Michal Parnas

Tolerant property testing and distance approximation

In this paper we study a generalization of standard property testing where the algorithms are required to be more tolerant with respect to objects that do not have, but are close to having, the property. Specifically, a tolerant property testing algorithm is required to accept objects that are e1 -close to having a given property P and reject objects that are e2-far from having P, for some parameters 0 ≤ e1 < e2 1. Another related natural extension of standard property testing that we study, is distance approximation. Here the algorithm should output an estimate e of the distance of the object to P, where this estimate is sufficiently close to the true distance of the object to P. We first formalize the notions of tolerant property testing and distance approximation and discuss the relationship between the two tasks, as well as their relationship to standard property testing. We then apply these new notions to the study of two problems: tolerant testing of clustering and distance approximation for monotonicity. We present and analyze algorithms whose query complexity is either polylogarithmic or independent of the size of the input.

Multi-index hashing for information retrieval

We describe a technique for building hash indices for a large dictionary of strings. This technique permits robust retrieval of strings from the dictionary even when the query pattern has a significant number of errors. This technique is closely related to the classical Turan problem for hypergraphs. We propose a general method of multi-index construction by generalizing certain Turan hypergraphs. We also develop an accompanying theory for analyzing such hashing schemes. The resulting algorithms have been implemented and can be applied to a wide variety of recognition and retrieval problems.<<ETX>>

Testing Basic Boolean Formulae

An ignition system for a multicylinder engine comprising a magneto having an igniting exciter coil to induce an AC voltage in synchronism with rotation of the engine; ignition coils having secondary coil portions respectively connected to ignition plugs for cylinders of the engine, to supply a high voltage to the respective ignition plugs; and a primary current controlling semiconductor switch connected to the igniting exciter coil in parallel therewith to be turned on in advance of respective igniting positions of the cylinders so that a current flows through the semiconductor switch. The ignition system further comprises a signal generator driven by the engine and having signal coils to generate timing signals at the respective igniting positions of the cylinders; and primary current conducting semiconductor switches provided between the respective primary coil portions of the ignition coils and the igniting exciter coil to be turned on by the respective timing signals from the signal generator to abruptly pass the primary current through the respective primary coil portions of the ignition coils whereby a high voltage is established across the secondary coil portions of the ignition coils to spark the ignition plugs.

Fast connected components algorithms for the EREW PRAM

We present fast and efficient parallel algorithms for finding the connected components of an undirected graph. These algorithms run on the exclusive-read, exclusive-write (EREW) PRAM. On a graph with n vertices and m edges, our randomized algorithm runs in O(logn) time using (m + n1+ǫ)/ logn EREW processors (for any fixed ǫ > 0). A variant uses (m + n)/ logn processors and runs in O(logn log logn) time. A deterministic version of the algorithm runs in O(log n) time using m + n EREW processors.

Testing membership in parenthesis languages

We continue the investigation of properties defined by formal languages. This study was initiated by Alon et al. [1], who described an algorithm for testing properties defined by regular languages. Alon et al. also considered several context free languages, and in particular Dyck languages, which contain strings of properly balanced parentheses. They showed that the first Dyck language, which contains strings over a single type of pairs of parentheses, is testable in time independent of n, where n is the length of the input string. However, the second Dyck language, defined over two types of parentheses, requires Ω (log n) queries. Here we describe a sublinear-time algorithm for testing all Dyck languages. Specifically, the running time of our algorithm is O(n2/3/e3), where e is the given distance parameter. Furthermore, we improve the lower bound for testing Dyck languages to Ω (n1/11) for constant e. We also describe a testing algorithm for the context free language LREV = {uurvvr : u, v ∈ Σ*}, where Σ is a fixed alphabet. The running time of our algorithm is O(√n/e), which almost matches the lower bound given by Alon et al. [1].

Efficient Dynamic Traitor Tracing

The notion of traitor tracing was introduced by Chor, Fiat, and Naor [Tracing Traitors, Lecture Notes in Comput. Sci. 839, 1994, pp. 257--270] in order to combat piracy scenarios. Recently, Fiat and Tassa [ Tracing Traitors, Lecture Notes in Comput. Sci. 1666, 1999, pp. 354--371] proposed a dynamic traitor tracing scenario, in which the algorithm adapts dynamically according to the responses of the pirate. Let n be the number of users and p the number of traitors. Our main result is an algorithm which locates p traitors, even if p is unknown, using a watermarking alphabet of size p+1 and an optimal number of

Fast Connected Components Algorithms for the EREW PRAM

\Theta(p^2 + p\log n)

Testing metric properties

rounds. This improves the exponential number of rounds achieved by Fiat and Tassa in this case. We also present two algorithms that use a larger alphabet: for an alphabet of size p+c+1,

Testing Parenthesis Languages

c\geq1

Neighborhood Preserving Hashing and Approximate Queries

, an algorithm that uses O(p2/c+ p log n) rounds; for an alphabet of size pc+1, an algorithm that uses O(p logcn) rounds. Our final result is a lower bound of