Introduction to the Interface of Probability and Algorithms

Probability and algorithms enjoy an almost boisterous interaction that has led to an active, extensive literature that touches fields as diverse as number theory and the design of computer hardware. This article offers a gentle introduction to the simplest, most basic ideas that underlie this development

Multicritical continuous random trees

We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this k-th order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-positive universal weights at the branching points together with fractional derivative couplings. We prove universality by rederiving the same results within a purely continuous axiomatic approach based on the resolution of a set of consistency relations for the multi-point correlations. The average profile is shown to obey a fractional differential equation whose solution involves hypergeometric functions and matches the integral formula of the discrete approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps

Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

Optimal spatial transportation networks where link-costs are sublinear in link-capacity

Consider designing a transportation network on $n$ vertices in the plane, with traffic demand uniform over all source-destination pairs. Suppose the cost of a link of length $\ell$ and capacity $c$ scales as $\ell c^\beta$ for fixed $0<\beta<1$. Under appropriate standardization, the cost of the minimum cost Gilbert network grows essentially as $n^{\alpha(\beta)}$, where $\alpha(\beta) = 1 - \frac{\beta}{2}$ on $0 < \beta \leq {1/2}$ and $\alpha(\beta) = {1/2} + \frac{\beta}{2}$ on ${1/2} \leq \beta < 1$. This quantity is an upper bound in the worst case (of vertex positions), and a lower bound under mild regularity assumptions. Essentially the same bounds hold if we constrain the network to be efficient in the sense that average route-length is only $1 + o(1)$ times average straight line length. The transition at $\beta = {1/2}$ corresponds to the dominant cost contribution changing from short links to long links. The upper bounds arise in the following type of hierarchical networks, which are therefore optimal in an order of magnitude sense. On the large scale, use a sparse Poisson line process to provide long-range links. On the medium scale, use hierachical routing on the square lattice. On the small scale, link vertices directly to medium-grid points. We discuss one of many possible variant models, in which links also have a designed maximum speed $s$ and the cost becomes $\ell c^\beta s^\gamma$.Comment: 13 page

Quantum speedup of classical mixing processes

Most approximation algorithms for #P-complete problems (e.g., evaluating the permanent of a matrix or the volume of a polytope) work by reduction to the problem of approximate sampling from a distribution $\pi$ over a large set $\S$. This problem is solved using the {\em Markov chain Monte Carlo} method: a sparse, reversible Markov chain $P$ on $\S$ with stationary distribution $\pi$ is run to near equilibrium. The running time of this random walk algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} \log 1/\pi_*)$ as shown by Aldous, where $\delta$ is the spectral gap of $P$ and $\pi_*$ is the minimum value of $\pi$. A natural question is whether a speedup of this classical method to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$, the diameter of the graph underlying $P$, is possible using {\em quantum walks}. We provide evidence for this possibility using quantum walks that {\em decohere} under repeated randomized measurements. We show: (a) decoherent quantum walks always mix, just like their classical counterparts, (b) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (c) the mixing time of the decoherent quantum walk on a periodic lattice $\Z_n^d$ is $O(n d \log d)$, which is indeed $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ and is asymptotically no worse than the diameter of $\Z_n^d$ (the obvious lower bound) up to at most a logarithmic factor.Comment: 13 pages; v2 revised several part

Random multi-index matching problems

The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.Comment: 22 pages, 16 figure

Prevalence and Trends in Transmitted and Acquired Antiretroviral Drug Resistance, Washington, DC, 1999-2014.

Background Drug resistance limits options for antiretroviral therapy (ART) and results in poorer health outcomes among HIV-infected persons. We sought to characterize resistance patterns and to identify predictors of resistance in Washington, DC. Methods We analyzed resistance in the DC Cohort, a longitudinal study of HIV-infected persons in care in Washington, DC. We measured cumulative drug resistance (CDR) among participants with any genotype between 1999 and 2014 (n = 3411), transmitted drug resistance (TDR) in ART-naïve persons (n = 1503), and acquired drug resistance (ADR) in persons with genotypes before and after ART initiation (n = 309). Using logistic regression, we assessed associations between patient characteristics and transmitted resistance to any antiretroviral. Results Prevalence of TDR was 20.5%, of ADR 40.5%, and of CDR 45.1% in the respective analysis groups. From 2004 to 2013, TDR prevalence decreased for nucleoside and nucleotide analogue reverse transcriptase inhibitors (15.0 to 5.5%; p = 0.0003) and increased for integrase strand transfer inhibitors (INSTIs) (0.0–1.4%; p = 0.04). In multivariable analysis, TDR was not associated with age, race/ethnicity, HIV risk group, or years from HIV diagnosis. Conclusions In this urban cohort of HIV-infected persons, almost half of participants tested had evidence of CDR; and resistance to INSTIs was increasing. If this trend continues, inclusion of the integrase-encoding region in baseline genotype testing should be strongly considered

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

Delay and poor diagnosis of Down syndrome in KwaZulu-Natal, South Africa: A retrospective review of postnatal cytogenetic testing

Background. Down syndrome (DS) is the most common chromosomal disorder in newborns. Until 20 years ago DS was considered rare in black African children in South Africa (SA). Lack of awareness of DS on the part of medical staff in SA, and difficulty in diagnosing it, appear to persist.Objectives. To establish an epidemiological profile of DS and investigate the ability of clinicians in KwaZulu-Natal Province (KZN), SA, to make accurate clinical diagnoses of DS.Methods. Records at the South African National Blood Service cytogenetic laboratory in Pinetown, KZN, were examined for all tests for clinically suspected DS undertaken during January 2009 - December 2013 and all cytogenetically proven DS test results. Age at diagnosis, the hospital from where the test was sent and type of chromosomal pattern for each confirmed DS test result were recorded.Results. Of a total of 1 578 tests requested, 875 confirmed DS, indicating that clinicians correctly clinically diagnosed DS 55.4% of the time. The average age of cytogenetic diagnosis of DS was 1 year and 20 days. The minimum population prevalence of DS was 0.8/1 000.Conclusions. The diagnosis of DS is a challenge in KZN, potentiating missed opportunities for early intervention. The relatively low population prevalence of DS may be attributable to a lack of confirmatory cytogenetic tests or missed clinical diagnoses. It may also be attributable to a high mortality rate for children with DS in the province