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If nodes have access to a global clock, such a centralized algorithm can be implemented in a distributed way, pro-vided that each node has global knowledge of network topology: in every

CHAPTER 24

Broadcasting in Radio Networks

ANDRZEJ PELC

Département d’Informatique, Université du Québec à Hull, Hull, Québec, Canada

24.1 INTRODUCTION

Broadcasting is one of the fundamental tasks in network communication Its goal is to transmit a message from one node of the network, called the source, to all other nodes Remote nodes get the source message via intermediate nodes, along paths in the net-work In this chapter we consider broadcasting in radio networks (Broadcasting in

oth-er types of networks, in particular point-to-point networks, has been extensively studied and is surveyed in [22, 26, 27].) A radio network is a collection of transmitter–receiver devices (referred to as nodes) Every node can reach a given subset of other nodes, de-pending on the power of its transmitter and on the topographic characteristics of the sur-rounding region

Two types of models of radio networks prevail in the literature The first one is a graph model Nodes of the graph represent nodes of the network and the existence of a directed

edge (u v) means that node v can be reached from u In this case, u is called a neighbor of

v If the power of all transmitters is the same, any node u can reach v, if and only if it can

be reached by v, i.e., the graph is symmetric The second type of model has a more

geo-metric flavor Each node of the radio network is represented by a point in the plane, and each of those points has a region associated with it, often a circle of given radius centered

at this point It is assumed that any node v of the network represented by a point within the

region associated to a given node u can be reached by the transmitter of u Again u is

called a neighbor of v in this case.

It is clear that the first type of model is more general than the second Given the geo-metric setting described above, it is easy to construct a graph on the set of points, in which

a directed edge from u to v exists if v is within the circle associated with u On the other

hand, it is not difficult to construct graphs that cannot be obtained in this way As for the applicability, each of the representations is appropriate in a different physical situation If the region in which the transmitter–receiver devices are situated is approximately flat and free of large obstacles, every transmitter reaches to the same distance in every direction, and consequently the geometric model with circular regions is appropriate, the radius of each circle depending on the power of the transmitter If, on the other hand, the topography

of the region is complicated by obstacles, either natural, such as mountains, or man-made,

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Copyright © 2002 John Wiley & Sons, Inc ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic)

such as buildings, then more complicated reachability graphs may be needed to model the network because obstacles obstruct radio waves in some directions

We assume that communication in a radio network proceeds in synchronous rounds In

every round every node acts either as a transmitter or as a receiver A node w acting as a

transmitter in a given round sends a message to all nodes within its reach (This means a

message is sent to all nodes to which there is an edge from w in the graph model, and all nodes within the region associated with w in the geometric model.) A node u acting as a

receiver in a given round gets a message if and only if exactly one of its neighbors trans-mits in this round If at least two neighbors v and v ⬘ of u transmit simultaneously in a

giv-en round, none of the messages is received by u in this round In this, case we say that a collision occurred at u.

One of the most important performance parameters of a broadcasting scheme is the to-tal time, i.e., the number of rounds used to inform all the nodes of the network In this chapter, we focus attention on this efficiency measure and show how to design fast broad-casting algorithms under various settings We also show lower bounds on time, which are intrinsic performance limitations of any broadcasting scheme

The previously mentioned characteristics of radio communication (multidirectional transmitting and inability to receive in the case of a collision) indicate the main difficulty

in designing a time-efficient broadcasting algorithm Although the fact that a node simul-taneously transmits a message to all nodes within its reach seems to speed up the broad-casting process, it is also the most important cause of slow-downs in many situations If

two nodes, u and v, have a common node w within their reach, they need to decide which

of them informs w; the other cannot transmit in this round This is a potential reason for

communication delay, as the waiting node may be the only one capable of transmitting the source message to some part of the network For this reason, scheduling a fast broadcast turns out to be a difficult task in many radio networks

This chapter is organized as follows In Section 24.2, we discuss several communica-tion scenarios most often studied in the literature In Seccommunica-tions 24.3 and 24.4, we present broadcasting algorithms and describe results concerning their running time, for the graph model and the geometric model, respectively In Section 24.5, we briefly mention some other variations of the problem: communication tasks related to but different from broad-casting and/or other communication models for radio networks Section 24.6 contains conclusions and open problems

24.2 COMMUNICATION SCENARIOS

In this section, we present various assumptions concerning the communication process in radio networks Their numerous combinations result in many communication scenarios used in the literature and significantly affecting the design of broadcasting algorithms and their efficiency

The first choice concerns the use of randomness in the communication process Ran-domized algorithms accomplish the broadcasting task with high probability but not al-ways On the other hand, we will see that they usually run much faster than deterministic

algorithms, require very little knowledge of the network, and are easy to implement in a distributed way, without any central monitor

The issue of centralized versus distributed control is crucial in all network communica-tion A centralized algorithm assumes the existence of a monitor having full knowledge of network topology and scheduling transmissions for all nodes If nodes have access to a global clock, such a centralized algorithm can be implemented in a distributed way, pro-vided that each node has global knowledge of network topology: in every round each node simply acts in the way it would be ordered to do so by the central monitor The situation becomes more complicated when each node has only limited knowledge of the network; for example, it knows only its close vicinity—the part of the network at a small distance from it, or, in the extreme case, only its own label With such limited information, central-ized algorithms requiring full knowledge of the network cannot be applied, and it becomes necessary to design distributed broadcasting schemes relying only on local knowledge available to nodes

The next feature that may significantly affect the communication process is that of adaptivity Nonadaptive algorithms have all transmissions scheduled ahead of time, prior

to the begining of broadcasting, whereas in adaptive algorithms, a node may schedule fu-ture transmissions on-line, depending on its previous history In a centralized algorithm with a known source of broadcasting, all transmissions can be scheduled off-line, before broadcasting begins In this case, adaptivity does not help, as nodes cannot learn any in-formation during broadcasting that could affect scheduling of future transmissions If the source is not known, adaptivity can help even when nodes have full knowledge of network topology The label of the source can be appended to the source message Upon receiving

it, a node can decide how to schedule retransmissions of the source message depending on its origin Adaptivity can help even more significantly in distributed broadcasting when nodes have only limited knowledge of network topology In this case, a node can receive, together with the source message, some precious information concerning the topology of remote parts of the network, which can help it to schedule retransmissions in a way that accelerates the rest of the broadcasting process

As mentioned above, a node can gain knowledge about the network from previously obtained messages There is, however, another potential way of learning useful informa-tion The availability of this method depends on what exactly happens during a collision,

i.e., when u acts as a receiver and two or more neighbors of u transmit simultaneously As previously mentioned, u does not get any of the messages in this case However, two sce-narios are possible Node u may either hear nothing (except for the background noise), or

it may receive interference noise different from any message received properly but also different from background noise These two scenarios are often referred to as the absence (or availability) of collision detection (cf., e.g., [5]) Which of the two scenarios occurs in

a particular situation may depend on technological characteristics of the transmitter/re-ceiver devices used by the nodes A discussion justifying both scenarios can be found in [5, 24] We will see that efficiency and even feasibility of a particular communication task are significantly influenced by the choice between these scenarios

Another issue concerning network communication in general, and broadcasting in ra-dio networks in particular, is that of fault tolerance Most algorithms are designed

assum-ing that the communication environment is fault-free However, this is not a realistic as-sumption, as the growing size and complexity of communication networks make them in-creasingly vulnerable to component failures A fault-tolerant broadcasting algorithm should guarantee that all fault-free nodes will be informed, under some assumptions on the number of faults (usually upper bounds on their number or the probability of their oc-curing), without knowing their location Faults can be of various types: omission (when a faulty node does not transmit messages) or Byzantine (when faulty nodes can corrupt messages arbitrarily), transient or permanent, and situated randomly or according to a worst-case distribution It is not surprising that there exist trade-offs between the degree of fault tolerance of a broadcasting algorithm (e.g., in terms of the maximum number of faults under which it still works correctly) and its speed The difficulty in designing good fault-tolerant broadcasting schemes consists in getting maximum efficiency while pre-serving a given degree of robustness with respect to faults

The assumptions about communication presented above can be applied to both models

of radio networks mentioned in the Section 24.1: to the graph model and to the geometric model In the rest of this chapter, we study algorithms and results concerning broadcasting

in both models, under communication scenarios resulting from various combinations of these assumptions

24.3 THE GRAPH MODEL

In this section, we discuss broadcasting in radio networks modeled by directed graphs

with a distinguished node s called the source We assume that there exists a directed path from s to all other nodes, otherwise broadcasting from s is impossible There are no other

restrictions on the topology of the graph Many authors, e.g [2, 5, 23, 30], use the model

of undirected connected graphs instead, which is a more restrictive assumption corre-sponding to the situation when the reachability graph is symmetric Hence, we will use the more general setting of directed graphs, pointing out cases when a given algorithm uses symmetry of the graph

Important parameters that influence the performance of broadcasting in radio networks are:

앫 The number n of nodes in the graph

앫 The maximum in-degree ⌬, i.e., the maximum number of neighbors of a node

앫 The eccentricity D of the source in the graph, i.e., the largest distance from the

source to any other node

The eccentricity D is a trivial lower bound on the time of any broadcasting algorithm.

24.3.1 Deterministic Algorithms

Early work on broadcasting in radio networks concentrated on deterministic algorithms One of the most natural questions is the following optimization problem in the context of

centralized broadcasting Given a graph and a designated source, find a broadcasting schedule using the shortest possible time It is shown in [8] that this problem is NP-hard

In the same paper, the authors propose a centralized broadcasting algorithm working in

time O(D⌬)

The first (centralized and deterministic) broadcasting algorithm whose running time is

slower than the lower bound D only by a factor polylogarithmic in the number of nodes is given in [10] Below, we present the main idea of this algorithm, which uses time O(D log2

n) The authors call their approach “wave expansion,” as the progress of broadcasting is

viewed as a wave front carrying the message: it starts at the source and advances farther away until all nodes are informed

At each round of the algorithm execution, we denote by X a subset of the set of in-formed nodes and by Y a subset of the set of uninin-formed nodes The front in this round is the set F of pairs (x, y), such that x 僆 X, y 僆 Y, and x is a neighbor of y The covered front

X F = {x 僆 X : (x, y) 僆 F, for some y 僆 Y} (or the uncovered front Y F = {y 僆 Y : (x, y) 僆 F, for some x 僆 X}) is the set of informed (or uninformed) nodes that belong to a couple in the front F We define the spokesmen set S 債 X Fas the set of those informed nodes in the

front that act as transmitters in the next round For any spokesmen set S, R S 債 Y Fdenotes

the set of nodes that receive the message correctly when exactly nodes from S transmit Hence R S consists of those nodes in Y F that have exactly one neighbor in S The main dif-ficulty of the algorithm is to choose S at each round in such a way that R Sis as large as possible and so that the choosing process is polynomial Clearly, inspection of all possible candidate sets is out of the question In [10], the following spokesmen election algorithm (SEA) is described

Algorithm SEA

Phase 1 Finding the size of the spokesmen set S.

For every 1 ⱕ i ⱕ n, let Si be the family of all i-element subsets of X F Let w ibe the

av-erage size of sets R S over all S 僆 S i Let k be the index i for which this average is maxi-mized This will be the size of the chosen set S.

Phase 2 Finding the elements of the spokesmen set S.

Elements of S are found one by one, in k iterations In the beginning S = 0/, N = X F The

mth iteration, for 1 ⱕ m ⱕ k, starts with S containing m – 1 nodes and N = X F \ S For each

x 僆 N, we define F S,x as the family of all sets of the form S 傼 {x} 傼 P, where P is any (k – m)-element subset of N \ {x} For any x 僆 N, let u S,x denote the average of |R T| over all

sets T in the family F S,x The element x 僆 N for which u S,xis the largest is transfered from

N to S, i.e., S := S 傼 {x} and N := N \ {x} The algorithm ends after k iterations, with the

It is shown in [10] that the averages w i in Phase 1 and u S,xin Phase 2 can be computed

in polynomial time, hence Algorithm SEA runs in polynomial time Moreover, the

spokes-men set S obtained by this algorithm satisfies the property |R S | > |Y F |/ln|X F| This means

that the choice of S guarantees that at least a fraction 1/ln|X F| of nodes that can potentially receive the source message for the first time are actually informed in a given round Algorithm SEA is used as a subroutine in the main algorithm called wave expansion

broadcast (WEB) This algorithm is structured according to layers in the graph, where the

ith layer L i is defined as the set of those nodes whose distance from the source is i

Clear-ly, the number of layers is D + 1, where D is the eccentricity of the source.

Algorithm WEB

The algorithm works in D phases called superwaves During the ith superwave, the front is formed from layers L i–1 and L i The ith superwave consists of a certain number of rounds called waves In the beginning of this superwave, X F = L i–1 and Y F = L i In consecutive

waves, the Algorithm SEA is applied to the current front, yielding the spokesmen set S Then all (newly informed) elements of the set R S are removed from Y F , and X Fremains

unchanged In the next wave, Algorithm SEA is applied to this new front Waves of the ith superwave are executed until Y F is exhausted This terminates the ith superwave The algo-rithm stops at the end of the Dth superwave 첸

The above-mentioned property of SEA guaranteeing that |R S | > |Y F |/ln|X F| permits us to

prove the following bound on the number t i of rounds (waves) in the ith superwave: t i<

ln|L i–1 |ln|L i| The first superwave clearly lasts one round Hence, the total running time of

Algorithm WEB is bounded by 1 + t2+ · · · + t D< 1 + ⌺D

i=2 ln|L i–1 |ln|L i| This number is

maximized when all layers are of equal size, thus giving running time O(D log2n) of

Al-gorithm WEB on an arbitrary graph

The order of magnitude O(D log2n) of the time of broadcasting cannot be improved in

general Indeed, in [2] the existence of a family of networks with D = 2 is proved, for

which any broadcast schedule requires time ⍀(log2 n) Hence, for these networks

Algo-rithm WEB from [10] is asymptotically optimal However, for networks whose source has large eccentricity, this is not always the case In [23] the authors show a centralized

deter-ministic algorithm that performs broadcasting in time O(D + log5n), and is thus

asymptot-ically optimal for networks with source eccentricity ⍀(log5n) The order of magnitude of

optimal broadcasting time for radio networks with D nonconstant but below ⍜(log5n)

re-mains an open problem

We now turn our attention to distributed broadcasting in the situation when nodes have only limited knowledge of the topology of the radio network We start with the most ex-treme scenario, when the knowledge of each node is restricted to its own label, and labels

are distinct integers between 1 and n (Note that all results remain valid when labels are distinct integers between 1 and M 僆 O(n).) Thus, the initial situation is that of complete

ignorance concerning the network: nodes do not know even their immediate neighborhood

and are unaware of global parameters such as the size n of the network or the eccentricity

D of the source On the other hand, the assumption about the existence of distinct labels is

necessary If the radio network is anonymous, it is clear that deterministic broadcasting cannot be done even in the 4-cycle The importance of designing efficient broadcasting al-gorithms that do not assume any knowledge that nodes may have about the rest of the net-work comes, e.g., from applications in mobile netnet-works whose topology and size may change over time

The lack of knowledge concerning the network raises the problem of precise definition

of the task of broadcasting and of its execution time In a centralized algorithm, time of broadcasting can be known in advance, and thus all nodes can be aware of the termination

of broadcasting as soon as it is completed A different situation occurs in the distributed setting with restricted knowledge Since even the size of the network is unknown, broad-cast can well be finished but no node need be aware of this fact Consequently, the follow-ing two communication tasks are distfollow-inguished in [12] In radio broadcastfollow-ing (RB) the goal is simply to communicate the source message to all nodes In acknowledged radio broadcasting (ARB) the goal is to achieve RB and inform the source about it This may be essential, e.g., when the source has several messages to disseminate, and none of the nodes are supposed to learn the next message until all nodes get the previous one

It is assumed that the algorithm starts in round 1 and the current round number is

indi-cated by the global clock An algorithm accomplishes RB in t rounds if all nodes know the source message after round t and no messages are sent after round t An algorithm accom-plishes ARB in t rounds, if it accomaccom-plishes RB in t rounds and if, after round t, the source

knows that all nodes know the source message

Distributed broadcasting in radio networks with unknown topology was first

investigat-ed in [5] Under this scenario, adaptivity of algorithms may be important, and hence it should be made precise if collision detection (as discussed previously) is or is not avail-able We first present results assuming the latter scenario This is the assumption made in [5] One of the main results of that paper is the lower bound ⍀(n) on deterministic broad-casting time, even for the restricted class of symmetric networks, and even when each node knows its immediate neighborhood The authors construct a class of symmetric net-works of bounded diameter, for which every deterministic broadcasting algorithm uses time ⍀(n) (Later it was shown in [28] that deterministic broadcasting time for this class

of networks is the same as for the class of arbitrary symmetric networks and is in fact

equal to n – 1.) A matching upper bound is established in [12]: the authors construct an al-gorithm accomplishing radio broadcasting in time O(n), for arbitrary symmetric networks,

under the most restrictive assumption that each node knows only its own label A subtle

point should be mentioned here The algorithm from [12] makes heavy use of spontaneous

transmissions: the ability of nodes that have not yet gotten the source message to transmit some control messages (The lower bound from [5] remains valid under this assumption.)

If this is precluded, linear time broadcasting is not possible any more: in [7] a class of

symmetric networks of diameter D is constructed for which any broadcasting requires

time ⍀(D log n) if spontaneous transmissions are forbidden In particular, for D linear in

n, this gives the lower bound of ⍀(n log n) on broadcasting time.

On the other hand, in [12] the authors prove the surprising result that acknowledged ra-dio broadcasting is impossible even in symmetric networks if collision detection is not available The idea of the proof is as follows Suppose that there exists an ARB-protocol P.

This protocol works in some time t for the graph that consists only of the source In [12] the authors construct a (large) symmetric graph G such that the protocol P, when run on

G, causes the source to obtain no messages in the first t rounds and does not inform some

nodes during these rounds Since during these first t rounds the source has the same input

as when P is run on the graph consisting only of the source, the protocol induces the

source to falsely conclude that ARB is accomplished on G after t rounds.

As opposed to the case of symmetric radio networks, for which an asymptotically opti-mal algorithm has been constructed, for arbitrary directed networks the problem is not completely solved We start by presenting lower bounds on broadcasting time in this

gen-eral case In [12], a family of directed graphs with source of eccentricity D is constructed,

for which any broadcasting algorithm requires time ⍀(D log n) This family is similar to

the one from [7], except that it is not symmetric and the lower bound holds even when spontaneous broadcasting is permitted In [15] this lower bound is sharpened to ⍀(n log

D) Although, in terms of the size of the network only, both results give the same bound

⍀(n log n), the result from [15] shows that linear time broadcasting is impossible even for

some networks with quite small eccentricity of the source

On the upper bound side, a series of recent papers establish broadcasting algorithms of increasing efficiency This series was initiated by Chlebus et al [12] who proposed the

following simple algorithm working in time O(n2) First suppose that all nodes know n.

Then broadcasting can be accomplished by the following procedure

Procedure Round-Robin (n)

The procedure works in n identical phases In each phase, all nodes that have the source message act as transmitters in turn: the node with label i is in the ith round of the phase 첸

If n is unknown, the above procedure should be applied many times using the following

doubling technique

Algorithm Simple-Sequencing

The algorithm works in phases numbered by positive integers In phase k, Procedure

Round-Robin (2k) is executed by all nodes with labels 1 to 2k, with the following modifi-cation: a node that obtained the source message and transmitted it in some round remains

In every round, at most one node acts as a transmitter, hence collisions are avoided It is easy to see that after phase log n all nodes get the source message, and the first log n

phases use a total of O(n2) rounds

In the same paper, a more sophisticated broadcasting algorithm is constructed, working

in time O(n11/6) This algorithm is based on the notion of a selective family of sets A fam-ily F of subsets of U is said to be k-selective for the set U iff for any X 債 U, |X| ⱕ k, there

is a set Y 僆 F satisfying |X 傽Y | = 1 The existence of a small sufficiently strongly

selec-tive family has to be proved, and this family is then used to construct appropriate sets of transmitters that avoid collisions

Subsequently, a series of faster broadcasting algorithms have been proposed, including

one constructive algorithm with execution time O(n3/2) [13], and three nonconstructive

al-gorithms based on probabilistic methods, with execution times O(n5/3log1/3n) [16], O(n3/2

兹lo苶g苶苶) [34], and O(n logn 2n) [14] All these algorithms, apart from the one in [13] which

uses finite geometries, are based on (a variation of) the concept of selective families It should be stressed that the nonconstructive algorithms are, in fact, deterministic Probabil-ity is used only to establish the existence of an appropriate selective family of sets, and given this family (which may, e.g., be found by all nodes off-line) the rest of the scheme is entirely deterministic Recall that broadcasting time is defined as the number of commu-nication rounds, and hence the time of local computations of nodes (used, e.g., to find an appropriate family of sets) is ignored

Below, we describe the idea of the fastest of these algorithms (and, in fact, the fastest currently known distributed deterministic broadcasting algorithm working for arbitrary

ra-dio networks with unknown topology): the O(n log2n) algorithm from [14] As before, we

assume that n is known This assumption can be removed by modifying the algorithm

us-ing the doublus-ing technique described in the context of Procedure Round-Robin

The following variation of the concept of a selective family is used in [14] An

m-ele-ment family S = {S0, S1, , S m–1 } of subsets of {1, , n} is called a w-selector, if it

sat-isfies the following property:

앫 For any two disjoint sets X, Y with w/2 ⱕ |X| ⱕ w and |Y| ⱕ w, there exists i for which |S i 傽 X| = 1 and S i 傽 Y = 0/

It is proved in [14] that for each n and each w ⱕ n/log n there exists an m-element

w-se-lector S = {S0, S1, , S m–1 } with m 僆 O(w log n).

The broadcasting algorithm is now defined as a sequence of transmission sets

specify-ing that nodes act as transmitters in a given round: if S is the transmission set correspond-ing to round t, nodes actcorrespond-ing as transmitters in round t are those that got the source message and whose labels are in S.

Let l = log (n/log n), w j= 2j , for each j = 1, , l, and S0= [{1}, {2}, , {n}] For j >

0, let S j be a w j -selector of size m j 僆 O(w j log n).

Algorithm DoBroadcast

The algorithm consists of stages, each of which consists of l + 1 僆 O(log n) rounds The transmission set in the jth round of stage s is defined as the set from S j with index s mod

It is proved in [14] that Algorithm DoBroadcast informs all nodes in time O(n log2n).

The above algorithm, as well as the previously mentionned broadcasting schemes pre-ceeding it, were designed to perform efficiently in arbitrary networks However, a few broadcasting algorithms have been also designed to work particularly fast for sparse net-works, i.e., those with small maximum degree ⌬ In [13] two such algorithms were

pro-posed: one working in time O冢n ·冢 冣·⌬ log n冣, and the other in time O[n⌬2log3

n/log( ⌬ log n)] However, they are both superlinear in n regardless of other parameters of

the network This has been further improved in [15]: the authors propose a broadcasting

algorithm working in time O(D⌬ log3 n), and hence sublinear in n for sparse networks

with small eccentricity of the source In particular, for D and ⌬ polylogarithmic in n, this

gives polylogarithmic broadcasting time, unlike any of the previous algorithms

The above results do not use the assumption about availability of collision detection Under the scenario with collision detection, broadcasting may be done faster in some cases For the class of strongly connected graphs, a radio broadcasting algorithm working

in time O(nD) is described in [12] This algorithm is thus faster than the other ones for small eccentricity D and large maximum degree ⌬ It is also observed how the collision

detection capability can be used to code messages Using noise and silence essentially as

log n

log ⌬

bits of the transmitted message, the authors show a simple scheme that broadcasts a

mes-sage of size l in time O(lD), hence they get an asymptotically optimal algorithm to broad-cast messages of size O(1), in arbitrary graphs.

The impact of collision detection is even more significant for the task of acknowledged radio broadcasting This problem is also investigated in [12] Although ARB is impossible

to achieve without this capability, availability of it permits us to perform this task rather

fast For symmetric graphs, the authors show an algorithm for ARB working in time O(n) for n-node graphs, and thus asymptotically optimal If the graph is nonsymmetric, in order

to make ARB possible, it must be at least strongly connected For such graphs, an

algo-rithm for acknowledged radio broadcasting working in time O(nD) is proposed in [12].

24.3.2 Randomized Algorithms

Randomized algorithms usually have the advantage of being simple and not relying on much knowledge available to nodes The first randomized broadcasting algorithm for arbi-trary radio networks was proposed in [5] Not only does it not assume any knowledge about the topology of the network and does not use collision detection, but (unlike for de-terministic broadcasting) nodes do not need to have distinct identities, and thus the algo-rithm works for anonymous networks as well The only knowledge available to nodes is the error bound ␧and the size n of the network (The result still holds if any polynomial upper bound on n is known instead of n itself.) The algorithm achieves broadcasting with

probability 1 – ␧and works in time O((D + log (n/␧))log n) (If the maximum degree ⌬ is additionally known to nodes, time can be improved to O((D + log (n/␧))log ⌬).)This per-formance closely matches known lower bounds: the previously mentioned lower bound

⍀(log2n) from [2] (the family of networks constructed in this paper does not admit any

faster broadcasting scheme, even randomized), and the lower bound ⍀(D log(n/D)) from [30] on randomized broadcasting time in any network Hence the algorithm from [5] is

as-ymptotically optimal for all D not very close to linear in n, e.g., for D 僆 O(n␣), for ␣< 1

For D linear in n, or, e.g., D 僆 ⍜(n/log n), a small gap remains between the performance

of the algorithm and the lower bounds

Below, we sketch the algorithm from [5] The algorithm is based on the following pro-cedure A set of nodes that already have the source message compete for a round in which exactly one of them transmits This can be achieved with positive probability in relatively few trials based on randomly decreasing the number of competitors At a call of the proce-dure, each node knows if it competes or not

Procedure Decay (k)

The procedure is executed in k rounds In each round, all competing nodes act as

transmit-ters and transmit the source message At the end of each round each competing node sets its variable coin randomly to 0 or 1 with probability 1/2 Those nodes with value 0 of coin

It is proved in [5] that if the number of competing nodes at the call of Procedure Decay

(k) is d then, for k ⱖ 2log d, the probability that there exists a round in the execution of