Shortest string containing all permutations

In this paper, we consider the problem of constructing a shortest string of {1,2,…,n} containing all permutations on n elements as subsequences. For example, the string 1 2 1 3 1 2 1 contains the 6 (=3!) permutations of {1,2,3} and no string with less than 7 digits contains all the six permutations. Note that a given permutation, such as 1 2 3, does not have to be consecutive but must be from left to right in the string.We shall first give a rule for constructing a string of {1,2,…,n} of infinite length and the show that the leftmost n²?2n+4 digits of the string contain all the n! permutations (for n≥3). We conjecture that the number of digits f(n) = n²?2n+4 (for n≥3) is the minimum.Then we study a new function F(n,k) which is the minimum number of digits required for a string of n digits to contain all permutations of i digits, i≤k. We conjecture that F(n,k) = k(n?1) for 4≤k ≤n?1.     

Algorithmic Folding Complexity

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is O(lg² n). We also show that a lower bound of the complexity of the problems is ??(lg² n/lg lg n). Next we show that almost all mountain-valley patterns require ??(n/lg n) folds, which means that the uniform and pleat foldings are relatively easy problems. We also give a general algorithm for folding an arbitrary sequence of length n in O(n/lg n) folds, meeting the lower bound up to a constant factor.     

Some extremal problems arising from discrete control processes

We consider a simple abstract model for a class of discrete control processes, motivated in part by recent work about the behavior of imperfect random sources in computer algorithms. The process produces a string ofn bits and is a “success” or “failure” depending on whether the string produced belongs to a prespecified setL. In an uninfluenced process each bit is chosen by a fair coin toss, and hence the probability of success is ¦L¦/2 ⁿ . A player called the controller, is introduced who has the ability to intervene in the process by specifying the value of some of the bits of the string. We answer the following questions for both worst and average case: (1) how much can the player increase the probability of success given a fixed number of interventions? (2) in terms of ¦L¦what is the expected number of interventions needed to guarantee success? In particular our results imply that if ¦L¦/2 ⁿ =1/Ω(n) where Ω(n) tends to infinity withn (so the probability of success with no interventions is 0(1)) then withO(√n logΩ(n)) interventions the probability of success is 1?0(1). Our main results and the proof techniques are related to well-known results of Kruskal, Katona and Harper in extremal set theory.     

Patterson-Wiedemann construction revisited

In 1983, Patterson and Wiedemann constructed Boolean functions on n=15 input variables having nonlinearity strictly greater than 2^n-1-2^(n-1)/2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n=15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all GF(2)-linear transformations of GF(2^ab) which acts on PG(2,2^a).     

Sorting Using Complete Subintervals and the Maximum Number of Runs in a Randomly Evolving Sequence

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0’s, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1’s? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n ^1/2, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n ^1/3. We also treat some variations, including priority queues and sock-sorting. The proofs use methods originally developed for random graphs.     

Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zⁿ of the n-dimensional Euclidean space Rⁿ. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zⁿ and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.     

Efficient Comparison Based String Matching

We study the exact number of symbol comparisons that are required to solve the string matching problem and present a family of efficient algorithms. Unlike previous string matching algorithms, the algorithms in this family do not "forget" results of comparisons, what makes their analysis much simpler. In particular, we give a linear-time algorithm that finds all occurrences of a pattern of length m in a text of length n in [formula] comparisons. The pattern preprocessing takes linear time and makes at most 2m comparisons. This algorithm establishes that, in general, searching for a long pattern is easier than searching for a short one. We also show that any algorithm in the family of the algorithms presented must make at least [formula] symbol comparisons, for m = 2^k − 1 and any integer k ≥ 1.     

Pattern avoidance for alternating permutations and Young tableaux

We define a class Ln,k of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give bijections between the set A2n(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape 〈n³〉, and between the set A2n+1(1234) and standard Young tableaux of shape 〈³n−1,2,1〉. This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previous work on doubly-alternating permutations (alternating permutations whose inverses are alternating) to our more general context.The set Ln,k may be viewed as the set of reading words of the standard Young tableaux of a certain skew shape. In the last section of the paper, we expand our study to consider pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape λ/μ whose reading words avoid 213 is a natural μ-analogue of the Catalan numbers (and in particular does not depend on λ, up to a simple technical condition), and that there are similar results for the patterns 132, 231 and 312.     

Connection relations and bilinear formulas for the classical orthogonal polynomials

Suppose we have an operator T that maps a set of orthogonal polynomials {P_n(x)}_{n = o}^∞ to another set of orthogonal polynomials. We show how such a mapping can be used to derive connection relations as well as bilinear formulas for the pre-images {P_n(x)}_{n = o}^∞. This method is carried out in detail for the Jacobi, Laguerre, and Hahn polynomials.     

Topologically inseparable functions II: Infinitary case

Given a set A and a function A: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f ⁿ , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A ⁿ → A.     

On rectangular visibility

Given a set of n points in the plane, two points are said to be rectangularly visible if the orthogonal rectangle with the two points as opposite vertices has no other point of the set in its interior. In this paper it is shown that all pairs of rectangularly visible points in a set of size n can be determined in O(n log n + k) time, where k is the number of reported pairs, using O(n) space. Also, we consider the query problem: Given a set V of points and an arbitrary point p, determine those points in V that are rectangularly visible from p. A dynamic data structure is described that uses O(n log n) space, has a query time of O(k + log²n) and an update time of O(log³ n). Additionally, we extend the results to the 3-dimensional case.     

Ensembles Intersectifs et Récurrence de Poincaré

We show that the intersective set and the set of Poincaré are equal and we apply this fact to the repartition modulo 1 of a certain sequence of type (xθ ⁿ) _n≧0.     

Computing optimal islands

Let S be a bicolored set of n points in the plane. A subset I of S is an island if there is a convex set C such that I=C∩S. We give an O(n³)-time algorithm to compute a monochromatic island of maximum cardinality. Our approach is adapted to optimize similar (decomposable) objective functions. Finally, we use our algorithm to give an O(logn)-approximation for the problem of computing the minimum number of convex polygons that cover a class region.     

The space clcv(? n ) with the Hausdorff-Bebutov metric and statistically invariant sets of control systems

We obtain conditions that allow one to evaluate the relative frequency of occurrence of the reachable set of a control system in a given set. If the relative frequency of occurrence in this set is 1, then the set is said to be statistically invariant. It is assumed that the images of the right-hand side of the differential inclusion corresponding to the given control system are convex, closed, but not necessarily compact. We also study the basic properties of the space clcv(? ⁿ ) of nonempty closed convex subsets of ? ⁿ with the Hausdorff-Bebutov metric.     

On the Enumeration of Non-Crossing Pairings of Well-Balanced Binary Strings

A non-crossing pairing on a binary string pairs ones and zeroes such that the arcs representing the pairings are non-crossing. A binary string is well-balanced if it is of the form ${1^{a_1} 0^{a_1}1^{a_2} 0^{a_2} . . .1^{a_r} 0^{a_r}}$ . In this paper we establish connections between non-crossing pairings of well-balanced binary strings and various lattice paths in plane. We show that for well-balanced binary strings with a ₁ ≤ a ₂ ≤  . . . ≤  a _r , the number of non-crossing pairings is equal to the number of lattice paths on the plane with certain right boundary, and hence can be enumerated by differential Goncarov polynomials. For the regular binary strings S =  (1 ^k 0 ^k ) ⁿ , the number of non-crossing pairings is given by the (k + 1)-Catalan numbers. We present a simple bijective proof for this case.     

On the maximum number of distinct factors of a binary string

In this note we prove that a binary string of lengthn can have no more than \(2^{k + 1} - 1 + \left( {\mathop {n - k + 1}\limits_2 } \right)\) distinct factors, wherek is the unique integer such that 2^k + k - 1 ≤ n < 2^k+1 + k. Furthermore, we show that for eachn, this bound is actually achieved. The proof uses properties of the de Bruijn graph.     

Sets of nonnegative matrices with positive inhomogeneous products

Let X be a set of k×k matrices in which each element is nonnegative. For a positive integer n, let P(n) be an arbitrary product of n matrices from X, with any ordering and with repetitions permitted. Define X to be a primitive set if there is a positive integer n such that every P(n) is positive [i.e., every element of every P(n) is positive]. For any primitive set X of matrices, define the index g(X) to be the least positive n such that every P(n) is positive. We show that if X is a primitive set, then g(X)?2^k?2. Moreover, there exists a primitive set Y such that g(Y) = 2^k?2.     

Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.     

New lower bounds for Hopcroft's problem

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $\mathbb{R}^d $ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n ^2/3m^2/3 + m logn) in two dimensions, or Ω(n logm + n ^5/6m^1/2 + n^1/2m^5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log^*(n+m)) of the best known upper bound, due to Matou?ek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.     

On the Order Dimension of Convex Geometries

We study the order dimension of the lattice of closed sets for a convex geometry. We show that the lattice of closed subsets of the planar point set of Erd?s and Szekeres from 1961, which is a set of 2 ^n???2 points and contains no vertex set of a convex n-gon, has order dimension n???1 and any larger set of points has order dimension at least n.