Multiple choice tries and distributed hash tables

In this article we consider tries built from n strings such that each string can be chosen from a pool of k strings, each of them generated by a discrete i.i.d. source. Three cases are considered: k = 2, k is large but fixed, and k ? clog n. The goal in each case is to obtain tries as balanced as possible. Various parameters such as height and fill‐up level are analyzed. It is shown that for two‐choice tries a 50% reduction in height is achieved when compared with ordinary tries. In a greedy online construction when the string that minimizes the depth of insertion for every pair is inserted, the height is only reduced by 25%. To further reduce the height by another 25%, we design a more refined online algorithm. The total computation time of the algorithm is O(nlog n). Furthermore, when we choose the best among k ≥ 2 strings, then for large but fixed k the height is asymptotically equal to the typical depth in a trie. Finally, we show that further improvement can be achieved if the number of choices for each string is proportional to log n. In this case highly balanced trees can be constructed by a simple greedy algorithm for which the difference between the height and the fill‐up level is bounded by a constant with high probability. This, in turn, has implications for distributed hash tables, leading to a randomized ID management algorithm in peer‐to‐peer networks such that, with high probability, the ratio between the maximum and the minimum load of a processor is O(1). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Randomly coloring sparse random graphs with fewer colors than the maximum degree

We analyze Markov chains for generating a random k‐coloring of a random graph G_n,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k‐coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Vertices of given degree in series‐parallel graphs

We show that the numbers of vertices of a given degree k ≥ 1 in several kinds of series‐parallel labeled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labeled planar graphs. The proof method is based on generating functions and singularity analysis. In particular, we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

De Bruijn cycles for covering codes

A de Bruijn covering code is a q‐ary string S so that every q‐ary string is at most R symbol changes from some n‐word appearing consecutively in S. We introduce these codes and prove that they can have size close to the smallest possible covering code. The proof employs tools from field theory, probability, and linear algebra. Included is a table of the best known bounds on the lengths of small binary de Bruijn covering codes, up to R = 11 and n = 13, followed by several open questions in this area. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

A constructive approach for the lower bounds on the Ramsey numbers R (s,t)

Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices K_k, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some K_k?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004     

Probabilistic behavior of asymmetric level compressed tries

Level‐Compressed (in short LC) tries were introduced by Andersson and Nilsson in 1993. They are compacted versions of tries in which, from the top down, maximal height complete subtrees are level compressed. We show that when the input consists of n independent strings with independent Bernoulli (p) bits, p ≠ 1/2, then the expected depth of a typical node is in probability asymptotic to where H ? p log p ? (1 ? p) log (1 ? p) is the Shannon entropy of the source, and H_?∞ = log (1 / min(p, 1 ? p)). The height is in probability asymptotic to where H₂ = log(1/(p² + (1?p)²)). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Simple Constructions of Almost k-wise Independent Random Variables

We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1)) (log log n + k/2 + log k + log 1/?), where ? is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ? < 1/(k log n)). An additional advantage of our constructions is their simplicity.     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Let F _q[X] denote a polynomial ring over a finite field F _q with q elements. Let 𝒫_n be the set of monic polynomials over F _q of degree n. Assuming that each of the qⁿ possible monic polynomials in 𝒫_n is equally likely, we give a complete characterization of the limiting behavior of P(Ω_n=m) as n→∞ by a uniform asymptotic formula valid for m≥1 and n−m→∞, where Ω_n represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in 𝒫_n. The distribution of Ω_n is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q⁻¹. Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerning P(Ω_n=m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Rényi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 17–47, 1998     

List circular coloring of trees and cycles

Suppose G=(V, E) is a graph and p ≥ 2q are positive integers. A (p, q)‐coloring of G is a mapping ?: V → {0, 1, …, p‐1} such that for any edge xy of G, q ≤ |?(x)‐?(y)| ≤ p‐q. A color‐list is a mapping L: V → ({0, 1, …, p‐1}) which assigns to each vertex v a set L(v) of permissible colors. An L‐(p, q)‐coloring of G is a (p, q)‐coloring ? of G such that for each vertex v, ?(v) ∈ L(v). We say G is L‐(p, q)‐colorable if there exists an L‐(p, q)‐coloring of G. A color‐size‐list is a mapping ? which assigns to each vertex v a non‐negative integer ?(v). We say G is ?‐(p, q)‐colorable if for every color‐list L with |L(v)| = ?(v), G is L‐(p, q)‐colorable. In this article, we consider list circular coloring of trees and cycles. For any tree T and for any p ≥ 2q, we present a necessary and sufficient condition for T to be ?‐(p, q)‐colorable. For each cycle C and for each positive integer k, we present a condition on ? which is sufficient for C to be ?‐(2k+1, k)‐colorable, and the condition is sharp. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 249–265, 2007     

Asymptotic enumeration of labeled multigraphs by vertices,edges, and degree parities

We show that the number of labeled (n, q)-multigraphs with some specified p vertices of odd degree is asymptotically independent of p and is in fact asymptotically 2^1?n times the number of (n, q)-multigraphs. We determine the asymptotic number of (n, q)-multigraphs.     

Functional limit theorems for digital expansions

The main purpose of this paper is to discuss the asymptotic behaviour of the difference s _q,k(P(n)) - k(q-1)/2 where s _q,k (n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q ₁-ary and q ₂-ary digital expansions where q ₁ and q ₂ are coprime. This revised version was published online in June 2006 with corrections to the Cover Date.     

Types of superregular matrices and the number of n‐arcs and complete n‐arcs in PG (r,q)

Based on the classification of superregular matrices, the numbers of non‐equivalent n‐arcs and complete n‐arcs in PG(r, q) are determined (i) for 4 ≤ q ≤ 19, 2 ≤ r ≤ q ? 2 and arbitrary n, (ii) for 23 ≤ q ≤ 32, r = 2 and n ≥ q ? 8<$>. The equivalence classes over both PGL (k, q) and PΓL(k, q) are considered throughout the examinations and computations. For the classification, an n‐arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non‐singular square submatrices. Four types of superregular matrices are studied and the non‐equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m′(r, q)—the smallest and the second largest size for complete arcs in PG(r, q)—are also reported, stating that m′(2, 31) = 22, m′(2, 32) = 24, t(3, 23) = 10, and m′(3, 23) = 16. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006     

Randomly coloring constant degree graphs

We study a simple Markov chain, known as the Glauber dynamics, for generating a random k ‐coloring of an n ‐vertex graph with maximum degree Δ. We prove that, for every ε > 0, the dynamics converges to a random coloring within O(nlog n) steps assuming k ≥ k₀(ε) and either: (i) k/Δ > α* + ε where α*≈? 1.763 and the girth g ≥ 5, or (ii) k/Δ >β * + ε where β*≈? 1.489 and the girth g ≥ 7. Our work improves upon, and builds on, previous results which have similar restrictions on k/Δ and the minimum girth but also required Δ = Ω (log n). The best known result for general graphs is O(nlog n) mixing time when k/Δ > 2 and O(n²) mixing time when k/Δ > 11/6. Related results of Goldberg et al apply when k/Δ > α* for all Δ ≥ 3 on triangle‐free “neighborhood‐amenable” graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Packing and covering constants for certain families of trees. I

If ? denotes a family of rooted trees, let p_k(n) and c_k(n) denote the average value of the k-packing and k-covering numbers of trees in ? that have n nodes. We assume, among other things, that the generating function y of trees in ? satisfies a relation of the type y = x?(y) for some power series ?. We show that the limits of p_k(n)/n and c_k(n)/n as n → ∞ exist and we describe how to evaluate these limits.     

The rank of sparse random matrices over finite fields

Let M be a random (n×n)-matrix over GF[q] such that for each entry M_ij in M and for each nonzero field element α the probability Pr[M_ij=α] is p/(q−1), where p=(log n−c)/n and c is an arbitrary but fixed positive constant. The probability for a matrix entry to be zero is 1−p. It is shown that the expected rank of M is n−𝒪(1). Furthermore, there is a constant A such that the probability that the rank is less than n−k is less than A/q^k. It is also shown that if c grows depending on n and is unbounded as n goes to infinity, then the expected difference between the rank of M and n is unbounded. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 407–419, 1997     

Second moments of Dirichlet L-functions weighted by Kloosterman sums

For the general modulo q ? 3 and a general multiplicative character χ modulo q, the upper bound estimate of |S(m, n, 1, χ, q)| is a very complex and difficult problem. In most cases, the Weil type bound for |S(m, n, 1, χ, q)| is valid, but there are some counterexamples. Although the value distribution of |S(m, n, 1, χ, q)| is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for k-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet L-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y.Yi, X.He: On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.     

Approximate Hamilton decompositions of random graphs

We show that if pn ? log n the binomial random graph G_n,p has an approximate Hamilton decomposition. More precisely, we show that in this range G_n,p contains a set of edge‐disjoint Hamilton cycles covering almost all of its edges. This is best possible in the sense that the condition that pn ? log n is necessary. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The mixing time of Glauber dynamics for coloring regular trees

We consider Metropolis Glauber dynamics for sampling proper q‐colorings of the n‐vertex complete b‐ary tree when 3 ≤ q ≤ b/(2lnb). We give both upper and lower bounds on the mixing time. Our upper bound is n^{O(b/ log b)} and our lower bound is n^{Ω(b/(q log b))}, where the constants implicit in the O() and Ω() notation do not depend upon n, q or b. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010