The dynamic dictionary problem is considered: provide an algorithm for storing a dynamic set, allowing the operations insert, delete, and lookup. A dynamic perfect hashing strategy is given: a randomized algorithm for the dynamic dictionary problem that takes O(1) worst-case time for lookups and O(1) amortized expected time for insertions and deletions; it uses space proportional to the size of the set stored. Furthermore, lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved. This class encompasses realistic hashing-based schemes that use linear space. Such algorithms have amortized worst-case time complexity \Omega(log n) for a sequence of n insertions and

In this paper, weinvestigate combinatorial properties and constructions of two recent topics of cryptographic interest, namely frameproof codes for digital ngerprinting, and traceability schemes for broadcast encryption. We rstgive combinatorial descriptions of these two objects in terms of set systems, and also discuss the Hamming distance of frameproof codes when viewed as error-correcting codes. From these descriptions, it is seen that existence of a c-traceability scheme implies the existence of a c-frameproof code. We then give several constructions of frameproof codes and traceability schemes by using combinatorial structures such as t-designs, packing designs, error-correcting codes and perfect hash families. We also investigate embeddings of frameproof codes and traceability schemes, which allow agiven scheme to be expanded at a later date to accommodate more users. Finally, we look brie y at bounds which establish necessary conditions for existence of these structures. 1

Motivated by boolean queries in text database systems, we consider the problems of finding the intersection, union, or difference of a collection of sorted sets. While the worst-case complexity of these problems is straightforward, we consider a notion of complexity that depends on the particular instance. We develop the idea of a proof that a given set is indeed the correct answer. Proofs, and in particular shortest proofs, are characterized. We present adaptive algorithms that make no a priori assumptions about the problem instance, and show that their running times are within a constant factor of optimal with respect to a natural measure of the difficulty of an instance. In the process, we develop a framework for designing and evaluating adaptive algorithms in the comparison model. 1 Introduction and Overview Our work can be seen in the general context of performing searches quickly in a database or data warehousing environment. The broad issue is that of characterizing what type ...

Frameproof codes were introduced by Boneh and Shaw as a method of "digital fingerprinting" which prevents a coalition of a specified size c from framing a user not in the coalition. Stinson and Wei then gave a combinatorial formulation of the problem in terms of certain types of extremal set sytems. In this paper, we study frameproof codes that provide a certain (weak) form of traceability. We extend our combinatorial formulation to address this stronger requirement, and show that the problem is solved by using (i; j)-separating systems, as defined by Friedman, Graham and Ullman. Using constructions based on perfect hash families, we give the first efficient explicit constructions for these objects for general values of i and j. We also review nonconstructive existence results that are based on probabilistic arguments. Then we look at two other, related concepts, namely key distribution patterns and non-adaptive group testing algorithms. We again approach these problems from the point...

Abstract — We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced in [1], [2] that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments put in evidence that our heuristics performs significantly better than state of the art heuristics. I.

We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced in [1], [2] that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments put in evidence that our heuristics performs significantly better than state of the art heuristics.

In a given dynamical system there are essentially two different types of information which could be of practical interest: on the one hand there is the need to describe the behavior of single trajectories in detail. This information is helpful for the analysis of transient behavior and also in the investigation of geometric properties of dynamical systems. On the other hand, if the underlying invariant set is generated by complicated dynamics then the computation of single trajectories may give misleading results. In this case there still exists important set related information covering both topological and statistical aspects of the underlying dynamical behavior. Within the DFG-Schwerpunkt we have focussed on the development of set oriented methods for the numerical approximation of --- invariant sets (e.g. invariant manifolds, global attractors, chain recurrent sets); --- (natural) invariant measures; --- almost invariant sets. The basic concept is a subdivision algorithm which is sim...

Contents 1 Introduction 1 2 The Computation of Invariant Sets 2 2.1 Brief Review on Invariant Sets . . . . . . . . . . . . . . . . . . 2 2.2 The Computation of Relative Global Attractors . . . . . . . . 3 2.3 Convergence Behavior and Error Estimate . . . . . . . . . . . 6 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 The Computation of Chain Recurrent Sets . . . . . . . . . . . 9 2.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The Computation of Invariant Manifolds 12 3.1 Description of the Method . . . . . . . . . . . . . . . . . . . . 13 3.2 Convergence Behavior and Error Estimate . . . . . . . . . . . 14 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 15 4 The Computation of SRB-Measures 18 4.1 Brief Review on SRB-Measures and Small Random Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Spectral Approximation for the Per

An (n; m;w)-perfect hash family is a set of functions F such that f : f1; : : : ; ng ! f1; : : : ; mg for each f 2 F , and for any X ` f1; : : : ; ng such that jX j = w, there exists at least one f 2 F such that f j X is one-to-one. Perfect hash families have been extensively studied by computer scientists for over 15 years, mainly due to their applications in database management. In particular, much attention has been given to finding efficient algorithms to construct perfect hash families. In this paper, we study perfect hash families from a combinatorial viewpoint, and describe some new recursive constructions for these objects. 1 Introduction We begin with a definition. An (n; m;w)-perfect hash family is a set of functions F such that f : f1; : : : ; ng ! f1; : : : ; mg for each f 2 F , and for any X ` f1; : : : ; ng such that jXj = w, there exists at least one f 2 F such that f j X is one-to-one. We will use the notation PHF(N ; n; m;w) for an (n; m;w)-perfect hash family wit...

Abstract. Dynamic programming is a classical programming technique, applicable in a wide variety of domains such as stochastic systems analysis, operations research, combinatorics of discrete structures, flow problems, parsing of ambiguous languages, and biosequence analysis. Little methodology has hitherto been available to guide the design of such algorithms. The matrix recurrences that typically describe a dynamic programming algorithm are difficult to construct, error-prone to implement, and, in nontrivial applications, almost impossible to debug completely. This article introduces a discipline designed to alleviate this problem. We describe an algebraic style of dynamic programming over sequence data. We define its formal framework, based on a combination of grammars and algebras, and including a formalization of Bellman’s Principle. We suggest a language used for algorithm design on a convenient level of abstraction. We outline three ways of implementing this language, including an embedding in a lazy functional language. The workings of the