Caching is widely recognized as an effective mechanism for improving the performance of the World Wide Web. One of the key components in engineering the Web caching systems is designing document pl...

A standard technique from the hashing literature is to use two hash functions h1(x) and h2(x) to simulate additional hash functions of the form gi(x) = h1(x) + ih2(x). We demonstrate that this tech...

We study the composition of random permutations drawn from a small family of O(n3) simple permutations on {0,1}n. Specifically, we ask how many randomly selected simple permutations need be compose...

In standard property testing, the task is to distinguish between objects that have a property and those that are e-far from , for some e > 0. In this setting, it is perfectly acceptable for the tes...

Let G = ( V , E ) be an undirected weighted graph on | V | = n vertices and | E | = m edges. A t -spanner of the graph G , for any t ý 1, is a subgraph ( V , E S ), E S ý E , such that the distance...

We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman and colleagu...

Let n be a positive integer and > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(, n) to be the graph with vertex set Vn, in ...

Level-Compressed (in short LC) tries were introduced by Andersson and Nilsson in 1993. They are compacted versions of tries in which, from the top down, maximal height complete subtrees are level compressed. We show that when the input consists of n independent strings with independent Bernoulli (p) bits, p ≠ 1/2, then the expected depth of a typical node is in probability asymptotic to where H − p log p − (1 − p) log (1 − p) is the Shannon entropy of the source, and H−∞ = log (1 / min(p, 1 − p)). The height is in probability asymptotic to where H2 = log(1/(p2 + (1−p)2)). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005

Abstract In a simple graph G without isolated nodes the following random experiment is carried out: each node chooses one of its neighbors uniformly at random. We say a rendezvous occurs if there are adjacent nodes u and v such that u chooses v and v chooses u; the probability that this happens is denoted by s ( G ). Métivier, Saheb, and Zemmari [“Randomized rendezvous,” Algorithms, trees, combinatorics, and probabilities, Eds. G. Gardy and A. Mokkadem, Trends in Mathematics, 183–194. Birkhäuser, Boston, 2000.] asked whether it is true that s ( G ) ≥ s ( K n ) for all n ‐node graphs G, where K n is the complete graph on n nodes. We show that this is the case. Moreover, we show that evaluating s ( G ) for a given graph G is a #P‐complete problem, even if only d ‐regular graphs are considered, for any d ≥ 5. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005

This article initiates the study of testing properties of directed graphs. In particular, the article considers the most basic property of directed graphs--acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency-matrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of O(1/e2), where e is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2/3, every graph whose adjacency matrix should be modified in at least e fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an Ω(|V|1/3) lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. Along with acyclicity, this article considers the property of strong connectivity. Contrasting upper and lower bounds are proved for the incidence list representation. In particular, if the testing algorithm can query on both incoming and outgoing edges at each vertex, then it is possible to test strong connectivity in O(1/e) time and query complexity. On the other hand, if the testing algorithm only has access to outgoing edges, then Ω(√N) queries are required to test for strong connectivity.