Refined quicksort asymptotics

The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Y_n is known to converge almost surely to a non‐degenerate random limit Y. This assumes a natural embedding of all Y_n on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown: where denotes a standard normal random variable. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 346–361, 2015     

Coloring graphs from random lists of fixed size

Let G = G(n) be a graph on n vertices with maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as ) of the existence of a proper coloring ? of G, such that for every vertex v of G. We show, for all fixed k and growing n, that if , then the probability that G has such a proper coloring tends to 1 as . A similar result for complete graphs is also obtained: if and L is a random ‐list assignment for the complete graph K_n on n vertices, then the probability that K_n has a proper coloring with colors from the random lists tends to 1 as .Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317‐327, 2014     

When does the K4‐free process stop?

The K₄‐free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K₄. Let G be the random maximal K₄‐free graph obtained at the end of the process. We show that for some positive constant C, with high probability as , the maximum degree in G is at most . This resolves a conjecture of Bohman and Keevash for the K₄‐free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has edges and is ‘nearly regular’, i.e., every vertex has degree . This answers a question of Erd?s, Suen and Winkler for the K₄‐free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least , which matches an upper bound obtained by Bohman up to a factor of . Our analysis of the K₄‐free process also yields a new result in Ramsey theory: for a special case of a well‐studied function introduced by Erd?s and Rogers we slightly improve the best known upper bound.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 355‐397, 2014     

Group representations that resist random sampling

We show that there exists a family of groups G_n and nontrivial irreducible representations ρ_n such that, for any constant t, the average of ρ_n over t uniformly random elements has operator norm 1 with probability approaching 1 as . More quantitatively, we show that there exist families of finite groups for which random elements are required to bound the norm of a typical representation below 1. This settles a conjecture of A. Wigderson. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 605–614, 2015     

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     

The diamond‐free process

Let denote the diamond graph, formed by removing an edge from the complete graph K₄. We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of . We show that, with probability tending to 1 as , the final size of the graph produced is . Our analysis also suggests that the graph produced after i edges are added resembles the uniform random graph, with the additional condition that the edges which do not lie on triangles form a random‐looking subgraph. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 513–551, 2014     

Increasing Hamiltonian paths in random edge orderings

Let f be an edge ordering of K_n: a bijection . For an edge , we call f(e) the label of e. An increasing path in K_n is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvátal and Komlós raised the question of estimating m(n): the minimum value of S(f) over all orderings f of K_n. The best known bounds on m(n) are , due respectively to Graham and Kleitman, and to Calderbank, Chung, and Sturtevant. Although the problem is natural, it has seen essentially no progress for three decades. In this paper, we consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, S(f) often takes its maximum possible value of n – 1 (visiting all of the vertices with an increasing Hamiltonian path). We prove that this occurs with probability at least about 1/ e. We also prove that with probability 1‐ o(1), there is an increasing path of length at least 0.85 n, suggesting that this Hamiltonian (or near‐Hamiltonian) phenomenon may hold asymptotically almost surely. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 588–611, 2016     

Mantel's theorem for random hypergraphs

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle‐free subgraph of K_n is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large p, every maximum triangle‐free subgraph of G(n, p) is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for for some constant K, and apart from the value of the constant this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by F₅, which is sometimes called the generalized triangle, the 3‐uniform hypergraph with vertex set and edge set . One of the first results in extremal hypergraph theory is by Frankl and Füredi, who proved that the maximum 3‐uniform hypergraph on n vertices containing no copy of F₅ is tripartite for n > 3000. A natural question is for what p is every maximum F₅‐free subhypergraph of w.h.p. tripartite. We show this holds for for some constant K and does not hold for . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 641–654, 2016     

Limit theorems for monochromatic stars

Let T(K_1,r,G_n) be the number of monochromatic copies of the r‐star K_1,r in a uniformly random coloring of the vertices of the graph G_n. In this paper we provide a complete characterization of the limiting distribution of T(K_1,r,G_n), in the regime where is bounded, for any growing sequence of graphs G_n. The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most r. Conversely, any limiting distribution of T(K_1,r,G_n) has a representation of this form. Examples and connections to the birthday problem are discussed.     

Note on a problem of M. Talagrand

Answering a question of M. Talagrand, we show that there is a fixed L with the following property. For positive integers and , if is the set of subgraphs of K_n containing at least copies of K_k, then there is a set of subgraphs of K_n such that (i) each member of contains a member of and (ii) (where means number of edges). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 663–668, 2015     

The thresholds for diameter 2 in random Cayley graphs

Given a group G, the model denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any and any family of groups G_k of order n_k for which , a graph with high probability has diameter at most 2 if and with high probability has diameter greater than 2 if . We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erd?s‐Renyi random graphs. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 218–235, 2014     

On connectivity,conductance and bootstrap percolation for a random k‐out,age‐biased graph

A uniform attachment graph (with parameter k), denoted G_n,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v ? 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of G_n,k to show that the conductance of G_n,k is of order . We also study the bootstrap percolation on G_n,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.     

On the maximal multiplicity of block sizes in a random set partition

We study the asymptotic behavior of the maximal multiplicity M_n = M_n(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with We^W = n. We show that, over subsequences {n_k}_{k ≥ 1} of the sequence of the natural numbers, , appropriately normalized, converges weakly, as k→∞, to , where Z₁ and Z₂ are independent copies of a standard normal random variable. The subsequences {n_k}_{k ≥ 1}, where the weak convergence is observed, and the quantity u depend on the fractional part f_n of the function W(n). In particular, we establish that . The behavior of the largest multiplicity M_n is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number and clique number is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that P_n contains any given graph H as an induced subgraph is asymptotically 0 or or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon .     

High degrees in random recursive trees

For , let T_n be a random recursive tree (RRT) on the vertex set . Let be the degree of vertex v in T_n, that is, the number of children of v in T_n. Devroye and Lu showed that the maximum degree Δ_n of T_n satisfies almost surely; Goh and Schmutz showed distributional convergence of along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in T_n. For any , let . Also, let be a Poisson point process on with rate function . We show that, up to lattice effects, the vectors converge weakly in distribution to . We also prove asymptotic normality of when slowly, and obtain precise asymptotics for when and is not too large. Our results recover and extends the previous distributional convergence results on maximal and near‐maximal degrees in RRT.     

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 .     

The time of graph bootstrap percolation

Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a “small” graph H and a “large” graph , in consecutive steps we obtain from G_t by adding to it all new edges e such that contains a new copy of H. We say that G percolates if for some , we have G_t = K_n. For H = K_r, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollobás and Morris considered graph bootstrap percolation for and studied the critical probability , for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time t for all © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 143–168, 2017     

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     

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     

Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when . We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i, j) is added with probability , where is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 408‐420, 2014