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A survey of maxtype recursive distributional equations
 Annals of Applied Probability 15 (2005
, 2005
"... In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent cop ..."
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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write
The Functional Equation of the Smoothing Transform
, 2010
"... Given a sequence T = (Ti)i≥1 of nonnegative random variables, a function f on the positive halfline can be transformed to E ∏ i≥1 f(tTi). Westudythefixedpointsofthistransformwithintheclass ofdecreasing functions. By exploiting the intimate relationship with general branching processes, a full descr ..."
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Cited by 17 (5 self)
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Given a sequence T = (Ti)i≥1 of nonnegative random variables, a function f on the positive halfline can be transformed to E ∏ i≥1 f(tTi). Westudythefixedpointsofthistransformwithintheclass ofdecreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on [0,∞), the results provide the full description of the set of solutions to the fixedpoint equation of the smoothing transform, X d = ∑ i≥1 TiXi, where d = denotes equality of the corresponding laws and X1,X2,... is a sequence of i.i.d. copies of X independent of T. Moreover, since leftcontinuous survival functions are covered as well, the results also apply to the fixedpoint equation X d = inf{Xi/Ti: i ≥ 1,Ti> 0}.
Distributional convergence for the number of symbol comparisons used by QuickSort
, 2012
"... Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic sourc ..."
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Cited by 10 (4 self)
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Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild “tameness ” condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by n. Additionally, under a condition that grows more restrictive as p increases, we have convergence of moments of orders p and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased (“fair”) bits, the mean exhibits periodic fluctuations of order n.
Analysis of the expected number of bit comparisons required by Quickselect
, 2009
"... When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard ..."
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Cited by 8 (4 self)
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When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank.
On a mintype stochastic fixedpoint equation related to the smoothing transformation
, 2008
"... This paper is devoted to the study of the stochastic fixedpoint equation X d = inf i≥1:Ti>0 Xi/Ti and the connection with its additive counterpart X d = ∑ i≥1 TiXi associated with the smoothing transformation. Here d = means equality in distribution, T def = (Ti)i≥1 is a given sequence of nonneg ..."
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Cited by 4 (4 self)
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This paper is devoted to the study of the stochastic fixedpoint equation X d = inf i≥1:Ti>0 Xi/Ti and the connection with its additive counterpart X d = ∑ i≥1 TiXi associated with the smoothing transformation. Here d = means equality in distribution, T def = (Ti)i≥1 is a given sequence of nonnegative random variables and X, X1,... is a sequence of nonnegative i.i.d. random variables independent of T. We draw attention to the question of the existence of nontrivial solutions and, in particular, of special solutions named αregular solutions (α> 0). We give a complete answer to the question of when αregular solutions exist and prove that they are always mixtures of Weibull distributions or certain periodic variants. We also give a complete characterization of all fixed points of this kind. A disintegration method which leads to the study of certain multiplicative martingales and a pathwise renewal equation after a suitable transform are the key tools for our analysis. Finally, we provide corresponding results for the fixed points of the related additive equation mentioned above. To some extent, these results have been obtained earlier by Iksanov [16].
On stochastic recursive equations of sum and maxtype
 Journal of Applied Probability
, 2006
"... In this paper we consider stochastic recursive equations of sumtype X d = �K i=1 AiXi+b and of maxtype X d = max(AiXi+bi; 1 ≤ i ≤ k) where Ai, bi, b are random and (Xi) are iid copies of X. Equations of this type typically characterize limits in the probabilistic analysis of algorithms, in combinat ..."
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Cited by 3 (2 self)
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In this paper we consider stochastic recursive equations of sumtype X d = �K i=1 AiXi+b and of maxtype X d = max(AiXi+bi; 1 ≤ i ≤ k) where Ai, bi, b are random and (Xi) are iid copies of X. Equations of this type typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems as well as in many other problems having a recursive structure. We develop some new contraction properties of minimal Lsmetrics which allow to establish general existence and uniqueness results for solutions without posing any moment conditons. As application we obtain a one to one relationship between the set of solutions of the homogeneous equation and the set of solutions of the inhomogeneous equation for sum and maxtype equations. We also give a stochastic interpretation of a recent transfer principle of Rösler (2003) from nonnegative solutions of sumtype to those of maxtype by means of random scaled Weibull distributions. 1
Exponential tail bounds for maxrecursive sequences
"... Exponential tail bounds are derived for solutions of maxrecursive equations and for maxrecursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise in the worst case analysis of divide and conquer alg ..."
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Cited by 2 (0 self)
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Exponential tail bounds are derived for solutions of maxrecursive equations and for maxrecursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise in the worst case analysis of divide and conquer algorithms, in parallel search algorithms or in the height of random tree models. For the proof we determine asymptotic bounds for the moments or for the Laplace transforms and apply a characterization of exponential tail bounds due to Kasahara (1978). 1
Bivariate uniqueness and endogeny for recursive distributional equations: Two examples
, 2004
"... In this work we prove the bivariate uniqueness property for two “maxtype” recursive distributional equations which then lead to the proof of endogeny for the associated recursive tree processes. Thus providing two concrete instances of the general theory developed by Aldous and Bandyopadhyay [3]. Th ..."
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In this work we prove the bivariate uniqueness property for two “maxtype” recursive distributional equations which then lead to the proof of endogeny for the associated recursive tree processes. Thus providing two concrete instances of the general theory developed by Aldous and Bandyopadhyay [3]. The first example discussed here deals with the construction of a frozen percolation process on a infinite regular binary tree. For this we prove that the construction do not involve any external randomness. It is also shown that same is true for any rregular tree and more interestingly for any infinite regular GaltonWatson branching process trees with mild moment condition on the progeny distribution. The second example is proving the endogeny for the Logistic recursive distributional equation which appears for studying the asymptotic limit of the random assignment problem using localweak convergence method. The two examples are quite unrelated and hence illustrate a broad range of applicability of the general methods of [3].
A Gaussian limit process for optimal FIND algorithms
, 2013
"... We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to b ..."
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We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as n → ∞, which depends on α. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties. AMS 2010 subject classifications. Primary 60F17, 68P10; secondary 60G15, 60C05, 68Q25. Key words. FIND algorithm, Quickselect, complexity, key comparisons, functional limit theorem,
Analysis of the Expected Number of Bit Comparisons Required by Quickselect ∗
"... When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard ..."
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When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this extended abstract, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank. 1