Showing papers in "Random Structures and Algorithms in 1991"
TL;DR: This article converts some of the applications of the Lovasz Local Lemma into polynomial time sequential algorithms (at the cost of a weaker constant factor in the “exponent”).
Abstract: The Lovasz Local Lemma is a remarkable sieve method to prove the existence of certain structures without supplying any efficient way of finding these structures. In this article we convert some of the applications of the Local Lemma into polynomial time sequential algorithms (at the cost of a weaker constant factor in the “exponent”). Our main example is the following: assume that in an n‐uniform hypergraph every hyperedge intersects at most 2n/48 other hyperedges, then there is a polynomial time algorithm that finds a two‐coloring of the points such that no hyperedge is monochromatic. © 1991 Wiley Periodicals, Inc.
312 citations
TL;DR: It is shown that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞, which settles a problem of Erdos who conjectured, in 1976, that A( G) = o(d2) as d →∞.
Abstract: A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two-colored cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞. This settles a problem of Erdos who conjectured, in 1976, that A(G) = o(d2) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d4/3/(log d)1/3); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two-colored cycle. All the proofs rely heavily on probabilistic arguments. © 1991 Wiley Periodicals, Inc.
243 citations
TL;DR: Here the Lovasz Local Lemma is modified and an algorithmic version that can be parallelized is achieved, thus obtaining deterministic NCl algorithms for several interesting algorithmic problems.
Abstract: The Lovasz Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck recently has found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of losing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NCl algorithms for several interesting algorithmic problems. © 1991 Wiley Periodicals, Inc.
215 citations
TL;DR: In this article, a simple proof for Szemeredi's regularity lemma and its generalization for k-uniform hypergraphs is given for fixed k, and there are altogether k -1 different versions of the regularality lemma for k -uniform Hypergraphs.
Abstract: We give a simple proof for Szemeredi's Regularity Lemma and its generalization for k-uniform hypergraphs. For fixed k, there are altogether k -1 different versions of the regularity lemma for k-uniform hypergraphs. The connection between regularity lemmas for hypergraphs and quasi-random classes of hypergraphs is also investigated.
103 citations
TL;DR: It turns out that there are asymptotically twice as many graphs not containing an induced quadrilateral than there are bipartite graphs.
Abstract: In this note we determine the structure of “almost all” graphs not containing a quadrilateral (i.e., a cycle of length four) as an induced subgraph. In particular, it turns out that there are asymptotically twice as many graphs not containing an induced quadrilateral than there are bipartite graphs.
80 citations
TL;DR: There are graphs for which this Randomized Greedy Algorithm (RGA) usually only obtains a matching close in size to that guaranteed by worst-case analysis (i.e., half the size of the maximum).
Abstract: We consider a randomized version of the greedy algorithm for finding a large matching in a graph. We assume that the next edge is always randomly chosen from those remaining. We analyze the performance of this algorithm when the input graph is fixed. We show that there are graphs for which this Randomized Greedy Algorithm (RGA) usually only obtains a matching close in size to that guaranteed by worst-case analysis (i.e., half the size of the maximum). For some classes of sparse graphs (e.g., planar graphs and forests) we show that the RGA performs significantly better than the worst-case. Our main theorem concerns forests. We prove that the ratio to maximum here is at least 0.7690…, and that this bound is tight.
75 citations
TL;DR: Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for w-dependent random variables.
Abstract: Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for w-dependent random variables. Examples include: the number of leaves (Ln), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply. © 1991 Wiley Periodicals, Inc.
74 citations
TL;DR: It is proved that a graph sequence (G,) is quasirandom if and only if in the SzemerCdi partitions of G, almost all densities are $ + o(1).
Abstract: In this paper we shall investigate the connection between the SzemerCdi Regularity Lemma and quasirandom graph sequences, defined by Chung, Graham, and Wilson, and also, slightly differently, by Thomason. We prove that a graph sequence (G,) is quasirandom if and only if in the SzemerCdi partitions of G, almost all densities are $ + o(1). Many attempts have been made to clarify when an individual event could be called random and in what sense. Both the fundamental problems of probability theory and some practical application need this clarification very much. For example, in applications of the Monte-Carlo method one needs to know if the random number generator used yields a sequence which can be regarded “pseudorandom” or not. The literature on this question is extremely extensive. Thomason [6-81 and Chung, Graham, and Wilson [2,3], and also Frankl, Rodl, and Wilson [4] started a new line of investigation, where (instead of regarding numerical sequences) they gave some characterizations of “randomlike” graph sequences, matrix sequences, and hypergraph sequences. The aim of this paper is to contribute to this question in case of graphs, continuing the above line of investigation. Let %(n, p) denote the probability space of labelled graphs on n vertices, where the edges are chosen independently and at random, with probability p. We shall say that “a random graph sequence (G,) has property P”
68 citations
TL;DR: It is proved that almost all graphs with n vertices and 1.44 n edges contain no subgraph with minimum degree at least three, and hence are 3-colorable.
Abstract: We establish a uniform asymptotic approximation of certain probabilities arising in the coupon collector's problem. Then we use this approximation to prove that almost all graphs with n vertices and 1.44 n edges contain no subgraph with minimum degree at least three, and hence are 3-colorable.
68 citations
TL;DR: The limit distribution is found of the length of shortest cycle contained in the largest component of G(n, M), as well as of the longest cycle outside it, and it is shown that the probability tending to 1 as n‐→∞ thelength of the shortest cycle in G( n, M) is of the order s2(n)/n.
Abstract: Let G(n, M) be a graph chosen at random from the family of all labelled graphs with n vertices and M(n) = 0.5n + s(n) edges, where s3(n)n−2→∞ but s(n) = o(n). We find the limit distribution of the length of shortest cycle contained in the largest component of G(n, M), as well as of the longest cycle outside it. We also describe the block structure of G(n, M) and derive from this result the limit probability that G(n, M) contains a cycle with a diagonal. Finally, we show that the probability tending to 1 as n‐→∞ the length of the longest cycle in G(n, M) is of the order s2(n)/n. © 1991 Wiley Periodicals, Inc.
61 citations
TL;DR: A nontrivial upper bound holds for almost all P ∈Ω(n, p), when p ≧ 1 /log n, which has the same form as the lower bound when p is constant.
Abstract: Let P = (X, 0, there exist δ, c > 0 so that: (1) if (log1+ϵn)/n n- cnlp log n for almost all P ∈Ω(n, p). The first inequality is best possible up to the value of the constant δ when p > (log2+en)/n. As to the accuracy of the second inequality, we have the trivial upper bound dim(P)≦n for all P∈Ω(n, p). We then develop a nontrivial upper bound which holds for almost all P ∈Ω(n, p), when p ≧ 1 /log n. This upper bound has the same form as the lower bound when p is constant. We also study the space ℒ(n) of all labelled ordered sets on n points and show that there exist positive constants c1, c2 so that n/4 - c1n/log n < dim(P)
TL;DR: This work analyzes the performance of a prototypical scheme for shared storage allocation: two initially empty stacks evolving in a contiguous block of memory of size m, with the case in which the stacks are more likely to shrink than grow, but with the probabilities of insertion and deletion allowed to depend arbitrarily on stack height.
Abstract: We analyze the performance of a prototypical scheme for shared storage allocation: two initially empty stacks evolving in a contiguous block of memory of size m. We treat the case in which the stacks are more likely to shrink than grow, but with the probabilities of insertion and deletion allowed to depend arbitrarily on stack height as a fraction of m. New results are obtained on the m → ∞ asymptotics of the stack collision time, and of the final stack heights. The results of Wentzell and Freidlin on the large deviations of Markov chains are used, and the relation of their formalism to the Hamiltonian formulation of classical mechanics is emphasized. Certain results on higher‐order asymptotics follow from WKB expansions. © 1991 Wiley Periodicals, Inc.
TL;DR: A storage allocation algorithm which permits one to maintain two stacks inside a shared memory area of a fixed size and the well-known banker algorithm which plays a fundamental role in parallel processing are analyzed.
Abstract: In this paper we analyze: (i) a storage allocation algorithm (D.E. Knuth, The Art of Computer Programming-Vol. I, Addison-Wesley, Reading, MA, 1969, Ex. 2.2.2.13) which permits one to maintain two stacks inside a shared (continuous) memory area of a fixed size, and (ii) the well-known banker algorithm which plays a fundamental role in parallel processing (J. Francon, Combinatoire et Parallelisme; Journees Informatique et Mathematique; Lumigny, October 15-17, 1987; A. N. Habermann, in Current Trends in Programming Methodology, Vol. 3, K. M. Chandy and R. T. Yeh, Eds., Prentice-Hall, Englewood Cliffs, NJ, 1987; J. Peterson and A. Silberschatz, Operating Systems Concepts, Addison-Wesley, Reading, MA, 1983). The natural formulation of the problems to be solved here is in terms of random walks. For (i) the random walk Ym(-) takes place in a triangle in a two-dimensional lattice space with two reflecting barriers along the axes (a deletion has no effect on an empty stack) and one absorbing barrier along the diagonal (the algorithm stops when the combined sizes of the stacks exhaust the available storage). For (ii) the random walk takes place in a rectangle with broken corner and has four reflecting barriers and one absorbing barrier (see Fig. 10). With the help of diffusion techniques, we obtain, asymptotically as m: ∞.
the distributions of the hitting place (Zm) and hitting time (Tm) on the absorbing boundary.
the joint distribution of Zm and Tm.
the distribution P[Ym(n) ≦ ym, n < Tm].
TL;DR: It is proved that, for any fixed x, T : R,(T)
Abstract: For T E GL,(F,), let Q , ( T ) be the number of irreducible factors that the characteristic polynomial of T has. We prove that, for any fixed x , # { T : R,(T)
TL;DR: This work considers first order sentences about two logical structures and characterize those p(n) for which a zero-one law holds.
Abstract: We consider first order sentences about two logical structures. First we consider 1,…, n with the successor relation and a random unary relation that points satisfy with probability p(n). We then replace the successor relation with less than. For both structures we characterize those p(n) for which a zero-one law holds. © 1991 Wiley Periodicals, Inc.
TL;DR: The first rigorous proofs of two of Kauffman's generalizations about a random Boolean cellular automaton are given: a large fraction of vertices stabilize quickly, consequently the length of cycles in the automaton's behavior is small compared to that of a random mapping with the same number of states.
Abstract: Based on computer simulations, Kauffman (Physica D, 10, 145-156, 1984) made several generalizations about a random Boolean cellular automaton which he invented as a model of cellular metabolism. Here we give the first rigorous proofs of two of Kauffman's generalizations: a large fraction of vertices stabilize quickly, consequently the length of cycles in the automaton's behavior is small compared to that of a random mapping with the same number of states; and reversal of the states of a large fraction of the vertices does not affect the cycle to which the automaton moves. © 1991 Wiley Periodicals, Inc.
TL;DR: It is shown that the probability, at the critical point λc, that n1, and n2 are connected satisfies a power law, in the sense that for n2 ≧ nt ≧ 1 for any δ > 0 and certain constants c1 and c2.
Abstract: We consider P(G is connected) when G is a graph with vertex set Z+ = {1,2, …}, and the edge between i and j is present with probability p(i, j) = min(λ h(i, j), 1) for certain functions h(i, j) homogeneous of degree -1. It is known that there is a critical value λc of λ such that
.
We show that the probability, at the critical point λc, that n1, and n2 are connected satisfies a power law, in the sense that for n2 ≧ nt ≧ 1
for any δ > 0 and certain constants c1 and c2.
TL;DR: A flow process in infinite graphs where vertices with large resources tend to attract resources from neighbors is studied, and it is proved that in each finite region all motion stops after a finite time.
Abstract: We study a flow process in infinite graphs where vertices with large resources tend to attract resources from neighbors. The initial resources are random. An interesting question is whether in each finite region all motion stops after a finite time. Under certain assumptions, we prove that this is true. For some other cases, we prove a weaker stability result. We pay attention mostly to the case of Z2, but several results can be easily generalized to Zd. © 1991 Wiley Periodicals, Inc.
TL;DR: If the occupancy probability is below the threshold 2 - √2 = 0.5857…, then the blocking probability tends to zero, whereas above this threshold it tends to one, which provides a theoretical explanation for results observed empirically in simulations by Bassalygo, Neiman, and Vvedenskaya.
Abstract: We determine the limiting behavior of the blocking probability for spider-web networks, a class of crossbar switching networks proposed by Ikeno. We use a probabilistic model proposed by the author, in which the busy links always form disjoint routes through the network. We show that if the occupancy probability is below the threshold 2 - √2 = 0.5857…, then the blocking probability tends to zero, whereas above this threshold it tends to one. This provides a theoretical explanation for results observed empirically in simulations by Bassalygo, Neiman, and Vvedenskaya.
TL;DR: The threshold for the existence of a spanning maximal planar subgraph in the random graph Gn, p is studied and it is shown that it is very near p = 1/n⅓.
Abstract: We study the threshold for the existence of a spanning maximal planar subgraph in the random graph Gn, p . We show that it is very near p = 1/n⅓ We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n½.
TL;DR: Given an arbitrary set of N points on the plane, one can two-color the points red and blue in such a way that the difference of the numbers of red andblue points in any half-plane has absolute value less than N1/4(log N)4.
Abstract: Given an arbitrary set of N points on the plane, one can two-color the points red and blue in such a way that the difference of the numbers of red and blue points in any half-plane has absolute value less than N1/4(log N)4. This is essentially best possible. © 1991 Wiley Periodicals, Inc.
TL;DR: Asymptotic results on the average number of ˆ-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family are given.
Abstract: A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ˆ-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.
TL;DR: A finite algorithm for obtaining the asymptotical solution for an arbitrary problem of such type with Turan-type problems for graphs with colored vertices is described.
Abstract: The “best” inequalities of type P{(ζ, η)⊂ E} ≧f(P{η⊂ D1}P{η⊂Dm}) for independent and identically distributed random elements ζ and η can be reduced to Turan-type problems for graphs with colored vertices. In the present work we describe a finite algorithm for obtaining the asymptotical solution for an arbitrary problem of such type. In the case of two colors we obtain the final form of asymptotic solution without using the algorithm.
TL;DR: An O((log n)2) time parallel algorithm, using n processors, for finding the lexicographically first depth first search tree in the random graph G n, p, with p fixed, is described.
Abstract: We describe an O((log n)2) time parallel algorithm, using n processors, for finding the lexicographically first depth first search tree in the random graph G n, p, with p fixed. The problem itself is complete for P, and so is unlikely to be efficiently parallelizable always.