Showing papers in "Random Structures and Algorithms in 1992"
TL;DR: This work presents three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent and their simplicity is their simplicity.
Abstract: 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/ϵ), where ϵ 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 ϵ < 1/(k log n)). An additional advantage of our constructions is their simplicity.
645 citations
TL;DR: It is shown that the Metropolis process takes super-polynomial time to locate a clique that is only slightly bigger than that produced by the greedy heuristic, which is one step above the greedy one in its level of sophistication.
Abstract: In a random graph on n vertices, the maximum clique is likely to be of size very close to 2 lg n. However, the clique produced by applying the naive “greedy” heuristic to a random graph is unlikely to have size much exceeding lg n. The factor of two separating these estimates motivates the search for more effective heuristics. This article analyzes a heuristic search strategy, the Metropolis process, which is just one step above the greedy one in its level of sophistication. It is shown that the Metropolis process takes super-polynomial time to locate a clique that is only slightly bigger than that produced by the greedy heuristic.
398 citations
TL;DR: The results imply that the evolution of a typical Qn process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.
Abstract: ***image*** be a random Qn”-process, that is let Q0 be the empty spanning subgraph of the cube Qn and, for 1 ⩽ t ⩽ M = nN/2 = n2n−1, let the graph Qt be obtained from Qt−1 by the random addition of an edge of Qn not present in Qt−1. When t is about N/2, a typical Qt undergoes a certain “phase transition'': the component structure changes in a sudden and surprising way. Let t = (1 + ϵ) N/2 where ϵ is independent of n. Then all the components of a typical Qt have o(N) vertices if ϵ 0 then, as proved by Ajtai, Komlos, and Szemeredi, a typical Qt has a “giant” component with at least α(ϵ)N vertices, where α(ϵ) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if η > 0 is fixed and t ⩽ (1 − n−η) N/2, then all components of a typical Qt have at most nβ(η) vertices, where β(η) > 0. More importantly, if 60(log n)3/n ⩽ ϵ = ϵn = o(1), then the largest component of a typical Qt has about 2ϵN vertices, while the second largest component has order O(nϵ−2). Loosely put, the evolution of a typical Qn process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices. © 1992 Wiley Periodicals, Inc.
95 citations
TL;DR: It is proved that there exist graphs G with arbitrarily large girth such that every proper edge coloring of G contains a rainbow cycle (i.e., a cycle having no pair of monochromatic edges).
Abstract: We prove that there exist graphs G with arbitrarily large girth such that every proper edge coloring of G contains a rainbow cycle (i.e., a cycle having no pair of monochromatic edges). This answers a problem raised by J. Spencer more than 10 years ago. © 1992 Wiley Periodicals, Inc.
51 citations
TL;DR: This work considers random PATRICIA trees constructed from n i.i.d. sequences of independent equiprobable bits, and studies the height Hn and minimal fill‐up level Fn, which define the minimum distance between the root and a leaf.
Abstract: We consider random PATRICIA trees constructed from n i.i.d. sequences of independent equiprobable bits. We study the height Hn (the maximal distance between the root and a leaf), and the minimal fill‐up level Fn (the minimum distance between the root and a leaf). We give probabilistic proofs of **image** and **image**. © 1992 Wiley Periodicals, Inc.
34 citations
TL;DR: The existence of sparse pseudorandom distributions which are not only sparse, but also have the property that no polynomial‐time algorithm may find an element in their support, except for a negligible probability, are proved independently of any intractability assumption.
Abstract: The existence of sparse pseudorandom distributions is proved. These are probability distributions concentrated in a very small set of strings, yet it is infeasible for any polynomial‐time algorithm to distinguish between truly random coins and coins selected according to these distributions. It is shown that such distributions can be generated by (nonpolynomial) probabilistic algorithms, while probabilistic polynomial‐time algorithms cannot even approximate all the pseudorandom distributions. Moreover, we show the existence of evasive pseudorandom distributions which are not only sparse, but also have the property that no polynomial‐time algorithm may find an element in their support, except for a negligible probability. All these results are proved independently of any intractability assumption. © 1992 Wiley Periodicals, Inc.
33 citations
TL;DR: The convergence of the annealing algorithm in the restricted parallel form is established, for an arbitrary network, a generalization of the unlimited parallelism for Boltzmann machines.
Abstract: This article introduces the notion of restricted parallelism for networks, a generalization of the unlimited parallelism for Boltzmann machines. The convergence of the annealing algorithm in the restricted parallel form is established, for an arbitrary network. © 1992 Wiley Periodicals, Inc.
31 citations
TL;DR: It is shown that there exists a a fully polynomial randomized approximation scheme for counting the number of Hamilton cycles in almost all directed graphs.
Abstract: We show that there exists a a fully polynomial randomized approximation scheme for counting the number of Hamilton cycles in almost all directed graphs.
23 citations
TL;DR: It is shown that the length Tm of the shortest tour through X1, …, Xm satisfies limm∞ E(Tm)/m1−1/d = β(d) for a certain number β( d) for some numerical constant K.
Abstract: Consider d ⩾ 2, and m points X1, …, Xm that are independent uniformly distributed in [0, 1]d. It is well known that the length Tm of the shortest tour through X1, …, Xm satisfies limm∞ E(Tm)/m1−1/d = β(d) for a certain number β(d). We show that for some numerical constant K,
.
15 citations
TL;DR: The threshold for Gn,p to be Hamiltonian, is the threshold for the 1-pancyclic property and a generalization of the pancyclic property is considered.
Abstract: We consider a generalization of the pancyclic property. A graph G is defined to be 1-pancyclic if there is some Hamilton cycle H in G such that we can find a cycle Cs of length s (3 ⩽ s ⩽ n − 1) using only the edges of H and one other edge es. We show that the threshold for Gn,p to be Hamiltonian, is the threshold for the 1-pancyclic property.
14 citations
TL;DR: The method used provides a better upper bound for this function than was previously known and is established as a tool to establish a theorem for projective planes.
Abstract: Say a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function fr(n) = max{min{|V(H)|: H (G)1r}: |V(G)| = n}, where H (G)1r indicates that H is vertex Ramsey to G, that is, any vertex coloring of H with r colors admits a monochromatic induced copy of G. The method used provides a better upper bound for this function than was previously known. As a tool, we establish a theorem for projective planes.
TL;DR: A Local limit theorem for the distribution of the number of components in random labelled relational structures of size n is proved as n∞, the case when the corresponding exponential generating functions diverge at their radii of convergence.
Abstract: A Local limit theorem for the distribution of the number of components in random labelled relational structures of size n (e.g., a type of random graphs on n vertices, random permutations of n elements, etc.) is proved as n∞. The case when the corresponding exponential generating functions diverge at their radii of convergence is considered.
TL;DR: It is proved that there is at least one block of each size between the smallest and largest block sizes of the set [n] if its block sizes form an interval.
Abstract: A set partition is called “gap-free” if its block sizes form an interval In other words, there is at least one block of each size between the smallest and largest block sizes Let B(n) and G(n), respectively, denote the number of partitions and the number of gap-free partitions of the set [n] We prove that ***image*** © 1992 Wiley Periodicals, Inc
TL;DR: In this paper, it was shown that under natural growth conditions on the function p = p(n), the ratio tends to 1 in probability as n tends to infinity, where n is the maximum number of paths of the routing R passing through any vertex of G and the minimum is taken over all the routings of G.
Abstract: A routing R of a graph G is a set of n(n − 1) elementary paths R(u, v) specified for all ordered pairs (u, v) of vertices of G. The vertex-forwarding index ξ(G) of G, is defined by ***image***
Where ξ(G, R) is the maximum number of paths of the routing R passing through any vertex of G and the minimum is taken over all the routings of G. Let Gp denote the random graph on n vertices with edge probability p and let m = np. It is proved among other things that, under natural growth conditions on the function p = p(n), the ratio ***image***
Tends to 1 in probability as n tends to infinity. © 1992 Wiley Periodicals, Inc.
TL;DR: It is shown that, by choosing the means of the Poisson distributions more carefully, an error quadratic in n−1b can be achieved, and that essentially nothing better is possible.
Abstract: The Ewens sampling formula is a family of probability distributions over the space of cycle types of permutations of n objects, indexed by a real parameter θ. In the case θ = 1, where the distribution reduces to that induced by the uniform distribution on all permutations, the joint distributions of the numbers of cycles of lengths less than b = o(n) is extremely well approximated by a product of Poisson distributions, having mean 1/j for cycle length j: the error is super-exponentially small with nb−1. For θ ≠ 1. the analogous approximation, with means adjusted to θ/j, is good, but with error only linear in n−1b. In this article, it is shown that, by choosing the means of the Poisson distributions more carefully, an error quadratic in n−1b can be achieved, and that essentially nothing better is possible.
TL;DR: For the families of t-ary trees and planted plane trees, the variance of the level numbers is determined using singularity analysis or a probabilistic approach using weak convergence and uniform integrability and some results on higher moments are presented.
Abstract: For the families of t-ary trees and planted plane trees, the variance of the level numbers (i.e., the depths of specified endnodes) is determined. We apply different types of techniques, such as singularity analysis (Kirschenhofer's diagonalization method) or a probabilistic approach using weak convergence and uniform integrability. In the case of binary trees, an exact variance formula for finite tree size is established; in the other cases, asymptotic equivalents are derived. Also some results on higher moments are presented.
TL;DR: This work considers the parallel greedy algorithm of Coppersmith, Raghavan, and Tompa for finding the lexicographically first maximal independent set of a graph and proves an Ω(log n) bound on the expected number of iterations for most edge densities.
Abstract: We consider the parallel greedy algorithm of Coppersmith, Raghavan, and Tompa (Proc. of 28th Annual IEEE Symp. on Foundations of Computer Science, pp. 260–269, 1987) for finding the lexicographically first maximal independent set of a graph. We prove an Ω(log n) bound on the expected number of iterations for most edge densities. This complements the O(log n) bound proved in Calkin and Frieze (Random Structures and Algorithms, Vol. 1, pp. 39–50, 1990). © 1992 Wiley Periodicals, Inc.
TL;DR: An exponential upper bound on the distribution of the Voronoi polygon having n hyperfaces is proved using a new integral formula for the Poisson process and the conditional distribution of volume of the fundamental region given n is found to be gamma distribution.
Abstract: An exponential upper bound on the distribution of the Voronoi polygon having n hyperfaces is proved. Using a new integral formula for the Poisson process, the conditional distribution of volume of the fundamental region given n is found to be gamma distribution. This yields an upper bound on distribution of the polygon's volume. © 1992 Wiley Periodicals, Inc.
TL;DR: It is proved that for every first-order sentence, the probability that the sentence is true for the random graph has an asymptotic limit.
Abstract: We consider random graphs with edge probability βn−α, where n is the number of vertices of the graph, β > 0 is fixed, and α = 1 or α = (l + 1) /l for some fixed positive integer l. We prove that for every first-order sentence, the probability that the sentence is true for the random graph has an asymptotic limit. © 1992 Wiley Periodicals, Inc.
IBM1
TL;DR: It is shown that there is an absolute constant c with the following property: for any two graphs G1 and G2 on the same set of vertices, the chromatic number of the graph G = (V, E1 U E2) is precisely cd.
Abstract: It is shown that there is an absolute constant c with the following property: For any two graphs G1 = (V, E1) and G2 = (V, E2) on the same set of vertices, where G1 has maximum degree at most d and G2 is a vertex disjoint union of cliques of size cd each, the chromatic number of the graph G = (V, E1 U E2) is precisely cd. The proof is based on probabilistic arguments. © 1992 Wiley Periodicals, Inc.
TL;DR: It is shown that, as n→∞, Vn→0 if N(n)⩽(k−ϵ)n →1 if N (n) ⩾ (k + ϵ) n, and that the same result holds for sampling uniformly from within the hypercube.
Abstract: Consider the hypercube [0, 1]n in Rn. This has 2n vertices and volume 1. Pick N = N(n) vertices independently at random, form their convex hull, and let Vn be its expected volume. How large should N(n) be to pick up significant volume? Let k=2/√≈1.213, and let ϵ > 0. We shall show that, as n→∞, Vn→0 if N(n)⩽(k−ϵ)n →1 if N(n) ⩾ (k + ϵ)n. A similar result holds for sampling uniformly from within the hypercube, with constant ***image***. © 1992 Wiley Periodicals, Inc.
TL;DR: The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymPTotic covariance structure is explicitly determined.
Abstract: We study the joint probability distribution of the number of nodes of outdegree 0, 1, and 2 in a random recursive tree. We complete the known partial list of exact means and variances for outdegrees up to two by obtaining exact combinatorial expressions for the remaining means, variances, and covariances. The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymptotic covariance structure is explicitly determined. It is also shown how to extend the results (at least in principle) to obtain a limiting multivariate normal distribution for nodes of outdegree 0, 1, …, k.
TL;DR: In this article, it was shown that the number of vertices that do not contain a fixed graph H as an induced subgraph is essentially determined by the maximum number of subgraphs of a single graph.
Abstract: In this article we study asymptotic properties of the class of graphs not containing a fixed graph H as an induced subgraph. In particular we show that the number Forbn★(H) of such graphs on n vertices is essentially determined by the number of subgraphs of a single graph. This implies that ***image***
Where t(H) is a graph-theoretic parameter generalizing the chromatic number and the clique covering number. This result complements a theorem of Erdos, Frankl, and Rodl [2] who showed that the number Forbn(H) of graphs on n vertices which do not contain H as a weak subgraph is given by ***image*** © 1992 Wiley Periodicals, Inc.
IBM1
TL;DR: It follows as a corollary that k ⩾ m/2 + cm3/4 is the maximum number of edges in a bipartite subgraph of G.
Abstract: Let G be a triangle‐free graph on n points with m edges and vertex degrees d1, d2,…, dn. Let k be the maximum number of edges in a bipartite subgraph of G. In this note we show that k ⩾ m/2 + Σ **image** √di. It follows as a corollary that k ⩾ m/2 + cm3/4. © 1992 Wiley Periodicals, Inc.
TL;DR: Various properties of C(n, q), the set of n‐vertex q‐edge labeled connected graphs are proved and the domain of validity of the asymptotic formula of Erdos and Renyi for |C( n, q)| is extended and the formula is seen to be the first term of an asymPTotic expansion.
Abstract: We prove various properties of C(n, q), the set of n‐vertex q‐edge labeled connected graphs. The domain of validity of the asymptotic formula of Erdos and Renyi for |C(n, q)| is extended and the formula is seen to be the first term of an asymptotic expansion. The same is done for Wright's asymptotic formula. We study the number of edges in a random connected graph in the random edge model 𝒢n,p. For certain ranges of n and q, we determine the probability that a random edge (resp. vertex) of a random graph in C(n, q) is a bridge (resp. cut vertex). We also study the degrees of random vertices. © 1992 Wiley Periodicals, Inc.
TL;DR: A linear expected time algorithm to color every graph which does not contain a clique on l + 1 vertices as a subgraph with a minimal number of colors is presented.
Abstract: This article presents a linear expected time algorithm to color every graph which does not contain a clique on l + 1 vertices as a subgraph with a minimal number of colors. This extends a result of Dyer and Frieze for l-colorable graphs. For the proof we develop a new method which allows us to precisely estimate the number of graphs with certain structural properties.
TL;DR: It is proved that Nm,n is the number of polygons of perimeter 2n and concavity m, and that the radius of convergence of the series counting all polygons with m = o(n) is equal to 1/4.
Abstract: We classify self-avoiding polygons on the square lattice according to a concavity measure, m, where 2m is the difference between the number of steps in the polygon and the perimeter of the minimal rectangle bounding the polygon. We generate series expansions for the perimeter generating functions Sm(x) for polygons of concavity m. We analyze the series Sm(x) for m = 0 to 3. If Nm,n denotes the number of polygons of perimeter 2n and concavity m, with m = o(n1/2), we prove that Nm,n ˜ 22n−m−7nm+1/m!, and that the radius of convergence of the series counting all polygons with m = o(n) is equal to 1/4. Our numerical data leads us to conjecture that in fact
for m = o(n1/2), a result confirmed for m = 0 and 1.
TL;DR: The power ofrete functional limitTheorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods are demonstrated.
Abstract: Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the Erdos-Turin Law for the log-order of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of distinct cycle sizes in a random permutation. @ 1992 John Wiley & Sons, Inc.
TL;DR: A tree partitioning process in which n elements are split into b at the root of a tree, the rest going recursively into two subtrees with a binomial probability distribution is considered, extending some familiar tree data structures of computer science like the digital trie and the digital search tree.
Abstract: Consider a tree partitioning process in which n elements are split into b at the root of a tree (b a design parameter), the rest going recursively into two subtrees with a binomial probability distribution. This extends some familiar tree data structures of computer science like the digital trie and the digital search tree. The exponential generating function for the expected size of the tree satisfies a difference-differential equation of order b, The solution involves going to ordinary (rather than exponential) generating functions, analyzing singularities by means of Mellin transforms and contour integration. The method is of some general interest since a large number of related problems on digital structures can be treated in this way via singularity analysis of ordinary generating functions. 0 1992 John Wiley & Sons, Inc.