Finding blocks and other patterns in a random coloring of ℤ

Let ξ = (ξ_k)_k∈? be i.i.d. with P(ξ_k = 0) = P(ξ_k = 1) = 1/2, and let S: = (S_k) be a symmetric random walk with holding on ?, independent of ξ. We consider the scenery ξ observed along the random walk path S, namely, the process (χ_k := ξ). With high probability, we reconstruct the color and the length of blockⁿ, a block in ξ of length ≥ n close to the origin, given only the observations (χ_k). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on blockⁿ and in the interval [?3ⁿ,3ⁿ]. Moreover, we reconstruct with high probability a piece of ξ of length of the order 3 around blockⁿ, given only 3 observations collected by the random walker starting on the boundary of blockⁿ. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On (ε,k)‐min‐wise independent permutations

A family of permutations of [n] = {1,2,…,n} is (ε,k)‐min‐wise independent if for every nonempty subset X of at most k elements of [n], and for any x ∈ X, the probability that in a random element π of , π(x) is the minimum element of π(X), deviates from 1/∣X∣ by at most ε/∣X∣. This notion can be defined for the uniform case, when the elements of are picked according to a uniform distribution, or for the more general, biased case, in which the elements of are chosen according to a given distribution D. It is known that this notion is a useful tool for indexing replicated documents on the web. We show that even in the more general, biased case, for all admissible k and ε and all large n, the size of must satisfy as well as This improves the best known previous estimates even for the uniform case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1}ⁿ, and k parties, who have noisy version of the source bits yⁱ ∈ {0, 1}ⁿ, when for all i and j, it holds that P [y = x_j] = 1 ? ?, independently for all i and j. That is, each party sees each bit correctly with probability 1 ? ?, and incorrectly (flipped) with probability ?, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions f_i: {0, 1}ⁿ → {0, 1} such that P [f₁(y¹) = … = f_k(y^k)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function f_i may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by f_i(yⁱ) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ?. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

On an online random k‐SAT model

Given n Boolean variables x₁,…,x_n, a k‐clause is a disjunction of k literals, where a literal is a variable or its negation. Suppose random k‐clauses are generated one at a time and an online algorithm accepts or rejects each clause as it is generated. Our goal is to accept as many randomly generated k‐clauses as possible with the condition that it must be possible to satisfy every clause that is accepted. When cn random k‐clauses on n variables are given, a natural online algorithm known as Online‐Lazy accepts an expected (1 ? )cn + a_kn clauses for some constant a_k. If these clauses are given offline, it is possible to do much better, (1 ? )cn + Ω( )n can be accepted whp . The question of closing the gap between a_k and Ω( ) for the online version remained open. This article shows that for any k ≥ 1, any online algorithm will accept less than (1 ? )cn + (ln 2)n k‐clauses whp , closing the gap between the constant and Ω( ). Furthermore we show that this bound is asymptotically tight as k → ∞. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Hamiltonian graphs involving neighborhood unions

Dirac proved that a graph G is hamiltonian if the minimum degree , where n is the order of G. Let G be a graph and . The neighborhood of A is for some . For any positive integer k, we show that every (2k ? 1)‐connected graph of order n ≥ 16k³ is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k ? 1 on connectivity is best possible in general while the lower bound 16k³ for n is conjectured to be unnecessary. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 83–100, 2006     

A bound on the chromatic number using the longest odd cycle length

It is known that for every integer k ≥ 4, each k‐map graph with n vertices has at most kn ? 2k edges. Previously, it was open whether this bound is tight or not. We show that this bound is tight for k = 4, 5. We also show that this bound is not tight for large enough k (namely, k ≥ 374); more precisely, we show that for every and for every integer , each k‐map graph with n vertices has at most edges. This result implies the first polynomial (indeed linear) time algorithm for coloring a given k‐map graph with less than 2k colors for large enough k. We further show that for every positive multiple k of 6, there are infinitely many integers n such that some k‐map graph with n vertices has at least edges. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 267–290, 2007     

Vertex pancyclic in‐tournaments

An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k‐pancyclicity of in‐tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in‐tournament is vertex k‐pancyclic for k ≤ 5 and k ≥ n − 3. In the latter case, we even show that the in‐tournaments in consideration are fully (n − 3)‐extendable which means that every vertex belongs to a cycle of length n − 3 and that the vertex set of every cycle of length at least n − 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in‐tournament of order n with minimum degree greater than is vertex k‐pancyclic for 5 < k < n − 3, and we present a family of examples showing that this bound would be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 84–104, 2001     

Spanning k‐arc‐strong subdigraphs with few arcs in k‐arc‐strong tournaments

Given a k‐arc‐strong tournament T, we estimate the minimum number of arcs possible in a k‐arc‐strong spanning subdigraph of T. We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k‐arc‐strong spanning subdigraph of T contains at least arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has arcs. This is best possible since it can be shown that every k‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than arcs. As our main result we prove that every k‐arc‐strong tournament contains a spanning k‐arc‐strong subdigraph with no more than arcs. We conjecture that for every k‐arc‐strong tournament T, the minimum number of arcs in a k‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k. We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004     

On extractors and exposure‐resilient functions for sublogarithmic entropy

We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n ‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n ‐ k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure‐resilient functions achieving these parameters are known. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On restricted min‐wise independence of permutations

A family of permutations ℱ ⊆ S_n with a probability distribution on it is called k-restricted min-wise independent if we have Pr[min π(X) = π(x)] = 1/|X| for every subset X ⊆ [n] with |X| ≤ k, every x ∈ X, and π ∈ ℱ chosen at random. We present a simple proof of a result of Norin: every such family has size at least Some features of our method might be of independent interest. The best available upper bound for the size of such family is 1 + ∑ (j − 1)(). We show that this bound is tight if the goal is to imitate not the uniform distribution on S_n, but a distribution given by assigning suitable priorities to the elements of [n] (the stationary distribution of the Tsetlin library, or self-organizing lists). This is analogous to a result of Karloff and Mansour for k-wise independent random variables. We also investigate the cases where the min-wise independence condition is required only for sets X of size exactly k (where we have only an Ω(log log n + k) lower bound), or for sets of size k and k − 1 (where we already obtain a lower bound of n − k + 2). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

A Class of 2‐(3n7, 3n−17, (3n−17−1)/2) Designs

Generalized Hadamard matrices are used for the construction of a class of quasi‐residual nonresolvable BIBD's with parameters . The designs are not embeddable as residual designs into symmetric designs if n is even. The construction yields many nonisomorphic designs for every given n ≥ 2, including more than 10¹⁷ nonisomorphic 2‐(63,21,10) designs. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 460–464, 2007     

Latin squares with no small odd plexes

A k‐plex in a Latin square of order n is a selection of kn entries in which each row, column, and symbol is represented precisely k times. A transversal of a Latin square corresponds to the case k = 1. We show that for all even n > 2 there exists a Latin square of order n which has no k‐plex for any odd but does have a k‐plex for every other . © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 477–492, 2008     

Subgraphs smaller than the girth

For graphs A, B, let () denote the number of subsets of nodes of A for which the induced subgraph is B. If G and H both have girth > k, and if () = () for every k-node tree T, then for every k-node forest F, () = (). Say the spread of a tree is the number of nodes in a longest path. If G is regular of degree d, on n nodes, with girth > k, and if F is a forest of total spread ≤k, then the value of () depends only on n and d.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

The color space of a graph

If k is a prime power, and G is a graph with n vertices, then a k‐coloring of G may be considered as a vector in $\mathbb{GF}$(k)ⁿ. We prove that the subspace of $\mathbb{GF}$(3)ⁿ spanned by all 3‐colorings of a planar triangle‐free graph with n vertices has dimension n. In particular, any such graph has at least n − 1 nonequivalent 3‐colorings, and the addition of any edge or any vertex of degree 3 results in a 3‐colorable graph. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 234–245, 2000     

Proximity‐preserving labeling schemes

This article considers informative labeling schemes for graphs. Specifically, the question introduced is whether it is possible to label the vertices of a graph with short labels in such a way that the distance between any two vertices can be inferred from inspecting their labels. A labeling scheme enjoying this property is termed a proximity‐preserving labeling scheme. It is shown that, for the class of n‐vertex weighted trees with M‐bit edge weights, there exists such a proximity‐preserving labeling scheme using O(M log n + log²n) bit labels. For the family of all n‐vertex unweighted graphs, a labeling scheme is proposed that using O(log² n · κ · n^1/κ) bit labels can provide approximate estimates to the distance, which are accurate up to a factor of In particular, using O(log³n) bit labels, the scheme can provide estimates accurate up to a factor of $\sqrt{2 \log n}$. (For weighted graphs, one of the log n factors in the label size is replaced by a factor logarithmic in the network's diameter.) © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 167–176, 2000     

Vertex‐disjoint cycles containing prescribed vertices

Enomoto 7 conjectured that if the minimum degree of a graph G of order n ≥ 4k ? 1 is at least the integer , then for any k vertices, G contains k vertex‐disjoint cycles each of which contains one of the k specified vertices. We confirm the conjecture for n ≥ ck² where c is a constant. Furthermore, we show that under the same condition the cycles can be chosen so that each has length at most six. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 276–296, 2003     

Critical graphs for subpancyclicity of 3‐connected claw‐free graphs

Let be the family of graphs G such that all sufficiently large k ‐connected claw‐free graphs which contain no induced copies of G are subpancyclic. We show that for every k≥3 the family is infinite and make the first step toward the complete characterization of the family . © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 263–278, 2009     

A randomized on–line algorithm for the k–server problem on a line

The k–server problem is one of the most important and well‐studied problems in the area of on–line computation. Its importance stems from the fact that it models many practical problems like multi‐level memory paging encountered in operating systems, weighted caching used in the management of web caches, head motion planning of multi‐headed disks, and robot motion planning. In this paper, we investigate its randomized version for which Θ(log k)–competitiveness is conjectured and yet hardly any <k competitive algorithms are known, even for the simplest of metric spaces of O(k) size. We present a –competitive randomized k–server algorithm against an oblivious adversary when the underlying metric space is given by n equally spaced points on a line . This algorithm is <k competitive for . Thus, it provides a super–linear bound for n with o(k)–competitiveness for the first time and improves the best results known so far for the range on . © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006