Statistics and Probability Letters 83 (2013) 2743–2749

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Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

Survivors in leader election algorithmsRavi Kalpathy a,∗, Hosam M. Mahmoud a, Walter Rosenkrantz b

a Department of Statistics, The George Washington University, Washington, DC 20052, USAb Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA

a r t i c l e i n f o

Article history:Received 2 July 2013Received in revised form 10 September2013Accepted 11 September 2013Available online 21 September 2013

MSC:60E0560F0568W40

Keywords:Randomized algorithmLeader electionStochastic recurrenceProbability metricsWeak convergence

a b s t r a c t

We consider the number of survivors in a broad class of fair leader election algorithms aftera number of election rounds. We give sufficient conditions for the number of survivors toconverge to a product of independent identically distributed randomvariables. The numberof terms in the product is determined by the round number considered. Each individualterm in the product is a limit of a scaled random variable associated with the splittingprotocol. The proof is established via convergence (to 0) of the first-order Wassersteindistance from the product limit. In a broader context, the paper is a case study of a classof stochastic recursive equations. We give two illustrative examples, one with binomialsplitting protocol (forwhichwe show that a normalized version is asymptoticallyGaussian)and one with uniform splitting protocol.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Randomized divide and conquer algorithms have numerous applications, and leader election (and associated digitaltrees) are an important class. In the context of distributed computing, leader election algorithms play a vital role. Thecost measures that appear in these algorithms have several important applications that can impact communication andcontrol in information processing systems. For instance, in digital trees, the size, height, depth, profile, etc., directly translateto memory usage and computing time. In the language of probability, these random manifestations appear as stochasticrecursive distributional equations for the cost measures.

In this note, we only concentrate on leader election algorithms. The classic case, introduced in Prodinger (1993), dealswith binomial splitting protocols. It was further researched in Fill et al. (1996), Janson and Szpankowski (1997), Louchardand Prodinger (2009, 2012), Louchard et al. (2012) and Kalpathy et al. (2011), among other references. Only very recently,research branched out from the specific case of binomial splitting to leader election with more general types of splittingprotocols; see Janson et al. (2008), Kalpathy (2013), and Kalpathy and Mahmoud (in press).

The applications of leader election algorithms are numerous. To name a few, we mention selection of a winner of acontest, a loser of bets, or a coordinator of a computer network in the case of failure of the existing central coordinator. Thetheme in these applications is the following. There are a number of contestants who will compete fairly to elect a winner. Insome variations the proceduremay result in nowinners. As an example, in book award competitions, occasionally the award

∗ Corresponding author.E-mail addresses: kalpathy@gwu.edu (R. Kalpathy), hosam@gwu.edu (H.M. Mahmoud), w_rosenkrantz@verizon.net (W. Rosenkrantz).

0167-7152/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.spl.2013.09.011

2744 R. Kalpathy et al. / Statistics and Probability Letters 83 (2013) 2743–2749

is withheld.1 The contestants keep going through elimination rounds, in which they generate events that decide whetheror not they advance to the next round, or alternatively a moderator generates these events on their behalf to fairly elect thecandidates at the next round. Fairness of leader election will be assumed throughout. Accordingly, all contestants have thesame chance (probability) to win.

Typical parameters in the analysis are the number of rounds till termination, i.e., the height of the underlying incompletetree (see Fill et al. (1996) and Janson and Szpankowski (1997)), the total cost till termination (see Prodinger (1993), Kalpathyet al. (2011), Kalpathy (2013) and Kalpathy and Mahmoud (in press)), and the number of rounds a contestant survives (seeKalpathy et al. (2011), Kalpathy (2013) and Kalpathy and Mahmoud (in press)), which is an important measure from thepoint of view of an individual contestant.

Another measure that can be important from the point of view of the moderator is the number of surviving contestantscontinuing after a certain stage. Starting with n contestants, we specify a number t (possibly dependent on n), and askhow many survivors are still in contention (a formal definition is given in the sequel) after t rounds. This parameter maybe important in planning ahead of time such resources as the space needed to hold the contestants. As of yet, this lattermeasure has not been investigated.

We alert the reader to distinguish our definition of the number of survivors from that in Louchard and Prodinger (2012)where the authors consider binomial-splitting leader election algorithms parameterized by τ ≥ 0, and the election isstopped, if the process ran τ rounds without change in the number of contestants. In Louchard and Prodinger (2012) thenumber of survivors is the number of contestants remaining at the time of termination. The class of leader election weconsider has τ = ∞, and again our number of survivors counts something different.

In this investigation, we take up the analysis of the number of survivors (in our sense), under fairly general assumptionson the splitting protocol. The nature of the application gives rise to a discussion of a new class of two-dimensional stochasticrecursive recurrence equations.

2. Leader election

Assume there are n contestants competing. They (or a contest moderator on their behalf) generate(s) a certain numberKn ∈ {0, 1, . . . , n}, possibly deterministic or random, of candidateswho remain in the contest and the rest of the contestantsare eliminated. The algorithm is then applied recursively on the remaining set of contestants, until a leader is elected or noone wins the contest. The generated events and the moderator are fair, in the sense that, given Kn = k, all subsets of size kare equally likely choices.

We call the set chosen to proceed to the next round the advancing set. We shall study the number of survivors after trounds (where t is the discrete time).

Let Sn,t be the remaining number of contestants (the survivors) after t rounds of competition. If the process is terminatedat time t ′ < t , we think of the situation as if the process continued after t ′, advancing 0 contestants in each subsequentround; that is, we define Sn,t = 0, for t > t ′. For example, suppose we started with 15 contestants, and five got eliminatedin the first round, another three got eliminated in the second round, and six got eliminated in the third round, producing awinner in the fourth round. Then, S15,0 = 15, S15,1 = 10, S15,2 = 7, S15,3 = 1, and 0 = S15,4 = S15,5 = S15,6 = · · ·. Here, wealso have K15 = 10 at the top level, and recursively the algorithm is repeated on a set of size K10 = 7, and so on – we havein advance a sequence of random variables, K0, K1, . . . , with Kj having a distribution on the set {0, 1, . . . , j}. We call thissequence of distributions the splitting protocol – whenever we need to generate winning candidates, say, when the contestsize becomes j, we use Kj from this list. That is, we generate a random value according to the distribution of Kj, and use it toadvance Kj contestants, and eliminate the rest.

In what follows, we use the notation D= for equality in distribution, and

D−→ and

a.s.−→ respectively for convergence in

distribution and almost surely. For n ≥ 2, we have a two-dimensional stochastic recurrence equation for the number ofsurvivors:

Sn,tD= SKn,t−1, (1)

with initial conditions S0,t = 0, for all t ≥ 0, and Sn,0 = n, for all n ≥ 0. Here, for any n ≥ 2, Kn is a random variable with agiven distribution in the set {0, . . . , n}; all the random variables are defined on the same probability space.

The rest of this short note is the main result and its proof (Section 3) and two illustrative examples (in two subsectionsof Section 4), and we conclude with some remarks in Section 5.

3. Scope and results

The size Kn of the advancing set is always a proportion of n, and we deal with cases where Kn/n converges in L1 to a limitK ∗, at a fast enough rate to aid the convergence of Sn,t/n. In the sequel, we shall give the formal definition of the class ofsplitting protocols dealt with in the main theorem. If Kn/n does not converge, there may be no convergence at all.

1 For instance, in 1949 the Nobel Committee for Literature did not select a winner for the Nobel Prize because none of the nominations met the selectioncriteria. The 1949 Nobel Prize in Literature was later given in 1950 to William Faulkner.

R. Kalpathy et al. / Statistics and Probability Letters 83 (2013) 2743–2749 2745

The proof tool is convergence of the Wasserstein distance. The Wasserstein distance of order k between two distributionfunctions F and G is defined by

∆k(F ,G) = inf ∥W − Z∥k,

where the infimum is taken over all coupled random variablesW and Z (defined on the same probability space) having therespective distribution functions F andG (with ∥·∥k being the usualLk norm). Inwhat follows, we use FY for the distributionfunction of a random variable Y . It is well known that, for a sequence of random variablesWn, the convergence of first-orderWasserstein distances between FWn and FW to 0 implies the convergence Wn

D−→W , as well as convergence of the first

absolute moment (see Bickel and Freedman (1981)).

Theorem 3.1. Suppose we conduct a leader election among n contestants, in which a fair selection of a subset of contestants ofa random size Kn advance to the next round, and the algorithm is applied recursively on that subset, till one leader is elected ornone. Assume for some α ∈ (0, 1), the advancing set size satisfies2

K ∗

n :=Kn

n= K ∗

+ OL1

1nα

,

for some limiting random variable K ∗, with distribution supported on [0, 1], with mean E[K ∗] < 1. Let Sn,t be the number of

survivors after t rounds of election. We then have

Sn,tn

D−→ V ∗

1 V∗

2 · · · V ∗

t ,

and {V ∗

i }∞

i=1 are totally independent, and each of which is distributed like K ∗.

Proof. Let us write the recurrence Eq. (1) in normalized form:

S∗

n,t :=Sn,tn

D=

Kn

n×

SKn,t−1

Kn= K ∗

n S∗

Kn,t−1. (2)

In view of the stated L1 convergence of K ∗n , the structure of the normalized Eq. (2) suggests that S∗

n,t converges to a limitingrandom variable, say S∗

t , satisfying, for t ≥ 1, the distributional recurrence

S∗

tD= K ∗S∗

t−1, (3)

with {S∗

i }∞

i=0 independent of {(Kj, K ∗)}∞j=0. We shall prove this first. That is, we prove that for each t ≥ 1, we have S∗n,t

D−→ S∗

t ,where S∗

t satisfies (3), and the initial condition S∗

0 = 1.Let us take the pairs (K ∗

n , K ∗), for n ≥ 0, to be independent. Assume the same for (S∗n,t , S

∗t ). We also assume that (S∗

n,t , S∗t )

are optimal couplings for all n ≥ 0. We then have

∆1(FS∗n,t

, FS∗t) ≤ E

K ∗

n S∗

Kn,t−1− K ∗S∗

t−1

:= bn.

It is clear that S∗t ≤ S∗

0 = 1, and thus for any t ≥ 0, S∗t is integrable. In fact E

S∗t

≤ 1. By the stated rate condition, we

have

bn ≤ EK ∗

n S∗

Kn,t−1 −

K ∗

n − OL1

1nα

S∗

t−1

≤ E

K ∗

n (S∗

Kn,t−1 − S∗

t−1) + E

S∗

t−1OL1

1nα

= E

K ∗

n (S∗

Kn,t−1 − S∗

t−1) + E

S∗

t−1

E

OL1

1nα

;

the separation of the expectations of the OL1 term (coming from K ∗n − K ∗) and S∗

t−1 follows from their independence.Subsequently, we have

bn ≤ EK ∗

n (S∗

Kn,t−1 − S∗

t−1) +

Anα

,

2 When we say a sequence of random variables Yn is OL1 (g(n)), we mean there exist a positive constant C and a positive integer n0 , such thatE[|Yn|] ≤ C |g(n)|, for all n ≥ n0 .

2746 R. Kalpathy et al. / Statistics and Probability Letters 83 (2013) 2743–2749

for some positive constant A, and some integer n0 > 0, and for all n ≥ n0. Note that the assumption that Kn converges to K ∗

in L1, with E[K ∗] < 1, implies that P(Kn = n) = 1 can occur only finitely many times. Suppose n′

0 is the largest index forwhich P(Kn = n) = 1. For n′′

0 ≥ max(n0, n′

0), we can write

bn ≤

1n

n−1k=0

kbkProb(Kn = k) +Anα

1 − Prob(Kn = n),

with a finite right-hand side, for all n ≥ n′′

0 .Let us start an induction to show that bn ≤ h/nα , for some constant h > 0. For 1 ≤ n ≤ n′′

0 , we have

bn ≤ max1≤j≤n

bj ≤

(n′′

0)α max

1≤j≤n′′0

bj

nα=:

h1

nα.

This can be a basis of induction, if h is taken at least as large as h1.Assume, for some n ≥ n′′

0 , and all k ∈ {1, . . . , n − 1}, that bk ≤ h/kα . We have

bn ≤

1n

nk=0

kh Prob(Kn = k)/kα− h Prob(Kn = n)/nα

+Anα

1 − Prob(Kn = n)

=h E[(Kn/n)1−α

]/nα− h Prob(Kn = n)/nα

+Anα

1 − Prob(Kn = n).

The induction step will be complete, if, for all n ≥ n′′

0 ,

h E[(K ∗n )1−α

] − h Prob(Kn = n) + A1 − Prob(Kn = n)

≤ h,

i.e., if

h ≥A

1 − E[(K ∗n )1−α]

.

Indeed, such a bound holds, if h is chosen high enough, specifically, if

h ≥ h2 :=A

1 − supn≥n′′

0

E[(K ∗n )1−α]

≥A

1 − E[(K ∗n )1−α]

, (4)

and the supremum involved is clearly less than 1.Inequality (4) holds if we take h high enough. Take h = max{h1, h2}, and by induction bn ≤ h/nα , for all n ≥ 1. The

induction is complete. Thus, we have ∆1(FSn,t , FS∗t) ≤ bn → 0. Hence S∗

n,tD

−→ S∗t , with S∗

t satisfying (3), which unwindsrecursively to a product of t independent random variables, each distributed like K ∗. �

Remark 3.1. By the strong law of large numbers, if t is large, the logarithm of the limiting product scaled by t can beapproximated by E[log K ∗

].

4. Examples

We give a couple of examples with splitting protocols that are rather natural, and likely to appear in applications.

4.1. Binomial splitting

This is a classic example and several of its properties have been studied (see Prodinger (1993), Fill et al. (1996), Janson andSzpankowski (1997) and Kalpathy et al. (2011)). Contestants flip coins, producing Heads with probability p, and producingTails with probability 1− p. Those who flip Heads advance to the next round; those who flip Tails are out of contention. Thecompetition continues, so long as there are at least two remaining contestants. Note that this splitting protocol may resultin nowinners. In this model, Kn has the binomial distribution underlying n independent identically distributed experiments,with success probability p per experiment. We use the notation Bin(n, p) to denote such a binomial random variable.

By the strong law of large numbers, Kn/na.s.

−→ p, at a rate of 1/√n (by the normal approximation to the binomial). We

can take α = 1/2 in the regularity condition of Theorem 3.1, and conclude thatSn,tn

D−→ pt .

R. Kalpathy et al. / Statistics and Probability Letters 83 (2013) 2743–2749 2747

Since the convergence is to a constant, it is also convergent in probability. Moreover, as we have a uniform bound Sn,t/n ≤ 1,we also have convergence in moments. We thus have the asymptotic mean, as n → ∞,

E[Sn,t ] ∼ ptn. (5)

Remark 4.1. Note that t = t(n) = log 1pn represents a phase change in the mean (and consequently in the limiting

distribution). As n → ∞, below this logarithmic threshold, E[Sn,t ] grows to+∞; at the logarithmic phase E[Sn,t ] approachesa constant; above this logarithmic threshold, E[Sn,t ] diminishes to 0.

A modified (winnerless) splitting protocol, also binomial, but differing only in its stopping rule, provides a transparentbenchmark, allowing us to extend the result in this example to second order asymptotics. That is, we specify a rate ofconvergence in the weak law already stated, which as we shall see takes the form of a central limit theorem.

Consider the binomial splitting protocol that behaves exactly as the one described, but if one contestant is left, shecontinues to flip coins and keeps advancing to further rounds as long as she flips Heads. With probability 1, ultimatelythat contestant gets eliminated when she gets Tails. Note that this is a ‘‘complete elimination process’’ that continues tillall contestants are out of contention, and thus it never produces a single winner. Let S̃n,t be the number of survivors after trounds of competition. Clearly, S̃n,t coincideswith Sn,t , till 1 or 0 candidates are left. In the original version, if one candidate isleft, that candidate is declared a winner and 0 contestants advance to later rounds after that point; whereas in the modifiedwinnerless version, that candidate stays a bit longer, till she gets eliminated. Thus point-wise in the sample space, we have

S̃n,t − 1 ≤ Sn,t ≤ S̃n,t . (6)

Two special properties of S̃n,t merit consideration:

(i) It has the same asymptotic behavior as Sn,t .(ii) A fact of independent interest is that the sequence of random variables S̃n,t/pt , t = 0, 1, . . . , is a martingale. To see this,

note that when S̃n,t−1 = k, then S̃n,t has the binomial distribution Bin(k, p); consequently, ES̃n,t | S̃n,t−1 = k

= kp,

equivalently ES̃n,t | S̃n,t−1

= pS̃n,t−1. Dividing both sides by pt yields

p−t ES̃n,t | S̃n,t−1

= p−tpS̃n,t−1 = p−(t−1)S̃n,t−1.

In other words, for t = 0, 1, . . . , we have

E

S̃n,tpt

S̃n,t−1

=

S̃n,t−1

pt−1.

This, and the fact that S̃n,t , t = 0, 1, . . . , is a Markov chain, together imply that S̃n,t/pt is a martingale, with E[S̃n,t/pt ] =

E[S̃n,0] = n, t = 0, 1, . . . .

The modified winnerless case has a simple structure amenable to exact calculation of the probability distribution.

Theorem 4.1. Suppose we conduct a leader election among n contestants, in which a fair selection of a subset of contestants of arandom size Kn = Bin(n, p) advance to the next round, and the algorithm is applied recursively on that subset, till all contestantsare eliminated (exactly as in elimination by coin flipping). In this process, the number of survivors, S̃n,t , has the binomial distributionof Bin(n, pt).

Proof. We prove by induction on t that φS̃n,t (u), the moment generating function of S̃n,t , is (1 − pt + pteu)n, which is that

of Bin(n, pt). At t = 0, S̃n,0 = n = Bin(n, p0), providing a basis for the induction. Suppose now that, for t ≥ 1, S̃n,t−1 isdistributed like Bin(n, pt−1). If exactly k contestants survive till round t − 1, of these Bin(k, p) will advance to compete inround t . Thus, letting q = 1 − p, we have the conditional expectation

EeuS̃n,t | S̃n,t−1 = k

=

kj=0

kj

pjqk−jeuj = (q + peu)k.

So, we have the unconditional expectation

φS̃n,t (u) =

nk=0

(q + peu)k Prob(S̃n,t−1 = k),

2748 R. Kalpathy et al. / Statistics and Probability Letters 83 (2013) 2743–2749

which gives (by the induction hypothesis)

φS̃n,t (u) =

nk=0

q + peu

k nk

pt−1k

1 − pt−1n−k

=(q + peu)pt−1

+ 1 − pt−1n= (1 − pt + pteu)n,

completing the induction. �

According to Theorem 4.1, for each n ≥ 0, and every t ≥ 0, E[S̃n,t ] = ptn. Whence, the analog of the asymptoticrelation (5) is actually exact in themodifiedwinnerlessmodel. As a consequence of Theorem4.1,when t is fixed and n → ∞,we have

S̃n,t − ptn√n

D−→ N

0, pt(1 − pt)

,

where N (0, σ 2) is a normally distributed random variate with mean 0 and variance σ 2. So, in view of (6), for the originalversion of binomial splitting, we have the following corollaries.

Corollary 4.1. When t is fixed, as n → ∞, we have

Sn,t − ptn√n

D−→ N

0, pt(1 − pt)

.

Corollary 4.2. When t = t(n) = log 1p

ncn

, for uniformly bounded cn, we have

Poisson(cn) − 1 ≤ Sn,t ≤ Poisson(cn),

where Poisson (λ) is the usual Poisson random variable with parameter λ. On the other hand, if t grows faster than log n, S̃n,tbecomes degenerate, and consequently Sn,t converges to 0 in probability, too.

4.2. Uniform splitting

In this example, we study the behavior of Sn,t , when the splitting protocol generates a set of size Kn following a discreteuniform distribution on {1, 2, . . . , n}, and Kn/n converges inL1 to a standard continuous uniform distribution on (0, 1). Theregularity condition of Theorem 3.1 is satisfied (we can take α =

12 ), yielding:

Sn,tn

D−→U1U2 · · ·Ut ,

and {Ui}ti=1 are totally independent and identically distributed continuous Uniform (0,1) random variables. Note that

− ln(U1U2 · · ·Ut) has the distribution of a Gamma(t, 1) random variable. Therefore, Sn,t/n converges in distribution to arandom variable with probability density

f (s) =

lnt−1(1/s)(t − 1)!

, if 0 < s ≤ 1, t > 0;

0, otherwise.

We have the asymptotic mean, as n → ∞,

E[Sn,t ] ∼ E[U1] · · · E[Ut ] n =n2t

.

Note that t = t(n) = log2 n represents a phase change similar to that in the binomial splitting protocol.

5. Concluding remarks

We studied the number of survivors in leader election algorithms, which is an example of the two-dimensional stochasticrecurrence equation

Sn,tD= SKn,t−1.

We concluded that S∗n,t := Sn,t/n converges in distribution to a random variable S∗

t , satisfying the distributional recurrence

S∗

tD= K ∗S∗

t−1.

R. Kalpathy et al. / Statistics and Probability Letters 83 (2013) 2743–2749 2749

This is to be contrasted with stochastic recursive equations of the type

YnD= YKn + Tn,

with a toll function Tn = O(n), studied in Kalpathy (2013) and Kalpathy andMahmoud (in press). Under appropriate scaling,many of these types of stochastic recursive equations lead to a limit Y ∗ that solves a stochastic fixed point equation

Y ∗ D= AY ∗

+ B,

with Y ∗ independent of (A, B). The random variable Y ∗ is called a perpetuity and is given implicitly by its distributionalfixed point characterization. There is an extensive existing body of published research on stochastic recurrence equationsof more general types, see Vervaat (1979), Embrechts et al. (1997) and Alsmeyer et al. (2009), and the references therein.Note the difference between the limiting perpetuity Y ∗ (which has a definition in terms of itself) and the one that appears inthe present paper (which is a recurrence on an auxiliary parameter). The self-definition in the perpetuity equation requiressophisticated methods (like contraction in complete metric spaces) to justify the uniqueness of the solution (distribution),whereas the type of limit we have in this note is defined recursively, and does not require special methods, as it is a memberof a family of recursively uniquely-defined random variables, all falling back on one well-defined boundary condition.

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