Sorting in linear expected time

A new sorting algorithm, Double Distributive Partitioning, is introduced and compared against Sedgewick's quicksort. It is shown that the Double Distributive Partitioning algorithm runs, for all practical purposes, inO(n) time for many distributions of keys. Furthermore, the combined number of comparisons, additions, and assignments required to sort by the new method on these distributions is always less than quicksort.     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.     

Symmetric edge-decompositions of hypercubes

We call a set of edgesE of the n-cubeQ _n a fundamental set for Q _n if for some subgroupG of the automorphism group ofQ _n , theG-translates ofE partition the edge set ofQ _n .Q _n possesses an abundance of fundamental sets. For example, a corollary of one of our main results is that if |E| =n and the subgraph induced byE is connected, then if no three edges ofE are mutually parallel,E is a fundamental set forQ _n . The subgroupG is constructed explicitly. A connected graph onn edges can be embedded intoQ _n so that the image of its edges forms such a fundamental set if and only if each of its edges belongs to at most one cycle.We also establish a necessary condition forE to be a fundamental set. This involves a number-theoretic condition on the integersa _j (E), where for 1 j n, a _j (E) is the number of edges ofE in thej ^th direction (i.e. parallel to thej ^th coordinate axis).     

Irrationality of linear combinations of eigenvectors

A givenn ×n matrix of rational numbers acts onC ^π and onQ ^π. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors forC ^π with the standard basis forQ ^π. Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.     

0,1 distribution in the highest level sequences of primitive sequences over<Emphasis Type="Italic">Z</Emphasis><Subscript>2e</Subscript>

In this paper, we discuss the 0,1 distribution in the highest level sequence a_e-1 of primitive sequence over Z_2e generated by a primitive polynomial of degreen. First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight fore relatively small ton. We also get an estimate which is suitable fore relatively large ton. Combining the two bounds, we obtain an estimate depending only onn, which shows that the largern is, the closer to 1/2 the proportion of 1 will be.     

Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.     

Distribution modulo 1 of some oscillating sequences

Sequences of the form (P(n)f(Q(n))) _n=1 ^∞ ,P andQ polynomials,f a “highly differentiable” periodic function, are considered. The results of [3] concerning the recurrence of this sequence to its value forn=0 are given a quantitative form. Density and uniform distribution modulo 1 are studied for specialQ’s.     

Bounds for the crossing number of the N-cube

Let Q_n denote the n-dimensional hypercube. In this paper we derive upper and lower bounds for the crossing number v(Q_n), i.e., the minimum number of edge-crossings in any planar drawing of Q_n. The upper bound is close to a result conjectured by Eggleton and Guy and the lower bound is a significant improvement over what was previously known. Let N = 2ⁿ be the number of vertices of Q_n. We show that v(Q_n) < 1/6N². For the lower bound we prove that v(Q_n) = Ω(N(lg N)^{c lg lg N}), where c > 0 is a constant and lg is the logarithm base 2. The best lower bound using standard arguments is v(Q_n) = Ω(N(lg N)²). The lower bound is obtained by constructing a large family of homeomorphs of a subcube with the property that no given pair of edges can appear in more than a constant number of the homeomorphs.     

Hypercube subgraphs with minimal detours

Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Q_n having the property that for vertices x, y of Q_n, distances are related by d_G(x, y) ≤ d_Qn(x, y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Q_n. After preliminary work on distances in subgraphs of product graphs, we show that The subgraphs we construct to establish this bound have the property that the longest distances are the same as in Q_n, and thus the diameter does not increase. We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Q_n, and end with conjectures and questions. © 1996 John Wiley & Sons, Inc.     

A note on sufficient conditions forQ 0 andQ 0 ⋂P 0 matrices

In this note we settle an open problem posed by Al-Khayyal on a condition being sufficient for a matrix to belong to the class ofQ ₀-matrices. The answer is in the affirmative and we further relax the condition and obtain a sufficient condition forQ ₀-matrices. The results yield a class of matrices for which the linear complementarity problems can be solved as simple linear programs.     

Poisson Convergence in the n-Cube

We consider two types of random subgraphs of the n-cube Q_n obtained by independent deletion the vertices (together with all edges incident with them) or the edges of Qⁿ, respectively, with a prescribed probability q = 1 — p. For these two probabilistic models we determine some values of the probability p for which the number of (isolated) k-dimensional subcubes or the number of vertices of a given degree k, respectively, has asymptotically a Poisson or a Normal distribution. The technique which will be used is that of Poisson convergence introduced by BARBOUR [1] (see also [4]).     

Large Deviations Rate Function for Polling Systems

In this paper, we identify the local rate function governing the sample path large deviation principle for a rescaled process n ^–1 Q _nt , where Q _t represents the joint number of clients at time t in a polling system with N nodes, one server and Markovian routing. By the way, the large deviation principle is proved and the rate function is shown to have the form conjectured by Dupuis and Ellis. We introduce a so called empirical generator consisting of Q _t and of two empirical measures associated with S _t , the position of the server at time t. One of the main step is to derive large deviations bounds for a localized version of the empirical generator. The analysis relies on a suitable change of measure and on a representation of fluid limits for polling systems. Finally, the rate function is solution of a meaningful convex program. The method seems to have a wide range of application including the famous Jackson networks, as shown at the end of this study. An example illustrates how this technique can be used to estimate stationary probability decay rate.     

On the construction of multi-dimensional embedded cubature formulae

Summary In order to compute an integralI[f], one needs at least two cubature formulaeQ _j ,j{1, 2}. |Q ₁[f]–Q ₂[f]| can be used as an error estimate for the less precise cubature formula. In order to reduce the amount of work, one can try to reuse some of the function evaluations needed forQ ₁, inQ ₂. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ ₁, drop (at least) one knot and calculate the weights such that the new formulaQ ₂ is exact for as much monomials as possible. We describe how such embedded formulae with positive weights can be found. The disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae. In this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m+1/2m–1. The method works for all regions ⁿ ,n2. We will also show the influence of structure on the cubature formulae.     

Quasihomogeneous Markov chains and their application to the analysis of queueing systems with “Warmup”

We study the question of recurrence and calculation of stationary probabilities of Markov chains in which the transition probability matrix depends on the indexn of the step, being equal toP fornkm (m=const) and equal toQ forn=km. An example is given of a queueing system described by such chains.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 80–84, 1987.     

New regular embeddings of n‐cubes Qn

We investigate highly symmetrical embeddings of the n‐dimensional cube Q_n into orientable compact surfaces which we call regular embeddings of Q_n. We derive some general results and construct some new families of regular embeddings of Q_n. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Q_n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004     

Spanning caterpillars of a hypercube

Spanning trees of the hypercube Q_n have been recently studied by several authors. In this paper, we construct spanning trees of Q_n which are caterpillars and establish quantitative bounds for a caterpillar to span Q_n. As a corollary, we disprove a conjecture of Harary and Lewinter on the length of the spine of a caterpillar spanning Qⁿ. © 1997 John Wiley & Sons, Inc.     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

A Uniform Approach to the Analysis of Trie Structures That Store Prefixing-Keys

Triesare data structures for storing sets where each element is represented by a key that can be viewed as a string of characters over a finite alphabet. These structures have been extensively studied and analyzed under several probability models. All of these models, however, preclude the occurrence of sets in which the key of one element is a prefix of that of another—such a key is called aprefixing-key. This paper presents an average case analysis of several trie varieties, which we generically calledprefixing-tries, for representing sets with “unrestricted” keys, that is, sets in which the key of one element may be a prefix of that of another. The underlying probability model, which we call theprefix model, _{h, n, m}assumes as equally likely alln-element sets whose keys are composed of at mosthcharacters from a fixed alphabet of sizem. For each of the trie varieties analyzed, we derive exact formulas for the expected space required to store such a set, and the average time required to retrieve an element given its key, as functions ofh,n, andm. Our approach to the analysis is of interest in its own right. It provides a unifying framework for computing the expectations of a wide class of random variables with respect to the prefix model. This class includes the cost functions of the trie varieties analyzed here.     

The asymptotic number of acyclic digraphs. I

We obtain an asymptotic formula forA _n,q , the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forA _n,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatA _n,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula. Research supported by NSF under grant MCS-8300414. Research supported by NSERC under grant A4067. Research supported by NSF under grant MCS-8302282. Research supported by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowship Scheme, while this author was at the University of Newcastle, Australia.     

Covering minimum spanning trees of random subgraphs

We consider the problem of finding a sparse set of edges containing the minimum spanning tree (MST) of a random subgraph of G with high probability. The two random models that we consider are subgraphs induced by a random subset of vertices, each vertex included independently with probability p, and subgraphs generated as a random subset of edges, each edge with probability p. Let n denote the number of vertices, choose p ∈ (0, 1) possibly depending on n, and let b = 1/(1 ? p). We show that in both random models, for any weighted graph G, there is a set of edges Q of cardinality O(n log_bn) that contains the minimum spanning tree of a random subgraph of G with high probability. This result is asymptotically optimal. As a consequence, we also give a bound of O(kn) on the size of the union of all minimum spanning trees of G with some k vertices (or edges) removed. More generally, we show a bound of O(n log_bn) on the size of a covering set in a matroid of rank n, which contains the minimum‐weight basis of a random subset with high probability. Also, we give a randomized algorithm that calls an MST subroutine only a polylogarithmic number of times and finds the covering set with high probability. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006