might be possible Unfortunately, this is not the case, not only forany fixed constant, but also for much larger ratios:
Theorem 3 [1] Unless NP = ZPP, broadcast scheduling of arbitrary graphs cannot be approximated to within O(n1/2–⑀) for any ⑀> 0
This result is tight since there is an algorithm (see the next section) having an
approxi-mation ratio that is O(n1/2)
16.4 CENTRALIZED ALGORITHMS
Since broadcast scheduling is NP-complete, in this section (and the next) we investigateapproximation algorithms that are alternatives to producing optimal schedules These al-gorithms are evaluated on the basis of their running times and approximation ratios
16.4.1 A Classification of Approximation Algorithms for
Broadcast Scheduling
An overview of approximation algorithms for broadcast scheduling is given in this tion Only a few particular algorithms are specifically described, and the reader is referred
sec-to [11, 25] for a more comprehensive treatment
In developing approximation methods for broadcast scheduling, the classic algorithm
P_Greedy takes a purely greedy approach That algorithm is an iterative method in which
a node is arbitrarily chosen from the as yet uncolored nodes and is greedily colored The
running time of P_Greedy is O(n2) on arbitrary graphs and O(n) on unit disk graphs
Trang 7The approximation ratio of P_Greedy is min(, n1/2) [26, 22] on arbitrary graphs, and 13
on unit disk graphs [18]
Aside from P_Greedy, a variety of centralized approximation algorithms have been
proposed for broadcast scheduling These algorithms can be placed into three general egories, using a classification adapted from [11]:
cat-1 Traditional algorithms that preorder the nodes according to a specified criterion,
and then color the nodes in a greedy fashion according to that ordering A
represen-tative of such methods is Static_min_deg_last [11] In this method, the nodes are
placed into descending order according to their degrees The nodes are then
greedi-ly colored according to that ordering The running time is O(n min(n, 2)) on
arbi-trary graphs and O(n log n + n) on unit disk graphs The algorithm is mate on arbitrary graphs, and 13-approximate on unit disk graphs [18]
-approxi-2 Geometric algorithms that involve projections of the network onto simpler ric objects, such as the line A representative of such methods is Linear_Projection
geomet-[11], in which the positions of the nodes are projected onto a line, and then an mal distance-2 coloring is computed for those projected points One effect of pro-jecting nodes onto a line is that the projections of nodes may now be within dis-tance-2, whereas the original nodes were not within distance-2 The algorithmselects a line for projection that minimizes the number of such “false” distance-2
opti-neighbors Linear_Projection runs in time O(n2) on arbitrary graphs There are noresults on the approximation ratio
3 Dynamic greedy methods that also color nodes in a greedy fashion, but in which the
order of the coloring is determined dynamically as the coloring proceeds A
repre-sentative of such methods is max_cont_color [16] This algorithm initially colors an
arbitrary node and then all of the one-hop neighbors of that node At each quent step, the algorithm chooses for coloring a node that is now most constrained
subse-by its distance-2 neighbors Results [16, 18] show that this algorithm has the bestsimulation performance among all existing broadcast scheduling algorithms A
careful implementation [18] yields a running time of O(nD) on arbitrary graphs and O(n) on unit disk graphs The algorithm is -approximate on arbitrary graphs, and13-approximate on unit disk graphs [18]
16.4.2 A Better Approximation Ratio
The best approximation ratio for arbitrary graphs of the methods cited above is
min(n1/2, ), which is also the ratio of the simplest of these algorithms, P_Greedy Below,
an algorithm of the “traditional greedy” variety is described that has an arguably stronger
ratio for most graphs The algorithm is similar to Static_min_deg_last but the nodes are
ordered in a “dynamic,” rather than static, fashion The term progressive is taken from [23]
Algorithm progressive_min_deg_last(G)
Labeler(G, n);
for j 씯 1 to n do
Trang 8let u be such that L(u) = j;
greedily color node u;
The function Labeler, which assigns a label between 1 and n to each node, is defined as
de-Theorem 4 For a planar graph, progressive_min_deg_last is 9-approximate
Proof: Consider the neighbors of an arbitrary node u, and suppose that k of those nodes
have labels smaller than L(u) Hence, there are up to – k neighbors of u with labels
larg-er than L(u)
It follows from the properties of planar graphs, and the specification of Labeler, that k
ⱕ 5 Each node with a label smaller than L(u) may have at most – 1 neighbors (not
in-cluding u) and hence those k nodes and their neighbors utilize at most k(– 1) + k = k
colors that may not be assigned to u
Now consider the up to – k nodes with labels larger than L(u) When u is colored,
none of these nodes are colored (recall that coloring is done in increasing order of labels)
However, the neighbors of these nodes may have lower labels than L(u) and the already signed colors of these nodes may not be assigned to u Since the minimum node degree in
as-a plas-anas-ar gras-aph is as-alwas-ays five or less, it follows from the specificas-ation of Las-abeler thas-at therecan be at most 4 · (– k) 2-hop neighbors of u that are already colored (i.e., four for each uncolored 1 – hop neighbor of u, not counting u itself)
Thus, u can be colored using no more than k+ 4 · (– k) + 1 colors, and with kⱕ 5,this is at most 9– 19 Since the minimum coloring uses at least + 1 colors, the approx-
For arbitrary graphs, the approximation ratio of progressive_min_deg_last depends on
the thickness of the graph That is, the minimum number of planar graphs into which thegraph may be partitioned Note that the algorithm does not compute the thickness (indeed,computing the thickness is NP-complete [21]), but rather only the bound depends on thatvalue Further, experimental results [25] establish that the thickness is generally much lessthan
Corollary 1 For an arbitrary graph of thickness , progressive_min_deg_last has an proximation ratio that is O()
Trang 9ap-The corollary follows from the prior proof by noting that in a graph of thickness , there is
at least one node of degree not exceeding 6– 1 [25]
The analysis in [14] establishes:
Theorem 5 For q-inductive graphs, progressive_min_deg_last has an approximation tio of 2q – 1
ra-Several classes of graphs, including graphs of bounded genus, are q-inductive See [12] for additional information on q-inductive graphs
In regard to running times, it is shown in [24]:
Theorem 6 For planar graphs, progressive_min_deg_last has a running time of O(n).For arbitrary graphs of thickness , progressive_min_deg_last has a running time of O(n)
16.4.3 A Better Ratio for Unit Disk Graphs
Among the methods cited above, several are 13-approximate when applied to unit diskgraphs These ratios follow from a general result of [18] on the performance of “greedy”algorithms The best approximation ratio relative to unit disk graphs belongs to the follow-ing algorithm (of the traditional greedy variety):
Algorithm Continuous_color(G):
Let u be an arbitrary node of G;
L 씯a list of the nodes of G sorted in increasing order of Euclidean distance to u; while L ⫽0/ do
Let v be the first node of L;
Greedily color v and remove v from L;
Theorem 7 For unit disk graphs, Continuous_color has an approximation ratio of seven
Key to the proof of this theorem is the following result that establishes that certainnodes in geographical proximity to one another must be distance-2 neighbors
Lemma 1 Given a node p, let b1and b2be two points on the boundary of p’s interference region (i.e., the circle of radius 2R centered at p) If |b1b2| ⱕ R, then any two nodes that areboth:
앫 in the unit subset of p, and
앫 within the area bounded by the line segments from p to b1and b2and by the
bound-ary of the interference region of p that runs between b1and b2(we refer to this area
as section pb1b2and show it as the shaded area in Figure 16.2)
are distance-2 neighbors
Trang 10The proof of Lemma 1 involves the extensive use of trigonometric functions to establishthe proximity of points, and may be found in [18].
Proof: From Continuous_color, when a node v is to be colored, the already colored
dis-tance-2 neighbors of v lie on at most half of v’s interference region The perimeter of that
half of an interference region can be partitioned into seven sections such that the distance
between the extreme perimeter points in each section does not exceed R Within each of
these sections, from Lemma 1, all of the nodes are distance-2 neighbors, hence each of the
nodes must receive a distinct color Let x be the largest number of nodes in any one of these sections Then x + 1 is a lower bound on the number of different colors assigned to v and its already colored neighbors Likewise, 7x + 1 is an upper bound on the number of different colors used by Continuous_color The theorem follows from the ratio of the up-
In regard to running time, it is easy to see that the running time of Continuous_color is O(n log n +nD) when applied to arbitrary graphs Since D may be as large as 2, we have:
Lemma 2 For arbitrary graphs, the running time of Continuous_color is O(n log n +
n2)
For unit disk graphs the running time is less Key to that result is the following, which
establishes that D and are linearly related:
Lemma 3 In unit disk graphs, Dⱕ 25
Proof: Consider the interference region of an arbitrary node s in a unit disk graph
Clear-ly, all distance-2 neighbors of s lie within the interference region of s Now, define an cycle to be a circle of radius R/2 (it could be centered anywhere) and note that all nodes lying within any given S-cycle are one-hop neighbors Thus, at most nodes lie within any
S-given S-cycle The lemma follows since the interference region of any node can be ered with 25 S-cycles as shown in Figure 16.3 (in that figure, S-cycles are shown both
Figure 16.2 The locations of b1and b2
Trang 11Corollary 2 For unit disk graphs, the running time of Continuous_color is O(n log n +
n)
16.4.4 An Adaptive Algorithm
An adaptive algorithm for broadcast scheduling is described in this section
16.4.4.1 The Effects of the Joining and Leaving of Nodes
In designing adaptive algorithms, it is important to understand how the schedule might beaffected by the joining or leaving of a node (the algorithm presented in this section focus-
es on these basic functionalities):
앫 The joining of a node may introduce conflicts among the node’s one-hop neighbors,thus turning a previously conflict-free schedule into a conflicting schedule Anadaptive broadcast scheduling algorithm must modify the existing schedule to re-move these conflicts
앫 The leaving of a node never introduces conflicts into the schedule, though theschedule may now be longer than necessary (though that is not easy to determine!)
Further, if A is the deleted node, that deletion may result in a neighbor of A with a
color larger than its number of distance-2 neighbors The use of such a color is
nev-er required
Algorithm_IR (Iteratively Remove) handles the joining of a node A as follows:
Algorithm IR_Station_Join(A)
increase the degree of the neighbors of A by 1;
update D(u) for each distance-2 neighbor u of A;
RECOLOR_LIST 씯 min_conflict_set(A);
Figure 16.3 Dⱕ 25
Trang 12uncolor the nodes (if any) in RECOLOR_LIST;
add A to RECOLOR_LIST;
while RECOLOR_LIST ⫽ 0/ do
v 씯 a node in RECOLOR_LIST with largest constraint number;
greedily color node v;
delete v from RECOLOR_LIST;
Within IR_Station_Join, function min_conflict_set(A) returns the minimum conflict set of node A, determined as follows: Let U be the set consisting of all one-hop neighbors of A that now conflict on a given color c (this occurs because the presence of A makes these nodes into 2-hop neighbors) Among the nodes in U, let u ibe a node with largest conflict
number Then U – u i is a c-conflict clique of A A minimum conflict set of A is the union
of its c-conflict cliques (one per color)
Algorithm_IR handles the leaving of a node as follows:
Algorithm IR_Station_Leave(A)
decrease the degree of the neighbors of A by 1;
update D(u) for each distance-2 neighbor u of A;
for each distance-2 neighbor u of A do
if COLOR(u) > D(u) + 1
greedily recolor u;
We note that Algorithm_IR as given in [19, 18] contains a third procedure, maintenance.
That procedure, which attempts a recoloring around a node of maximum color after each
joining and leaving, is essential for the excellent performance of Algorithm_IR as
mea-sured by simulations That additional procedure does not however affect (positively ornegatively) the approximation ratio, hence its omission from this chapter
To determine the approximation ratio of Algorithm_IR, we begin with the following lower
bound on the length of an optimal schedule:
Lemma 4 For unit disk graphs, an optimal schedule uses at least 1 + D/13 colors
Proof: Recall that D is the distance-2 degree of the graph Given a node v in a unit disk
graph, all of its distance-2 neighbors lie within the interference region of v Partition the
interference region of v into 13 sections of equal angle around v As in the proof of
Theo-rem 7, each section satisfies the conditions of Lemma 1, hence all of the distance-2 bors of v within a given section are within distance-2 of one another It follows that each
neigh-of the nodes in a given section must be assigned different colors Thus, any coloring must
use at least 1 + D( v)/13 colors in coloring v and its distance-2 neighbors Hence, an mal schedule for the graph uses at least 1 + D/13 colors 씲
opti-Theorem 8 For unit disk graphs, Algorithm_IR has an approximation ratio of 13