[Paper Review] Dependence between External Path-Length and Size in Random Tries

Michael Fuchs, Hsien‐Kuei Hwang|arXiv (Cornell University)|Apr 29, 2016

Algorithms and Data Compression14 references3 citations

TL;DR

This paper investigates the dependence between the size and external path length (EPL) in random tries under biased and symmetric bit distributions. Using Poissonization, Mellin transforms, and the contraction method, it shows that EPL and size are asymptotically independent in the asymmetric case but exhibit strong, periodic dependence in the symmetric case—providing a rare example of bivariate normal limit laws with correlation 0, 1, or oscillating due to periodic fluctuations.

ABSTRACT

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results that the internal path length is totally positively correlated to the size and that both tend to the same normal limit law. These two examples provide concrete instances of bivariate normal distributions (as limit laws) whose correlation is $0$, $1$ and periodically oscillating.

Motivation & Objective

• To analyze the asymptotic dependence structure between the size and external path length (EPL) in random tries under varying bit bias.
• To determine whether EPL and size are asymptotically uncorrelated or correlated in the limit, depending on the bias parameter p.
• To identify the nature of the limiting bivariate distribution of size and EPL, particularly in the symmetric (p = 1/2) versus asymmetric (p ≠ 1/2) cases.
• To establish that the correlation between size and EPL can be 0, 1, or periodically oscillating in the limit, contrasting with known results on internal path length.

Proposed method

• Poissonization is applied to the recurrence relations of size $ S_n $, EPL $ K_n $, and internal path length $ N_n $, transforming the problem into a more tractable form.
• Mellin transform techniques are used to analyze the asymptotic moments and variance of $ S_n $ and $ K_n $, especially in the symmetric case where $ p = 1/2 $.
• The contraction method is employed to prove asymptotic joint normality of $ (S_n, K_n) $, showing convergence to a bivariate normal distribution with identity covariance matrix after centering and normalization.
• Periodic fluctuations in the symmetric case are captured via the function $ ancyscript{F}[g](n) $, which depends on the rationality of $ \log p / \log q $.
• De-Poissonization techniques are used to transfer results from the Poissonized model back to the original model with fixed $ n $.
• The correlation coefficient between $ S_n $ and $ K_n $ is analyzed via the asymptotic behavior of $ \mathrm{Cov}(S_n, K_n) $, with special attention to the case $ p = 1/2 $, where the variance of $ K_n $ is of linear order rather than $ n \log n $.

Experimental results

Research questions

• RQ1 How does the correlation between size and external path length in random tries behave asymptotically when the bit distribution is biased (p ≠ 1/2)?
• RQ2 What is the nature of the dependence between size and EPL in the symmetric case (p = 1/2), and why does it differ from the asymmetric case?
• RQ3 Can the joint limit distribution of size and EPL be characterized as bivariate normal, and if so, what is the limiting correlation structure?
• RQ4 Why does the variance of EPL scale as $ \Theta(n) $ in the symmetric case, while it is $ \Theta(n \log n) $ in the asymmetric case?

Key findings

• In the asymmetric case (p ≠ 1/2), the size $ S_n $ and external path length $ K_n $ are asymptotically uncorrelated, with correlation coefficient tending to 0.
• In the symmetric case (p = 1/2), the correlation between $ S_n $ and $ K_n $ is strongly positive and exhibits small periodic fluctuations due to the rationality of $ \log p / \log q $.
• The joint distribution of $ (S_n, K_n) $, after centering and normalization, converges in distribution to a bivariate normal with identity covariance matrix, confirming asymptotic joint normality.
• The limiting correlation between $ S_n $ and $ K_n $ is 0 in the asymmetric case, 1 in the symmetric case with periodic fluctuations, providing a concrete example of bivariate normal limits with correlation 0, 1, and oscillating values.
• The variance of $ K_n $ is of order $ n \log n $ when p ≠ 1/2, but only of order $ n $ when p = 1/2, due to the vanishing of the coefficient $ \lambda = \frac{pq \log^2(p/q)}{h^3} $ at p = 1/2.
• The contraction method and Poissonization technique successfully establish the asymptotic joint normality of $ (S_n, K_n) $, with convergence proven via $ L_3 $-convergence and almost sure convergence of matrix coefficients.