Suppose someone gives us an extremely fast program P that we can call as a black box to compute a function f . Should we trust that P works correctly? A self-testing/correcting pair allows us to: (1) estimate the probability that P (x) 6= f(x) when x is randomly chosen; (2) on any input x, compute f(x) correctly as long as P is not too faulty on average. Furthermore, both (1) and (2) take time only slightly more than Computer Science Division, U.C. Berkeley, Berkeley, California 94720, Supported by NSF Grant No. CCR 88-13632. y International Computer Science Institute, Berkeley, California 94704 z Computer Science Division, U.C. Berkeley, Berkeley, California 94720, Supported by an IBM Graduate Fellowship and NSF Grant No. CCR 88-13632. the original running time of P . We present general techniques for constructing simple to program selftesting /correcting pairs for a variety of numerical problems, including integer multiplication, modular multiplication, matrix multiplicatio...

, Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [24]). The sources considered output strings according to probability distributions in which no single string is too probable. The new model provides a fruitful viewpoint on problems studied previously as: ffl Extracting almost perfect bits from sources of weak randomness: the question of possibility as well as the question of efficiency of such extraction schemes are addressed. ffl Probabilistic Communication Complexity: it is shown that most functions have linear communication complexity in a very strong probabilistic sense. ffl Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [27]). The paper has appeared in SIAM Journal o...

a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NP-hard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Be-and timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�-colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3-colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first non-trivial approximation result as a function of the maximum degree. This result can be generalized to�-colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�-function. 1

The CDMA channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods. Furthermore, its analysis provides a comparison baseline for CDMA channels with deterministic signature waveforms spanning one symbol period. We analyze the spectral efficiency (total capacity per chip) as a function of the number of users, spreading gain, and signal-to-noise ratio, and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading. White Gaussian background noise and equal-power synchronous users are assumed. The following receivers are analyzed: a) optimal joint processing, b) single-user matched filtering, c) decorrelation, and d) MMSE linear processing.

Decision trees have proved to be valuable tools for the description, classification and generalization of data. Work on constructing decision trees from data exists in multiple disciplines such as statistics, pattern recognition, decision theory, signal processing, machine learning and artificial neural networks. Researchers in these disciplines, sometimes working on quite different problems, identified similar issues and heuristics for decision tree construction. This paper surveys existing work on decision tree construction, attempting to identify the important issues involved, directions the work has taken and the current state of the art. Keywords: classification, tree-structured classifiers, data compaction 1. Introduction Advances in data collection methods, storage and processing technology are providing a unique challenge and opportunity for automated data exploration techniques. Enormous amounts of data are being collected daily from major scientific projects e.g., Human Genome...

In this paper we study a Bayesian or average-case model of concept learning with a twofold goal: to provide more precise characterizations of learning curve (sample complexity) behavior that depend on properties of both the prior distribution over concepts and the sequence of instances seen by the learner, and to smoothly unite in a common framework the popular statistical physics and VC dimension theories of learning curves. To achieve this, we undertake a systematic investigation and comparison of two fundamental quantities in learning and information theory: the probability of an incorrect prediction for an optimal learning algorithm, and the Shannon information gain. This study leads to a new understanding of the sample complexity of learning in several existing models. 1 Introduction Consider a simple concept learning model in which the learner attempts to infer an unknown target concept f , chosen from a known concept class F of f0; 1g-valued functions over an instance space X....

We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of random bits and are robust with respect to the quality of the random bits. Using these sampling procedures to sample random inputs of polynomial time probabilistic algorithms, we can simulate the performance of some probabilistic algorithms with less random bits or with low quality random bits. We obtain the following results: 1. The error probability of an RP or BPP algorithm that operates with a constant error bound and requires n random bits, can be made exponentially small (i.e. 2 \Gamman ), with only (3 + ffl)n random bits, as opposed to standard amplification techniques that require \Omega\Gamma n 2 ) random bits for the same task. This result is nearly optimal, since the informati...

We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performance ratio of 2 - (1 - o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for k-uniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree k-uniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.

Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distance function as a “black box”. The structure is able to speed up nearest neighbor searching in a variety of settings, for example: points in low-dimensional or structured Euclidean space, strings under Hamming and edit distance, and bit vector data from an OCR application. The data structures are observed to need linear space, with a modest constant factor. The preprocessing time needed per site is observed to match the query time. The data structure can be viewed as an application of a “kd-tree ” approach in the metric space setting, using Voronoi regions of a subset in place of axis-aligned boxes. 1

This paper views clusterings as elements of a lattice. Distances between clusterings are analyzed in their relationship to the lattice. From this vantage point, we first give an axiomatic characterization of some criteria for comparing clusterings, including the variation of information and the unadjusted Rand index. Then we study other distances between partitions w.r.t these axioms and prove an impossibility result: there is no “sensible” criterion for comparing clusterings that is simultaneously (1) aligned with the lattice of partitions, (2) convexely additive, and (3) bounded. 1.