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In this note we analyze the performance of a simple root-finding algorithm in uniform attachment trees. The leaf-stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with probability $1 - \epsilon$, the set of remaining vertices contains the root and has a size only depending on $\epsilon$ but not on the size of the tree., Comment: 9 pages, 5 figures

We study the connected components in critical percolation on the Hamming hypercube $\{0,1\}^m$. We show that their sizes rescaled by $2^{-2m/3}$ converge in distribution, and that, considered as metric measure spaces with the graph distance rescaled by $2^{-m/3}$ and the uniform measure, they converge in distribution with respect to the Gromov-Hausdorff-Prokhorov topology. The two corresponding limits are as in critical Erd\H{o}s-R\'enyi graphs., Comment: 50 pages, 5 figures, comments are welcome

We give an explicit construction of the scaling limit of the minimum spanning tree of the complete graph. The limit object is described using a recursive construction involving the convex minorants of a Brownian motion with parabolic drift (and countably many i.i.d. uniform random variables); we call it the Brownian parabolic tree. Aside from the new representation, this point of view has multiple consequences. For instance, it permits us to prove that its Hausdorff dimension is almost surely 3. It also intrinsically contains information related to some underlying dynamics: one notable by-product is the construction of a standard metric multiplicative coalescent which couples the scaling limits of random graphs at different points of the critical window in terms of the same simple building blocks. The above results actually fit in a more general framework. They result from the introduction of a new family of continuum random trees associated with functions via their convex minorants, that we call convex minorant trees. We initiate the study of these structures in the case of Brownian-like paths. In passing, we prove that the convex minorant tree of a Brownian excursion is a Brownian continuum ranndom tree, and that it provides a coupling between the Aldous--Pitman fragmentation of the Brownian continuum random tree and its representation by Bertoin., Comment: 56 pages, 2 figures

We consider random temporal graphs, a version of the classical Erd\H{o}s--R\'enyi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We study the asymptotics for the distances in such graphs, mostly in the regime of interest where np is of order log n. We establish the first order asymptotics for the lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, the maxima of these lengths from a given vertex, as well as the maxima between any two vertices; this covers the (temporal) diameter., Comment: 26 pages

We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdos-Renyi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$ graphon. In particular, we define a notion of the critical window in this setting. The $L^3$ assumption ensures that the model is in the Erdos-Renyi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel's RGIV model [9] inside the critical window. Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs [52]. In [10], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Rucinski and Wormald's random graph process with degree restrictions [56] provided an additional technical condition about the barely subcritical behavior of this model can be proved., Comment: 67 pages, 1 figure, to appear in Transactions of the American Mathematical Society. The universality principle (Theorem 3.4) from arXiv:1411.3417 has now been included in this paper

We consider a pruning of the inhomogeneous continuum random trees, as well as the cut trees that encode the genealogies of the fragmentations that come with the pruning. We propose a new approach to the reconstruction problem, which has been treated for the Brownian CRT in [Electron. J. Probab. vol. 22, 2017] and for the stable trees in [Ann. IHP B, vol 55, 2019]. Our approach does not rely upon self-similarity and can potentially apply to general L\'evy trees as well.

Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$ are attached sequentially such that the parent of vertex $n$ is $n-Z_n$, where the $Z_n$ are i.i.d.\ random variables taking values in $\mathbb N$. We study several characteristics of the resulting random forest. In particular, we establish bounds for the expected tree sizes, the number of trees in the forest, the number of leaves, the maximum degree, and the height of the forest. We show that for all distributions of the $Z_n$, the forest contains at most one infinite tree, almost surely. If ${\mathbb E} Z_n < \infty$, then there is a unique infinite tree and the total size of the remaining trees is finite, with finite expected value if ${\mathbb E}Z_n^2 < \infty$. If ${\mathbb E} Z_n = \infty$ then almost surely all trees are finite.

In Batched Multi-Armed Bandits (BMAB), the policy is not allowed to be updated at each time step. Usually, the setting asserts a maximum number of allowed policy updates and the algorithm schedules them so that to minimize the expected regret. In this paper, we describe a novel setting for BMAB, with the following twist: the timing of the policy update is not controlled by the BMAB algorithm, but instead the amount of data received during each batch, called \textit{crowd}, is influenced by the past selection of arms. We first design a near-optimal policy with approximate knowledge of the parameters that we prove to have a regret in $\mathcal{O}(\sqrt{\frac{\ln x}{x}}+\epsilon)$ where $x$ is the size of the crowd and $\epsilon$ is the parameter error. Next, we implement a UCB-inspired algorithm that guarantees an additional regret in $\mathcal{O}\left(\max(K\ln T,\sqrt{T\ln T})\right)$, where $K$ is the number of arms and $T$ is the horizon., Comment: 31 pages

We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-R\'enyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 1997]. In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic [EJP 1998] have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain L\'evy-type processes. We, instead, look at the metric structure of these components and prove their Gromov-Hausdorff-Prokhorov convergence to a class of random compact measured metric spaces that have been introduced in a companion paper. Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general "critical" regime, and relies upon two key ingredients: an encoding of the graph by some L\'evy process as well as an embedding of its connected components into Galton-Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of L\'evy trees, which allows us to give an explicit construction of the limit objects from the excursions of the L\'evy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via regime-specific proofs, for instance: the case of Erd\H os-R\'enyi random graphs obtained by Addario-Berry, Goldschmidt and B. [PTRF 2012], the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang [PTRF 2017], or the power-law case as considered by Bhamidi, Sen and van der Hofstad [PTRF 2018]., Comment: 81 pages. arXiv admin note: substantial text overlap with arXiv:1804.05871

A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element $U$. The value of the root is the key quantity of interest in general. In this first study, all node values and function values are in a finite set $S$. In this note, we describe the limit behavior when the leaf values are drawn independently from a fixed distribution on $S$, and the tree $T_n$ is a random Galton--Watson tree of size $n$., Comment: 13 pages, 1 figure

Motivated by limits of critical inhomogeneous random graphs, we construct a family of sequences of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the Erd\H{o}s--R\'enyi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model). At the discrete level, the construction relies on a new point of view on (discrete) inhomogeneous random graphs that involves an embedding into a Galton--Watson forest. The new representation allows us to demonstrate that a processus that was already present in the pionnering work of Aldous [Ann. Probab., vol.~25, pp.~812--854, 1997] and Aldous and Limic [Electron. J. Probab., vol.~3, pp.~1--59, 1998] about the multiplicative coalescent actually also (essentially) encodes the limiting metric: The discrete embedding of random graphs into a Galton--Watson forest is paralleled by an embedding of the encoding process into a L\'evy process which is crucial in proving the very existence of the local time functionals on which the metric is based; it also yields a transparent approach to compactness and fractal dimensions of the continuous objects. In a companion paper, we show that the continuous L\'evy graphs are indeed the scaling limit of inhomogeneous random graphs., Comment: 48 pages; paper has been split into two

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for $2$-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an old question of Chanzy, Devroye and Zamora-Cura [\emph{Acta Inf.}, 37:355--383, 2000], Comment: 24 pages, 8 figures

The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let $(V_i)_{1\le i\le n}$ be an i.i.d.\ family of random variables with uniform distribution on $[0,1]$; shoot $n$ bullets one after another at times $1,2,\dots, n$, where the $i$th bullet has speed $V_i$. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalisation of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate., Comment: 29 pages

We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed-points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially efficient to treat cases for which the mass measure on the real tree induced by natural encodings only provides weak estimates on the Hausdorff dimension., Comment: 49 pages including an appendix of 8 pages, 5 figures

We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer $k$, take a sequence of random (rooted) trees of increasing size, say $(t_n)_{n\ge 1}$, and label each of these random trees uniformly at random in order to get a random Boolean expression on $k$ variables. We prove that, under rather weak local conditions on the sequence of random trees $(t_n)_{n\ge 1}$, the distribution induced on Boolean functions by this procedure converges as $n$ tends to infinity. In particular, we characterise two different behaviours of this limit distribution depending on the shape of the local limit of $(t_n)_{n\ge 1}$: a degenerate case when the local limit has no leaves; and a non-degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton--Watson trees).

A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large class of models, the nature of this emergence is similar to that of the Erd\H{o}s-R\'enyi random graph, in the sense that (a) the sizes of the maximal components in the critical regime scale like $n^{2/3}$, and (b) the structure of the maximal components at criticality (rescaled by $n^{-1/3}$) converges to random fractals. To date, (a) has been proven for a number of models using different techniques. This paper develops a general program for proving (b) that requires three ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent, (ii) scaling exponents of susceptibility functions are the same as that of the Erd\H{o}s-R\'enyi random graph, and (iii) macroscopic averaging of distances between vertices in the barely subcritical regime. We show that these apply to two fundamental random graph models: the configuration model and inhomogeneous random graphs with a finite ground space. For these models, we also obtain new results for component sizes at criticality and structural properties in the barely subcritical regime., Comment: 68 pages. The section on bounded-size rules has been omitted. It will be included in a future paper

We consider graphs with vertices of degree 1 or 2 and prove that the numbers of components of sizes 2 to q have a limit normal distribution for any q > 1. The result is also extended to multigraphs., Comment: p. 12

We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $n\to +\infty$. We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous [Ann. Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time $\lambda$, converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $(B_t+\lambda t - t^2/2, t\geq 0)$, but we also construct a version of the standard augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel., to appear] using an additional Poisson point process., Comment: 29 pages, 3 figures

Consider the Aldous--Pitman fragmentation process [Ann Probab, 26(4):1703--1726, 1998] of a Brownian continuum random tree ${\cal T}^{\mathrm{br}}$. The associated cut tree cut$({\cal T}^{\mathrm{br}})$, introduced by Bertoin and Miermont [Ann Appl Probab, 23:1469--1493, 2013], is defined in a measurable way from the fragmentation process, describing the genealogy of the fragmentation, and is itself distributed as a Brownian CRT. In this work, we introduce a shuffle transform, which can be considered as the reverse of the map taking ${\cal T}^{\mathrm{br}}$ to cut$({\cal T}^{\mathrm{br}})$., Comment: 22 pages, 13 figures

We study a fragmentation of the $\mathbf p$-trees of Camarri and Pitman [Elect. J. Probab., vol. 5, pp. 1--18, 2000]. We give exact correspondences between the $\mathbf p$-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the ICRTs (scaling limits of $\mathbf p$-trees) and give distributional correspondences between the ICRT and the tree encoding the fragmentation. The theorems for the ICRT extend the ones by Bertoin and Miermont [Ann. Appl. Probab., vol. 23(4), pp. 1469--1493, 2013] about the cut tree of the Brownian continuum random tree., Comment: 44 pages, 6 figures

We consider vertex partitions of the binomial random graph $G_{n,p}$. For $np\to\infty$, we observe the following phenomenon: in any partition into asymptotically fewer than $\chi(G_{n,p})$ parts, i.e. $o(np/\log np)$ parts, one part must induce a connected component of order at least roughly the average part size. Stated another way, we consider the $t$-component chromatic number, the smallest number of colours needed in a colouring of the vertices for which no monochromatic component has more than $t$ vertices. As long as $np \to \infty$, there is a threshold for $t$ around $\Theta(p^{-1}\log np)$: if $t$ is smaller then the $t$-component chromatic number is nearly as large as the chromatic number, while if $t$ is greater then it is around $n/t$. For $0 < p <1$ fixed, we obtain more precise information. We find something more subtle happens at the threshold $t = \Theta(\log n)$, and we determine that the asymptotic first-order behaviour is characterised by a non-smooth function. Moreover, we consider the $t$-component stability number, the maximum order of a vertex subset that induces a subgraph with maximum component order at most $t$, and show that it is concentrated in a constant length interval about an explicitly given formula, so long as $t = O(\log \log n)$. We also consider a related Ramsey-type parameter and use bounds on the component stability number of $G_{n,1/2}$ to describe its basic asymptotic growth., Comment: 23 pages, 1 figure; v2 accepted to Journal of Combinatorics

A random geometric irrigation graph $\Gamma_n(r_n,\xi)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_1,\ldots,X_n$ in the unit square $[0,1]^2$. Each point $X_i$ selects $\xi_i$ neighbors at random, without replacement, among those points $X_j$ ($j\neq i$) for which $\|X_i-X_j\| < r_n$, and the selected vertices are connected to $X_i$ by an edge. The number $\xi_i$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_i$ such that $\xi_i$ satisfies $1\le \xi_i \le \kappa$ for a constant $\kappa>1$. We prove that when $r_n = \gamma_n \sqrt{\log n/n}$ for $\gamma_n \to \infty$ with $\gamma_n =o(n^{1/6}/\log^{5/6}n)$, then the random geometric irrigation graph experiences explosive percolation in the sense that when $\mathbf E \xi_i=1$, then the largest connected component has size $o(n)$ but if $\mathbf E \xi_i >1$, then the size of the largest connected component is with high probability $n-o(n)$. This offers a natural non-centralized sparsification of a random geometric graph that is mostly connected.

Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant competitiveness which follows vertex adjacencies in the Delaunay triangulation. We call this strategy cone walk. We prove that given $n$ uniform points in a smooth convex domain of unit area, and for any start point $z$ and query point $q$; cone walk applied to $z$ and $q$ will access at most $O(|zq|\sqrt{n} +\log^7 n)$ sites with complexity $O(|zq|\sqrt{n} \log \log n + \log^7 n)$ with probability tending to 1 as $n$ goes to infinity. We additionally show that in this model, cone walk is $(\log ^{3+\xi} n)$-memoryless with high probability for any pair of start and query point in the domain, for any positive $\xi$. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border.

Consider a random geometric graph defined on $n$ vertices uniformly distributed in the $d$-dimensional unit torus. Two vertices are connected if their distance is less than a "visibility radius" $r_n$. We consider {\sl Bluetooth networks} that are locally sparsified random geometric graphs. Each vertex selects $c$ of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of $n^{-(1-\delta)/d}$ for some $\delta > 0$, then a constant value of $c$ is sufficient for the graph to be connected, with high probability. It suffices to take $c \ge \sqrt{(1+\epsilon)/\delta} + K$ for any positive $\epsilon$ where $K$ is a constant depending on $d$ only. On the other hand, with $c\le \sqrt{(1-\epsilon)/\delta}$, the graph is disconnected, with high probability.

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its vertices. We show that the resulting space converges in distribution, as n tends to infinity, to a random measured metric space in the Gromov-Hausdorff-Prokhorov topology. We additionally show that the limit is a random binary R-tree and has Minkowski dimension 3 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erd\"os-R\'enyi random graph. We exploit the explicit description of the scaling limit of the Erd\"os-R\'enyi random graph in the so-called critical window, established by the first three authors in an earlier paper, and provide a similar description of the scaling limit for a "critical minimum spanning forest" contained within the MST., Comment: 60 pages, 4 figures

In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224-2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov-Hausdorff sense to a limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms a conjecture of Curien and Le Gall., Comment: Published in at http://dx.doi.org/10.1214/13-AOP894 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and $k$-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_n(\xi )$ to visit in order to report the items matching a random query $\xi$, independent and uniformly distributed on $[0,1]$, satisfies $\mathbf {E}[{C_n(\xi )}]\sim\kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(s)$ of any fixed query $s\in[0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit us to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $\mathbf {E}[{\max_{x\in[0,1]}C_n(x)}]\sim \gamma n^{\beta}$; this settles a question of Devroye [Pers. Comm., 2000]., Comment: Published in at http://dx.doi.org/10.1214/12-AAP912 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1107.2231

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton-Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny $n$. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton-Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge., Comment: Published in at http://dx.doi.org/10.1214/13-AAP978 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The degree sequence $(n_i(t),i\geq 0)$ of $t$ satisfies $\sum_{i\ge 0} n_i(t)=1+\sum_{i\ge 0} in_i(t)=|t|$, where $|t|$ denotes the number of nodes in $t$. In this paper, we consider trees sampled uniformly among all trees having the same degree sequence $\ds$; we write $`P_\ds$ for the corresponding distribution. Let $\ds(\kappa)=(n_i(\kappa),i\geq 0)$ be a list of degree sequences indexed by $\kappa$ corresponding to trees with size $\nk\to+\infty$. We show that under some simple and natural hypotheses on $(\ds(\kappa),\kappa>0)$ the trees sampled under $`P_{\ds(\kappa)}$ converge to the Brownian continuum random tree after normalisation by $\nk^{1/2}$. Some applications concerning Galton--Watson trees and coalescence processes are provided.

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quad trees and k-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_n(\xi)$ to visit in order to report the items matching an independent and uniformly on $[0,1]$ random query $\xi$ satisfies $\Ec{C_n(\xi)}\sim \kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(x)$ of any fixed query $x\in [0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $E{\max_{x\in [0,1]} C_n(x)} \sim \gamma n^\beta$., Comment: 12 pages, 2 figures

We study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that control the model: the radius $r$ that determines the "visible neighbors" of each node and the number of edges $c$ that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters $r, c$ and completely characterize the connectivity threshold (in $c$) for values of $r$ close the critical value for connectivity in the underlying random geometric graph., Comment: 21 pages, 5 figures

We consider the model of random trees introduced by Devroye [SIAM J. Comput. 28 (1999) 409-432]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths of the items) is a natural measure of the efficiency of the algorithm/data structure. Using renewal theory, we prove convergence in distribution of the total path length toward a distribution characterized uniquely by a fixed point equation. Our result covers, using a unified approach, many data structures such as binary search trees, m-ary search trees, quad trees, median-of-(2k+1) trees, and simplex trees., Comment: Published in at http://dx.doi.org/10.1214/11-AAP812 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random DAG. This completes the study of natural distances in random DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also obtain large deviation bounds for the minimum of a branching random walk with constant branching, which can be seen as a simplified version of our main result., Comment: 21 pages, 2 figures

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height.

We study a class of hypothesis testing problems in which, upon observing the realization of an $n$-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been ``contaminated,'' that is, have a mean different from zero. We establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk. The combinatorial and geometric structure of the class of sets is shown to play a crucial role. The bounds are illustrated on various examples., Comment: Published in at http://dx.doi.org/10.1214/10-AOS817 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a branching random walk for which the maximum position of a particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n --> 0 as n --> infinity. We further remove ("kill") any particle whose displacement is negative, together with its entire descendence. The size $Z$ of the set of un-killed particles is almost surely finite. In this paper, we confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log Z]=infinity. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks., Comment: 21 pages, 1 figure

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, that is when p=1/n+ lambda*n^{-4/3}, for some fixed lambda in R. Then, as a metric space with the graph distance rescaled by n^{-1/3}, the sequence of connected components G(n,p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n,p) rescaled by n^{-1/3} converges in distribution to an absolutely continuous random variable with finite mean., Comment: 34 pages, 5 figures

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about \alpha^* \log n$ edges where \alpha^* ~ 3.5911 is the unique solution of the equation $alpha log(alpha) - \alpha =1. This answers a question posed by Janson (1999)., Comment: 21 pages; minor corrections and clarifications

This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height., Comment: 14 pages

We investigate the effective resistance $R_n$ and conductance $C_n$ between the root and leaves of a binary tree of height $n$. In this electrical network, the resistance of each edge $e$ at distance $d$ from the root is defined by $r_e=2^dX_e$ where the $X_e$ are i.i.d. positive random variables bounded away from zero and infinity. It is shown that $\mathbf{E}R_n=n\mathbf{E}X_e-(\operatorname {\mathbf{Var}}(X_e)/\mathbf{E}X_e)\ln n+O(1)$ and $\operatorname {\mathbf{Var}}(R_n)=O(1)$. Moreover, we establish sub-Gaussian tail bounds for $R_n$. We also discuss some possible extensions to supercritical Galton--Watson trees., Comment: Published in at http://dx.doi.org/10.1214/08-AAP572 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)