Leader Election in Unreliable Radio Networks
I C A L P
Leader Election in Unreliable Radio Networks
Mohsen Ghaffari 0
Calvin Newport 0
0 MIT , Cambridge, MA , USA Georgetown University , Washington, DC , USA
The dual graph model describes a radio network that contains both reliable and unreliable links. In recent years, this model has received significant attention by the distributed algorithms community [12, 4, 8, 10, 9, 16, 1, 14]. Due to results in [10], it is known that leader election plays a key role in enabling efficient computation in this difficult setting: a leader can synchronize the network in such a manner that most problems can be subsequently solved in time similar to the classical radio network model that lacks unreliable links. The feasibility of efficient leader election in the dual graph model, however, was left as an important open question. In this paper, we answer this question. In more detail, we prove new upper and lower bound results that characterize the complexity of leader election in this setting. By doing so, we reveal a surprising dichotomy: (1) under the assumption that the network size n is in the range 1 to N , where N is a large upper bound on the maximum possible network size (e.g., the ID space), leader election is fundamentally hard, requiring ?? (?N ) rounds to solve in the worstcase; (2) under the assumption that n is in the range 2 to N , however, the problem can be solved in only O?(D) rounds, for network diameter D, matching the lower bound for leader election in the standard radio network model (within log factors) [7].
and phrases Radio Networks; Leader Election; Unreliability; Randomized Algorithms

1998 ACM Subject Classification C.2.1 Wireless Communication
Introduction
The dual graph model is a generalization of the classical radio network model [5, 3] that
augments a connected network topology of reliable links with an additional set of unreliable
links. These latter links can come and go on a roundbyround basis as determined links.
These latter links can come and go on a roundbyround basis as determined by an unknown
adversarial process. The model is motivated by the observation that in real radio networks
link quality can change unpredictably due to many different factors (e.g., [17]). Upper bounds
proved in this general model are therefore more robust in real world deployments, while lower
bounds help formalize the complexity induced by link unreliability.
The dual graph model was introduced in [11] and has since been wellstudied in the
distributed algorithms community [12, 4, 8, 10, 9, 16, 1, 14]. In this paper, we present new
upper and lower bounds for the problem of leader election in this setting. As we elaborate
below, by doing so we resolve an important open question from [10] and provide a primitive
that enables efficient solutions to many problems in this difficult model.
The Power of Rerandomization. In more detail, the dual graph model describes a radio
network consisting of n processes connected by two network topology graphs G = (V, E)
ET
A
CS
and G0 = (V, E0), where the nodes in V correspond to the n processes, E describes reliable
links, and E0 describes unreliable links. We assume G is connected and use D to describe
its diameter. We consider randomized algorithms executing in synchronous rounds. In each
round, the network topology is fixed to include all edges in E and some subset of the edges in
E0, where the latter decision is made by an unknown adversary. Once the topology is fixed,
communication follows the standard multihop multiple access channel rules (see Section 2).
An important assumption in the dual graph model is the strength of the adversary that
selects the unreliable links to include in each round. Early work on this model assumed the
adversary was strongly adaptive (i.e., it knows the processes? random bits), which led, not
surprisingly, to pessimistic lower bounds [11, 12, 4, 8, 16]. In response to these negative
results, we argued in [10] that it makes more sense to assume an oblivious adversary, as this
still allows for unpredictable link behavior, but it removes the unrealistic assumption that
link quality can somehow adapt to the processes? private bits.
To validate the claim that the oblivious dual graph model is tractable, we demonstrated
that if you can synchronize a group of processes with new random bits generated after
the execution begins (and therefore unknown to the adversary), these processes can then
efficiently communicate. The highlevel idea is that transmitters use the shared bits to
randomly permute the broadcast probabilities used by the standard Decay contention
management routine [3] in a manner that is hard for the adversary to predict and therefore
is hard for the adversary to impede with its selection of unreliable links.
The key to enabling efficient communication in the dual graph model, therefore, is to
efficiently spread shared random bits among the processes in the network. In [10], we took a
first step toward this goal by showing that if you are provided a single leader in the network,
this leader can generate the bits itself, and then efficiently disseminate them to all processes
in O(D log n + log2(n)) rounds (with high probability in n)?nearly matching the lower
bound of ?(D log (n/D) + log2 n) for onetoall broadcast in the standard radio network
model [2, 13, 15]. The highlevel idea is that as the new bits spread, the processes that learn
them are synchronized and can therefore run the efficient Decay variation to efficiently spread
them to the next hop.
This strategy reduces the challenge of efficient communication in the dual graph model
to electing a single leader. Whether or not it is possible to efficiently solve leader election in
this setting, however, was left as key open question in [10]. In this paper, we answer it.
Our Results. We prove lower and upper bounds that characterize the complexity of leader
election in the dual graph model with an oblivious adversary. In doing so, we reveal a
surprising dichotomy: (1) under the assumption that the network size n is in the range 1
to N , where N is a large upper bound on the maximum possible network size (e.g., the
ID space), leader election is fundamentally hard, requiring ?? (?N ) rounds to solve in the
worstcase; (2) under the assumption that n is in the range 2 to N , however, the problem can
be solved in only O?(D) rounds, matching the lower bound for leader election in the standard
radio network model (within log factors) [7].
Put another way: the promise that you are not alone in the world renders the problem of
leader election vastly more tractable. We are unaware of other similar settings for which the
split between n ? 1 and n ? 2 holds such significance. In the standard radio network model,
for example, the best known leader election algorithms (which nearly match the relevant
lower bounds) work equally well for all network sizes [7].
Our lower bound focuses on the easier problem of loneliness detection, in which the goal is
for each process in a network to correctly determine whether or not it is alone. We construct
a network of maximum size N that consists of ? ?N outsider processes each connected to an
insider process in G. We then connect the insider processes to satisfy the model requirement
that G is connected. The graph G0 describing unreliable links is fully connected. We then
demonstrate how an oblivious adversary can construct a schedule for adding and removing
G0 edges between insider and outsider processes that delays some outsider processes from
receiving messages for a long period. During this period, these outsiders cannot distinguish
this execution from those in networks where they are alone. The challenge in constructing this
schedule is that the insider processes might coordinate after the execution begins, creating
correlated behavior that is hard to predict when constructing our G0 schedule. To sidestep
this problem we connect each insider process to a line of length ? ?N such that all these
lines connect at their far ends. This allows us to keep insider behavior independent for the
?(?N ) rounds that pass before any two insider processes can learn common information.
Our upper bound, by contrast, leverages a novel strategy that necessitates that n ? 2. In
particular, the algorithm works in phases. In each phase, it attempts to elect a single leader
by having each process flip a weighted coin and become a candidate leader if it comes up
heads. For each weight tried, the processes run a series of experiments to detect whether
they succeeded in electing a single leader.
At a highlevel, in these experiments, each candidate leader begins to grow a territory of
processes that it synchronizes with new random bits. These bits allow them to communicate
efficiently using the rerandomization strategy from [10] described above. For territory A to
detect if it neighbors some different territory B (indicating the election failed), processes
within each territory watch the success rate of their internal communication. If the collision
rate is higher than expected, this is evidence that another territory, using different random
bits, exists nearby. This strategy requires that each candidate leader has at least one neighbor
to recruit for this testing?generating our requirement that n ? 2. We emphasize that this
cooperative approach to interference detection is new in the dual graph literature.
Related Work. A close predecessor to the dual graph model of unreliable radio
communication was introduced by Clementi et al. [6]. The model in its current form was identified
by Kuhn et al. [11]. It has since been wellstudied by the distributed algorithms
community [12, 4, 8, 10, 9, 16, 1, 14]. Under the pessimistic assumption of a strongly adaptive
adversary controlling the unreliable links, results are known for global broadcast [11, 12],
local broadcast [8], and graph structuring algorithms [4, 16]. In [10], we suggested the
assumption of an oblivious adversary and produced near optimal bounds for global broadcast.
We also proved that local broadcast could not be solved in o(D) rounds?establishing the
O?(D) round strategy for enabling efficient communication presented in this paper as optimal
with respect to its dependence on D. Recent work by Lynch and Newport [14] explored a
different route to efficient communication in this setting by describing how to enable efficient
communication in o(D) time under the additional assumption that G and G0 satisfy strong
geographic constraints. The algorithm described here makes no assumption on the network
topologies and the graphs used in our lower bound do not satisfy the constraints of [14].
2
Model and Problem Statement
Model. We study the dual graph model of unreliable networks with an oblivious adversary.
In more detail, we model a network consisting of n processes connected by a topology
consisting of both reliable and unreliable links. We describe the reliable links with a
connected graph G = (V, E), and the unreliable links with a graph G0 = (V, E0). We use
D to describe the diameter of G. An algorithm in our setting assigns one computational
process to each node in V , and we assume processes are provided no advance information
about G or G0. In the following, we will use the terms node and process interchangeably.
Executions are divided into synchronous rounds. At the beginning of an execution, an
oblivious adversary determines which edges from G0 to include in the network topology
for all upcoming rounds (e.g., it generates an infinite sequence of graphs, G1, G2, ..., where
each Gi = (V, Ei), where Ei includes every edge in E and a subset of edges from E0). The
adversary does not have access to the nodes? private random bits in generating this sequence.
In each round r, we call the edges in Gr the active edges for the round. At the beginning of
the round, each node decides to listen or transmit. A node u receives a message m in r if
and only if: (a) u decides to listen; and (b) exactly one neighbor of u in the active edges
(i.e., neighbor in Gr) transmits in r and it transmits m. In this case, u learns whether the
message came from a reliable neighbor. Put another way, if two or more neighbors transmit,
all messages are lost at u due to collision. We assume that collisions cannot be distinguished
from silence (i.e., no collision detection).
Problem Statements. The problem of leader election requires each node v to output a
binary value bv such that, with high probability, exactly one node outputs 1. This node is
called the leader. This problem specification suffices for our lower bound. For our algorithmic
result, we strengthen the problem by also requiring that each node other than the leader
receive at least one message from the leader. The problem of loneliness detection requires
that each node v outputs a binary value b0v such that, with high probability, if v is not alone
in the network, b0v = 0, and if v is alone, then b0v = 1.
3
The Loneliness Detection Lower Bound
Here, we show that detecting whether a node is alone in the network or not requires ?? (?N )
rounds, where N is the best known upper bound on n, e.g., the size of the ID space. That
is, if all that nodes know about the current number n of nodes in the network is that
n ? [1, N ], then leader election in the dual graph model requires much more than its analogue
in the reliable model, where O?(D) round suffices. Note that in the reliable model, detecting
loneliness can be solved easily in O(log2 N ) rounds, with high probability, using the classical
Decay protocol. In contrast, as we show here, the same problem in the unreliable model
requires ?? (?N ) rounds. Formally, we prove the following result:
I Theorem 1. Loneliness detection in the obliviousadversary dual graph model requires
?? (?n) rounds. More specifically, there is a dual graph radio network (G, G0) with at most
N nodes in which, for any execution of any distributed algorithm A, some nodes cannot
distinguish this execution from one in which they are alone.
We first start with describing the structure of the dual graph radio network (G, G0)
that proves this lower bound. We then provide intuitive discussions about the adversary?s
oblivious strategy for each given algorithm A. At the end, we provide the formal attack
strategy and prove that it ensures that some of the nodes in the network experience an
execution identical to one in the setting where they are alone in the network.
The Lower Bound Network. We first explain the reliable graph G. We have b?N c paths,
each consisting of b N??N1 c ? ?N nodes. We call these support lines, and we label them 1, 2,
..., b?N c. For each line, we call one endpoint ?right? and the other ?left?. We connect all
the right endpoints to one node t. For each line i, we call the first node ai and the second
node bi. The unreliable graph G0 is then defined as a complete bipartite graph between
A = {aii ? [b?N c]} and B = {bii ? [b?N c]}, except for the first path edges ai?bi, for each
path i, which are reliable and in G. For each i, we refer to ai and bi as pairs of each other.
Figure 1 shows an illustration of this network.
3.1
Intuitive Discussions about the Adversary?s Strategy
The adversary?s plan is to ensure that by round O(?N / log N ), with high probability, there
are still at least half of the nodes of A which have not received any message from any other
node. Thus, for each of them, the execution is identical to one in which the node is alone.
The Adversarial Strategy ? Take 1. We start with an insufficient but instructive strategy.
For the moment, think of bi nodes as detached from the rest of their respective lines and let
us focus on just the bipartite graph A ? B. Let us call a node v ? A ? B compromised if it
has received at least one message from some other node of A ? B or some node in A ? B has
received a message of v. Otherwise, we call v uncompromised.
The adversary uses its knowledge of the algorithm A to calculate the expected number of
nodes in A ? B that will transmit in each given round. If this expectation is greater than
?(log N ), then the adversary will aim for creating a collision for all nodes, by making all
edges of G0 active. The idea is that, if we have independence between the random decision
of nodes in A ? B transmitting in this round, then due to this high expectation, with high
probability, two or more nodes transmit. That would mean all nodes in A ? B experience
collisions and none of them receives a message. Thus, in these highexpectation rounds, no
nodes gets compromised, with high probability. On the other hand, if the expected number
of transmitting nodes in A ? B is at most O(log N ), then the adversary will work in the other
extreme, and it does not activate any of the unreliable G0 edges. Assuming independence
between different nodes in A ? B transmitting, with high probability, at most O(log N ) nodes
transmit. This means, with high probability, there are at most O(log N ) many ai?bi pairs for
which some message exchange happens between ai and bi. Hence, in these lowexpectation
rounds, at most O(log N ) nodes get compromised. Therefore, per round, O(log N ) nodes
get compromised, with high probability. That means, even after O(?N / log N ) rounds?for
small enough constants?at least half of A remain uncompromised.
The Problem in the Take 1 Strategy. The key difficulty with the above strategy is that,
whether different nodes of A ? B transmit or not in a given round are not independent.
This creates a challenge for concentration arguments around the expectation. For instance,
there can be dependency between nodes that are compromised as there might be some
communication that happened between them. Furthermore, note that the adversary cannot
know which nodes are compromised in each round, as that depend on the random decisions
of nodes with respect to transmitting or not.
The Adversarial Strategy ? Take 2. The adversary will base its decision on the expected
number Xr of uncompromised nodes that transmit, in each given round. Notice that the
adversary does not know which nodes are compromised. Hence, in this expectation, whether
a node is compromised or not is also a random event. Note that Xr = Pi xrai + xbri where, for
instance, xrai is an indicator random variable for the event of ai remaining uncompromised
up to round r and then transmitting in round r.
The reason that we can focus the attack on only uncompromised nodes is that the
behavior of compromised nodes will not impact the new nodes that get compromised, with
high probability. Particularly, if Xr = ?(log N ) and we have independence, then with high
probability, two or more uncompromised nodes transmit. Thus activating all G0 edges makes
every node in A ? B experience collision. On the other hand, if Xr = O(log N ) and we
have independence, then with high probability, at most O(log N ) uncompromised nodes
transmit. Hence, if no G0edge is activated, only nodes which their pair transmitted can be
compromised. Thus, per round, at most O(log N ) new nodes get compromised. Hence, even
after O(?N / log N ) rounds, at least half of the nodes in A remain uncompromised.
The Problem in the Take 2 Strategy. It remains to explain how the adversary computes
the probabilities of the indicator random variables xrai and xbrj . Furthermore, the events of
these variables are not precisely independent, because there is dependency between different
nodes becoming compromised. For instance, if a node transmits alone in a highexpectation
round in A ? B, all nodes are compromised. Also, if a node gets compromised in a
lowexpectation round, so does its pair. We need to explain that we have sufficient independence
between the random variables to use concentration arguments.
3.2
The Formal Adversarial Strategy and its Analysis
We now describe our exact adversarial strategy for determining the active edges of G0 per
round. We describe this process algorithmically. We later prove that for any loneliness
detection algorithm A, under this oblivious adversarial strategy, at least half of the Anodes
remain uncompromised. the schedule produced by this process works well.
In more detail, let T = c?n/ log n, for a sufficiently small constant c > 0, to be fixed
later. As mentioned before, the adversary uses only two choices regarding which edges of
G0 are active: The first option is include no extra G0edges. We call these rounds sparse.
The second option is to include all G0 edges. We call these rounds dense. We can, therefore,
describe the adversary?s behavior with a binary array S of size T , where S[r] = 0 indicates
round r will be sparse and S[r] = 1 indicates it will be dense.
Oblivious Link Schedule Generation for Duration T
/ / Initialization
? is a positive constant that we use in defining our sparse/dense threshold
A is binary array of size T initialized to 0 in all positions
N (u), for u ? A ? B, is a real value initialized to 1
Pr(u), for u ? A ? B and r ? [T ], is the probability node u transmits in r if it is still
uncompromised at r (this depends on the definition of A)
/ / Assigning Values to Array S
for r = 1 to T :
x ? Pu?A?B N (u) ? Pr(u)
if x ? ? log n then
S[r] ? 1 / / round r is dense (so we assume no compromises)
else
S[r] ? 0 / / round r is sparse
foreach u ? A ? B
N (u) ? N (u) ? [(1 ? Pr(u)) ? (1 ? Pr(v)) + Pr(u) ? Pr(v)]
/ / updating the probability that u is compromised,
where v is the pair of u in the respective path.
We must first verify that the above schedule can be generated by the oblivious adversary
prior to the execution. The only nontrivial information in this effort is how the adversary
generates Pr(u) in advance, which we discuss next.
Computing Transmission Probabilities of Uncompromised Nodes in A ? B. For u ? A,
this follows directly from the definition of the algorithm: given the algorithm and a single
node u, it is straightforward to calculate the probability that the algorithm broadcasts in
any round r, given that node u is isolated.
For u ? B, where u = bi the adversary computes Pr(u) by recursively simulating the
following modified graph under a fixed pattern of delivering messages. The graph is simply
the line starting from bi and ending with the left endpoint of the path. The simulation is
done (recursively, round by round) under the setting that whenever S[r] = 1, node bi does not
receive any message even if it is listening and its left neighbor transmits, whenever S[r] = 0,
the reception of bi is the same as it when this detached path is running alone. Simulating
this detached line under this fixed reception strategy to bi allows us to compute the Pr(bi)
for each round r ? T . This is because, if bi is not compromised, the only difference between
this simulated ditached line and the line in our network would be the connection to node t
at the far right end. However, that connection cannot affect the behavior of bi in the first
T = c?N / log N rounds, as it is ?N hops away from bi and due to the synchronous rounds,
the causal dependency propagates with a speed of one hop per round at most.
Analysis. Next we analyze the performance of any algorithm A when executed in our
network with the G0 edges following the adversarial strategy captured by the array S as
generated above.
I Lemma 2. With high probability, in each of the first T = c?N / log N rounds, at most
O(?N ) new nodes get compromised.
Proof. By induction on r, we show that with high probability, (1) in each dense round, all
nodes in A ? B experience collision and thus no new node gets compromised, and (2) in each
sparse round, at most O(log N ) nodes get compromised. This directly proves the lemma.
The claim is trivially correct for round r = 1 because in this round, all nodes are
uncompromised and N (u) = 1 for all of them. So, the parameter x in the adversary?s
calculations is the correct expectation of the number of transmitting nodes in A ? B in this
round. Since we have independence between these events in the first round, Chernoff bound
gives that with high probability, dense rounds are in collision for all nodes in A ? B, and
sparse rounds compromise at most O(log N ) nodes.
Next, we prove the inductive step. Suppose that the claim is true for all rounds r0 ? r ? 1.
Note that the claim implies that, modulo an inversepolynomially small term (for the negligible
change of the claim being broken), in round r, the variable N (u) in the pseudocode correctly
captures the probability of node u remaining uncompromised. This is because, the calculation
ignores the dense rounds, in which we know the u would not get compromised there, and it
takes the transmission probability of the node and its pair into account in sparse rounds, as
only the transmission of exactly one of these two nodes can compromise u.
Define Nr be random variable indicating the set of uncompromised nodes in r. Also,
define the random variable Xr to be the number of uncompromised nodes in A ? B that
transmit in r. Let Yr(u) be the indicator variable that is 1 if and only if u is uncompromised
and transmits in r. We clearly have E[Xr] = Pu?A?B Yr(u) = Pu?A?B Pr(u) ? Pr[u ? Nr].
In the following, we use the notation xr to refer to the value of x in the pseudocode run
by our oblivious scheduler algorithm in round r. By the argument above, and thanks to the
hypothesis of induction, we know that the computations of N (u) for the probability of not
being compromised was correct up to round r ? 1.
I Claim 3. By way of induction, assume that in every dense round r0 ? r ? 1, all nodes of
A ? B experience collision and thus, none of them receives a message. It follows that for each
v, u ? A ? B where (u, v) ?/ G, and each r ? T , the events Yr(u) and Yr(v) are independent.
Claim Proof. Given the inductive assumption that collisions are enforced in dense rounds
r0 ? r ? 1, for each u ? A ? B, we have the following: If it is in A, until u is compromised,
it sees an execution identical to those in which it is alone. Also, if it is in B, until u is
compromised, it experiences an execution identical to those in which the related detached
line is under the reception adversary discussed above when computing Pr(u). Particularly,
u has received no information from outside its support line, and therefore its transmission
decisions are independent of nodes in other lines.
Hence, the only way for a node to be compromised for the first time is if the node u or
its pair transmit during a sparse round, in the past. However, these transmissions (prior to
being compromised) are independent events between nodes of different pairs (i.e., lines). J
By the independence provided by the above claim, we can prove that the inductive
assumption of having collisions in dense rounds is correct also for the next round r. This
is because, under the inductive assumption that dense rounds prior to round r were all in
collisions, the calculations N (u) in the pseudocode correctly captures the probability of a
node u being uncompromised up to and including round r. This is equal to the probability
of remaining uncompromised up to round r ? 1, which we know was correctly computed
by indcution, and then having either none of u and its pair v transmit, or both transmit.
Now, given that these events Yr(u) for different u ? A ? B are independent, except for
each depending on one other event in the same set, we can finish the argument as follows:
we use the standard extensions of Chernoff bound to settings with bounded (constant)
dependencygraph degree and conclude the following: if the expectation E[Xr] is ?(log N ),
then with high probability, Xr ? 2. Hence, with high probability, the dense round r also
is in collision. Similarly, if the expectation E[Xr] is O(log N ), then with high probability
Xr ? O(log N ) and thus, in round r, at most O(log N ) new nodes get compromised. This
finishes the proof of the inductive proof and thus also the proof of the lemma. J
4
The Algorithm
Here, we explain a distributed algorithm that computes a leader in O?(D) rounds, in the dual
graph model with an oblivious adversary, when nodes are given the promise that n ? 2.
I Theorem 4. There is a distributed algorithm that in any dual graph network (G, G0) with
an oblivious adversary, and where n ? 2 and G has diameter D, elects a leader in at most
O(D log4 N ) rounds, with probability at least 1 ? 1/N c, for any desired constant c ? 2.
4.1
Outline
Suppose that all nodes know an upper bound D on the diameter of the reliable graph G, that
is within a constant factor of the diameter. We will explain how to remove this assumption
using the standard doubling technique, at the end of this section.
We will try O(log N ) estimates of the form ? = 2i for n, where i ? [1, log N ]. For each of
these, we run O(log N ) experiments, where in each experiment each node is marked with
probability 1/? = 2?i. Each marked node is considered as a candidate for leadership and an
experiment is successful if and only if there is exactly one candidate. It is easy to see that,
with high probability, there is at least one experiment in which there is exactly one marked
node. The challenge is to identify experiments with exactly one candidate.
We run the O(log2 N ) experiments in parallel. Particularly, we divide time into epochs
of length O(log2 N ) and the jth round of each epoch belongs to the jth experiment. Let us
focus on one experiment. The problem we need to solve in a single experiment is to detect
whether in this experiment, there is exactly one candidate or not, in O?(D) rounds.
For now, we focus on one fixed arbitrary experiment. In this experiment, we perform
a special broadcast procedure starting from each of the candidates. At any point in time,
we call the set of nodes that receive (only) the identifier of a candidate v the territory of v.
Nodes that have received two or more candidate identifiers in this experiment are called in
conflict, and nodes that have not received any candidate identifier in this experiment are
called leaderless. We define the conflict and leaderless territories accordingly.
We first explain in Section 4.2 the simple broadcast scheme that starts from each candidate
and tries to grow its territory. The guarantee will be that, if the candidate is alone, the
message reaches all nodes in O(D log2 N ) rounds. In Section 4.3, we then explain how to
detect whether the broadcast has already reached all nodes or not. The guarantee will be
that an experiment terminates if and only if there is exactly one candidate, and in that case,
it terminates in ?(D log2 N ) rounds. Hence, once an experiment terminates, if we wait for a
(sufficiently large) constant factor more time, all those experiment that can terminate do so.
At the end, among these successful experiments, the one with the smallest experimentnumber
wins. That means the single candidate of that experiment is elected as the leader.
4.2
Growing Territories
Each candidate v initiates a special singlemessage broadcast procedure. Included in this
message is the identifier of the candidate v, as well as O(log2 N log log N ) bits of randomness.
The broadcast procedure then proceeds in D phases, where in each phase, we perform the
following procedure:
Rerandomized Decay Protocol. Each phase consists of L = ? log2 N consecutive rounds,
for a large enough constant ?. In each round of the phase, all nodes that have received the
message of a candidate by the start of the phase use the randomness shared in the message
to pick a shared random number j ?U [1, 2 log N ]. The details of this part will be discussed
shortly. Then, each node decides to transmit with probability 2?j, using its own private
randomness, and remains silent otherwise. At the end of the phase, each node only keeps the
messages that it received from its reliable neighbors, or those that it had before. Note that
the former uses the assumption that each node is able to distinguish successful receptions
from reliable and unreliable neighbors. This will later simplify the process of removing the
assumption of knowing an upper bound on the diameter D.
We note that the above is an adaptation of the classical Decay procedure[3], with the key
change being in the publicly random choice of the transmission probability. This property
helps crucially in defeating the oblivious adversary.
We show that if we have a single candidate, after O(D log2 N ) rounds, its message
reaches all nodes, w.h.p. This follows immediately from the following simple lemma, which
particularly implies that if there is only one candidate, its broadcast grows at a speed of one
Ghop per round, thus reaching all nodes in O(D log2 N ) rounds.
I Lemma 5. Suppose that node w has at least one Gneighbor that had received the message
of a candidate v by the start of the phase. Furthermore, assume that no G0neighbor of w
has received a message from any candidate other than v. Then, regardless of the behavior of
the adversary, w.h.p., node w receives the message of candidate v during this phase.
Proof. Consider each round r of the phase and let Gr be the graph fixed by the adversary
for round r. Let dr be the number of active neighbors of w in round r that have received
the message of candidate v, by the start of the phase. Notice that all potentiallyactive
neighbors of w that transmit have received the message of candidate v and act according to
the randomness bits received in the related message. Particularly, this shared randomness,
they pick a common j ?U [1, 2 log N ] for this round r. With probability 2 lo1g N , the random
value j ?U [1, 2 log N ] will be such that j = dlog dre. If that happens, the probability that
exactly one active neighbor of w transmit in round r is at least d2rj (1 ? 21j )dr?1 ? 1/5. Thus,
per round, the probability that node w receives the message of candidate v from one of its
1
active neighbors is at least 10 log N . By a simple Chernoff bound, this means that during the
? log2 N rounds of the phase, node w receives the message of v with high probability. J
4.3
Detecting Loneliness for Candidates
Note that if in an experiment, we have two or more candidates, their territories will grow
and eventually reach each other. At that point, the territories can obstruct each other from
growing and covering all nodes. The key task will be to determine if the broadcast of a
candidate has reached all nodes or not.
We call a node u in the territory of candidate v a boundary node of this territory if u has
at least one Gneighbor outside the territory of v. Notice that, if the broadcast of v has not
reached all nodes, then there is at least one boundary node in the territory of node v. On
the other hand, if v was the only candidate, once the broadcast reaches all nodes, there will
not be any boundary nodes.
The boundary nodes have the responsibility for detecting whether the broadcast has
reached all nodes or not. They will deliver their indications to the candidate, which then
decides whether the experiment was successful or not. The existence of boundary nodes will
be taken as an indication that the territory does not cover all nodes. Let us first explain the
key idea for detecting boundary nodes.
Boundary Detection. As the lower bound from the previous section suggests, even the
rudimentary problem of each node receiving a single message from any nearby node can be
hard. Thus we cannot rely on the boundary nodes receiving a message from a node outside
their territory. We turn the problem around. Instead of insisting on actually receiving a
message from outside the territory, we take lack of a frequent receptions from the node?s own
territory as an indication for collisions, implying that the node is potentially a boundary
node. More concretely, the following two are indications for the bound w being boundary
and thus that the related broadcast has not reached all nodes: (1) if w receives a message
from the territory of a different leader, or more cruicially (2) if w does not receive messages
from its own territory frequently enough. Let us make this formal.
The boundary detection algorithm consists of a single phase?i.e., L = ? log2 N rounds.
Each node that has received a message from exactly one candidate v performs one phase
of its Rerandomized Decay Protocol, exactly as explained above. On the other hand, all
other nodes (those in conflict or leaderless) transmits a special NOISE message in all of the
rounds of the phase. A node will consider itself boundary if it receives messages from its own
territory in less than ? log N of the rounds of this phase. The next two lemmas, the second of
15
which is the key lemma in this whole approach, explain why this criterion correctly identifies
the boundary nodes:
I Lemma 6. If a node w is not alone in its territory and it does not have a G0neighbor out
of its territory, then throughout the phase, it will receive at least ? log N messages from its
15
own territory.
Proof. The proof is similar to Lemma 5. We easily see that per round, w receives a message
from its own territory with probability at least log N . Thus, by a Chernoff bound, node w
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receives messages from its own territory in at least ? l1o5g N rounds, with high probability. J
I Lemma 7. [Key Lemma] If w has at least one Gneighbor out of its territory, then w.h.p.,
it will either receive a message from a different territory, or it receives less than ? log N
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messages from its own territory.
Proof. First note that if w has a Gneighbor in the conflict or leaderless territory, it will not
receive any message from its own territory and hence it detects that it is a boundary. Next,
suppose that each Gneighbor of w has received a message from exactly one marked node.
Consider a given round r of the phase, and fix the edges incident on w made active for this
round by the adversary. Let Ar be the set of active neighbors of w in its own territory, and
let Br be the set of active neighbors of w in other territories. Note that by the assumption
that w has at least one Gneighbor out of its territory, Br is nonempty. Suppose that the
set Br is composed of nodes of t territories, Br = Br1 ? Br2 ? . . . Brt , where without loss of
generality, Brt is not the conflict or leaderless territory. Define pr to be the probability of
the event that no node in Br1 to Brt?1 transmits. We have the following simple observations
regarding the transmission probabilities in these sets:
(1) The probability that no node in Ar transmits is at least 1/(2e).
5 .
(2) The probability that exactly one node of Ar transmits is in at most log N
(3) The probability that no node in any of Br1 to Brt transmits is at most pr.
(4) The probability that exactly one node of Br transmits, and it is in Brt , is at least 10 lporg N .
Note that the events of different territories are independent. Hence, by (2) and (3), we have
that the probability that w receives a message from its own territory is at most lo5gprN . Also,
by (1) and (4), we have that the probability that w receives a message from another territory
pr
is at least 20e log N .
Therefore, over all the L rounds of the phase, the expected number of messages that w
receives from other territories is at least PrL=1 20eplorg N . If PrL=1 20eplorg N ? 5 log N , then
with high probability, w receives a message from another territory. On the other hand, if
PL pr
r=1 20e log N ? 5 log N , the expected number of messages that w receives from its own
territory is at most PrL=1 lo5gprN ? l3o0g N , as the constant ? is chosen to be large enough.
Thus, in this case, with high probability, the number of rounds in which w receives a message
from its own territory is less than ? l2o5g N . This completes the proof. J
Delivering the Boundary Signal to the Candidate. If for a candidate v, there is a node u
in the territory of v for which the condition of Lemma 7 is satisfied, then v knows that its
experiment was not successful. We next describe how to let the candidate v know whether
there is such a ?boundary? node u in its territory. The basis for this again will be mimicking
collision detection, where lack of frequent receptions from a node?s own territory is regarded
as a collision.
We now explain how we deliver the indicator about the existence of boundaries to the
candidate, and how they react. Note that the broadcast algorithm, explained in Section 4.2,
grows the territory at a speed of one Ghop per phase, when there is a single candidate, and
at a (potentially) slower speed, when there are two or more. Hence, all nodes of the territory
of a candidate v are within its O(D) hops in G. After one phase of boundary detection,
if the territory of v does not include all nodes, there will be at least one node u in this
territory that satisfies the condition of Lemma 7. Thus, this node provides an excuse for not
announcing the experiment successful, and potentially continuing the broadcast.
To deliver this information to the candidate, we run O(D) additional phases of broadcast.
Here, if a node has noticed a boundary (or is in conflict or leaderless), it constantly transmits
NOISE in all the rounds of the phase. Otherwise, the node broadcasts simply according to
its rerandomized decay protocol, using the shared randomness provided in the candidate?s
message. At the end of each phase, each node w believes that there is a boundary node in
its territory if it believed so before or if receives a message from outside territory or a NOISE
message, or more critically, if it does not receive messages from the node?s own territory
frequently enough (less than ? log N times). If this happens, node w constantly transmits
25
NOISE in the rounds of the next phases. The following lemma follows by straightforward
application of Lemma 7.
I Lemma 8. If there is at least one boundary in the territory of candidate v, then by the
end of these O(D) phases, candidate v will be informed about it. Moreover, if the territory
includes all nodes, then v will not receive such an indication.
Finally, if after all these ?(D) phases, a candidate v received no indication of a boundary, it
considers this experiment successful. It then initiates a ?successful experiment? announcement.
Other candidate do not initiate any announcement. We run a broadcast from the candidates,
in O?(D) rounds, spreading this success announcement. If the experiment was successful,
then with high probability, the candidate is alone and thus, this success announcement gets
delivered to all nodes of the network.
Removing the knowledge of diameter. Recall that above, we assumed that nodes know a
constant factor upper bound D on the diameter of graph G. Here, we remove this assumption
using a standard doubling method.
Effectively, we try 2factor estimates of diameter. We divide the time into log N scales,
where the kth is made of O?(2k) consecutive rounds. In the kth scale, we imagine 2k to
be the upper bound on diameter. We thus grow the territories up to a radius of 2k by
running the broadcast (from scratch) for 2k phases. Then we have one boundary detection
phase, and then we spend 2k additional phases to deliver the indicator about existence
of boundaries to the candidate. A final set of 2k spreads the potential announcements of
successful experiments. Note that since the length of the scales form a geometric sum, and
as when 2k reaches D we will have the first successful experiment, we will observe the first
successful experiment in O(D) phases, considering all the scales. In the other experiments
prior to this scale, the broadcasts will not be able to reach all nodes, as it spreads at a speed
of one Ghop per round. Hence, those experiments will not be announced successful.
Consider the first scale for which some candidate announces a successful experiment.
In this scale, we take the candidate of these successful experiments that has the smallest
experiment number as the global leader.
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