And/or trees: A local limit point of view

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 , 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 , the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of : 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).     

Selection by rank in K‐dimensional binary search trees

In this work we show how to augment general purpose multidimensional data structures, such as K‐d trees, to efficiently support search by rank (that is, to locate the i‐th smallest element along the j‐th coordinate, for given i and j) and to find the rank of a given item along a given coordinate. To do so, we introduce two simple, practical and very flexible algorithms – Select‐by‐Rank and Find‐Rank – with very little overhead. Both algorithms can be easily implemented and adapted to several spatial indexes, although their analysis is far from trivial. We are able to show that for random K‐d trees of size n the expected number of nodes visited by Find‐Rank is for or , and for (with ), where depends on the dimension K and the variant of K‐d tree under consideration. We also show that Select‐by‐Rank visits nodes on average, where is the given rank and the exponent α is as above. We give the explicit form of the functions and , both are bounded in [0, 1] and they depend on K, on the variant of K‐d tree under consideration, and, eventually, on the specific coordinate j for which we execute our algorithms. As a byproduct of the analysis of our algorithms, but no less important, we give the average‐case analysis of a partial match search in random K‐d trees when the query is not random. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 14–37, 2014     

Finding Adam in random growing trees

We investigate algorithms to find the first vertex in large trees generated by either the uniform attachment or preferential attachment model. We require the algorithm to output a set of K vertices, such that, with probability at least , the first vertex is in this set. We show that for any ε, there exist such algorithms with K independent of the size of the input tree. Moreover, we provide almost tight bounds for the best value of K as a function of ε. In the uniform attachment case we show that the optimal K is subpolynomial in , and that it has to be at least superpolylogarithmic. On the other hand, the preferential attachment case is exponentially harder, as we prove that the best K is polynomial in . We conclude the paper with several open problems. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 158–172, 2017     

The CRT is the scaling limit of random dissections

Abstract–We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform p‐angulations. As their number of vertices n goes to infinity, we show that these random graphs, rescaled by , converge in the Gromov–Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 304–327, 2015     

Spectra of large diluted but bushy random graphs

We compute an asymptotic expansion in of the limit in of the empirical spectral measure of the adjacency matrix of an Erd?s‐Rényi random graph with vertices and parameter . We present two different methods, one of which is valid for the more general setting of locally tree‐like graphs. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 160–184, 2016     

The threshold for combs in random graphs

For let denote the tree consisting of an ‐vertex path with disjoint ‐vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 794–802, 2016     

Graph limits of random graphs from a subset of connected k‐trees

For any set Ω of non‐negative integers such that , we consider a random Ω‐k‐tree G_n,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that G_n,k, scaled by where H_k is the kth harmonic number and σ_Ω > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐k‐tree to an infinite but locally finite random Ω‐k‐tree G_∞,k.     

Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process

This paper studies a higher dimensional generalization of Frieze's ‐limit theorem on the d‐Linial–Meshulam process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in . © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 315–340, 2017     

Cliques in dense inhomogeneous random graphs

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant of the Erd?s–Rényi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of for each fixed n , and give examples of graphons for which exhibits wild long‐term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably‐partite. © The Authors Random Structures & Algorithms Published byWiley Periodicals, Inc. Random Struct. Alg., 2016 © 2017 The Authors Random Structures & Algorithms Published by Wiley Periodicals, Inc. Random Struct. Alg., 51, 275–314, 2017     

Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial‐Meshulam model for random simplicial complexes on n vertices, where each d‐cell is added independently with probability p to the complete ‐skeleton. Under the assumption , we prove that the spectral gap between the smallest eigenvalues and the remaining eigenvalues is with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi‐Komlós‐type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 506–537, 2017     

Sizes of the largest clusters for supercritical percolation on random recursive trees

We consider Bernoulli bond‐percolation on a random recursive tree of size , with supercritical parameter for some fixed. We show that with high probability, the largest cluster has size close to whereas the next largest clusters have size of order only and are distributed according to some Poisson random measure. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 29–44, 2014     

Dependence and phase changes in random m‐ary search trees

We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all root‐key distances or over all root‐node distances) in random m‐ary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when , but becomes of higher order when . Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when , but is periodically oscillating for larger m, and we also prove asymptotic independence when . Such a less anticipated phenomenon is not exceptional and our results can be extended in two directions: one for more general shape parameters, and the other for other classes of random log‐trees such as fringe‐balanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 353–379, 2017     

The size of the giant high‐order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any and .     

Persistence of centrality in random growing trees

We investigate properties of node centrality in random growing tree models. We focus on a measure of centrality that computes the maximum subtree size of the tree rooted at each node, with the most central node being the tree centroid. For random trees grown according to a preferential attachment model, a uniform attachment model, or a diffusion processes over a regular tree, we prove that a single node persists as the tree centroid after a finite number of steps, with probability 1. Furthermore, this persistence property generalizes to the top K ≥ 1 nodes with respect to the same centrality measure. We also establish necessary and sufficient conditions for the size of an initial seed graph required to ensure persistence of a particular node with probability , as a function of ϵ: In the case of preferential and uniform attachment models, we derive bounds for the size of an initial hub constructed around the special node. In the case of a diffusion process over a regular tree, we derive bounds for the radius of an initial ball centered around the special node. Our necessary and sufficient conditions match up to constant factors for preferential attachment and diffusion tree models.     

Rainbow hamilton cycles in random graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erd?s‐Rényi random graph G_n,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs. In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of G_n,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014     

Online purchasing under uncertainty

Suppose there is a collection of independent uniform random variables, and a hypergraph of target structures on the vertex set . We would like to purchase a target structure at small cost, but we do not know all the costs x_i ahead of time. Instead, we inspect the random variables x_i one at a time, and after each inspection, choose to either keep the vertex i at cost x_i, or reject vertex i forever. In the present paper, we consider the case where is the edge‐set of a complete graph (or digraph), and the target structures are the spanning trees of a graph, spanning arborescences of a digraph, the paths between a fixed pair of vertices, perfect matchings, Hamilton cycles or the cliques of some fixed size.     

Improved bounds for the randomized decision tree Complexity of recursive majority

We consider the randomized decision tree complexity of the recursive 3‐majority function. We prove a lower bound of for the two‐sided‐error randomized decision tree complexity of evaluating height h formulae with error . This improves the lower bound of given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero‐error randomized decision tree algorithm that has complexity at most . The previous best known algorithm achieved complexity . The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel “interleaving” of two recursive algorithms. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 612–638, 2016     

The Brownian plane with minimal neck baby universe

For each , let be a uniform rooted quadrangulation, endowed with an appropriate measure, of size n conditioned to have r(n) vertices in its root block. We prove that for a suitable function r(n), after rescaling graph distance by converges to a random pointed non‐compact metric measure space , in the local Gromov‐Hausdorff‐Prokhorov topology. The space is built by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 729–752, 2017     

Local reconstruction of low‐rank matrices and subspaces

We study the problem of reconstructing a low‐rank matrix, where the input is an n × m matrix M over a field and the goal is to reconstruct a (near‐optimal) matrix that is low‐rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ‐close to a matrix of rank (together with d and ), this algorithm computes with high probability a rank‐d matrix that is ‐close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry by reading only the data structure and 2d additional entries of M. We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and , we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see: http://eccc.hpi-web.de/report/2015/128/ © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 607–630, 2017     

The height of depth‐weighted random recursive trees

In this paper, we introduce a model of depth‐weighted random recursive trees, created by recursively joining a new leaf to an existing vertex . In this model, the probability of choosing depends on its depth in the tree. In particular, we assume that there is a function such that if has depth then its probability of being chosen is proportional to . We consider the expected value of the diameter of this model as determined by , and for various increasing we find expectations that range from polylogarithmic to linear.