On a leader election algorithm: Truncated geometric case study

1. Background Randomized divide and conquer algorithms have several manifestations. In particular, leader election problems are an interesting class, because randomized elections have a rich mathematical and algorithmic history. Prodinger (1993) precisely analyzed the average behavior of several characteristic properties for a certain leader election algorithm that flips unbiased coins. He coined the terminology incomplete trie for the tree structure underlying the elimination process. Using analytic methods, the exact and asymptotic average for the size of the tree, i.e., the number of nodes, the depth (also known as the height in the literature) or the number of rounds, and the cost (measured in terms of the total number of coin flips) were obtained. Fill et al. (1996) used analytic and probabilistic methods to obtain the oscillating distribution of the height of a random incomplete trie constructed using unbiased coins. Janson and Szpankowski (1997) analyzed the height for biased coins using analytic techniques. Mohamed (2006) also investigated the biased-case scenario for the height, but used probabilistic methods. More recently, Louchard and Prodinger (2009) used analytic methods to study the number of rounds in a coin flipping selection algorithm that occurs in the presence of a demon (who randomly eliminates some contestants). Louchard et al. (2012), complementing the previous paper, precisely analyzed the distribution and all moments of the number of survivors in a selection process that occurs in the presence of a demon. Louchard et al. (2011) studied another variant called the Swedish leader election protocol; they analyzed several parameters, e.g., the probability of success, the expected ∗ Corresponding author. E-mail addresses: [email protected] (R. Kalpathy), [email protected] (M.D. Ward). 0167-7152/$ – see front matter© 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.12.020 R. Kalpathy, M.D. Ward / Statistics and Probability Letters 87 (2014) 40–47 41 number of rounds, the expected number of players still playing by the time the protocol fails, etc., using analytic methods. Kalpathy et al. (2011) used analytic and probabilistic techniques to study a leader election algorithm using biased coins for the duration a particular player survives in the competition and the total cost involved in the selection process. Somemore general, broad frameworks (which unify some of the above results) have also been proposed, for the study of leader election algorithms. One such framework was proposed by Janson et al. (2008), giving a theory for the cost associated with the number of rounds (equivalently, the height of the underlying incomplete tree). Another framework was proposed by Kalpathy et al. (2013), giving the number of survivors in a broad class of fair leader election algorithms, using the theory of probabilitymetrics. That framework yields product randomvariables as the limit distribution. The classical example of leader election is via binomial splitting (where candidates advance in rounds of coin flipping). The binomial splitting protocol is only one of many possible strategies. More recently, Kalpathy and Mahmoud (in press) provided a broader framework to cover several splitting strategies of interest at once. They obtain very general results for general strategies, like uniform splitting, ladder tournaments, etc. They study (a) the duration of a particular contestant, i.e., the number of rounds a randomly selected contestant stays in the competition, and (b) the total cost of selection. Using probabilitymetrics and the contractionmethod, they present a unifying treatment for leader election algorithms, and they show how perpetuities naturally come about (see Kalpathy (2013) for a detailed discussion). Kalpathy and Mahmoud (in press) look at a set of mild sufficient conditions, which are easily met by most practical choices of splitting protocols, such as binomial, uniform, ladders, power laws, etc. Their theory, however, produces only trivial asymptotic results for the duration of election for some distributions, such as a truncated geometric distribution. In the case of a truncated geometric distribution, the limiting distribution of the duration of contestants is degenerate, and the method of Kalpathy and Mahmoud (in press) is insufficient for obtaining precise asymptotics. Thus, the truncated geometric distribution corresponds to a type of leader election example in which sharper tools are needed, to give necessary precision to the asymptotics. We opted to use the q-series methodology, because it provides sufficient sharpness, and it can produce such exact asymptotic results. Wealso emphasize that this type of asymptotic analysis is usually handled bybreaking the classification of the parameters p and q = 1− p into three separate cases: (1) p = q = 1/2; (2) ln p ln q is a rational number; (3) ln p ln q is an irrational number. Our analysis is sufficiently precise to handle all p’s and q’s. 2. Asymptotic notation We use theΘ notation from Knuth (1976), which describes the asymptotic growth of functions; we quote from page 20: ‘‘Θ(f (n)) denotes the set of all g(n) such that there exist positive constants C, C , and n0 with C f (n) ≤ g(n) ≤ C ′ f (n) for all n ≥ n0’’. This notation is quite standard in asymptotic analysis of algorithms. We also use another standard notation to indicate that f grows at a strictly smaller rate than g , namely, f (n) = o(g(n)) if, for all C > 0, there exists n0 (depending on C) such that |f (n)| ≤ C |g(n)| for all n ≥ n0. 3. Duration for truncated geometric splitting Let Dn be the duration of a specific contestant (say Bob) when starting the election with n contestants, i.e., the number of rounds Bob participated. Note that this is equivalent to the depth of the underlying incomplete tree. We set D0 = D1 = 0. At the start of a round, if n contestants are present, then Kn contestants advance to the next round. We handle the case where Kn is a truncated geometric random variablewith parameters p and q := 1 − p. We have P(Kn = l) = cpq, for some constant c. Since 1 = n l=0 P(Kn = l) = c n l=0 pq , it follows that c = 1 1−qn+1 . Thus the mass of Kn is P(Kn = l) = pq 1 − qn+1 , for l = 0, 1, . . . , n. Suppose we conduct a leader election among n contestants, in which a fair selection of a subset of contestants of a random size Kn advances to the next round, and the algorithm is applied recursively on that subset, till one leader or none is elected. We get the following result. Theorem 3.1. Let Dn be the duration of a specific contestant that begins with n participants. If the number of contestants who advance to the next round is a truncated geometric random variable with parameters p and q, then the first and second moments of Dn are E[Dn] = 1 + 1 n n 