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1,355
Expander Codes
 IEEE Transactions on Information Theory
, 1996
"... We present a new class of asymptotically good, linear errorcorrecting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both random ..."
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Cited by 275 (10 self)
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We present a new class of asymptotically good, linear errorcorrecting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both randomized and explicit constructions for some of these codes. Experimental results demonstrate the extremely good performance of the randomly chosen codes. 1. Introduction We present a new class of error correcting codes derived from expander graphs. These codes have the advantage that they can be decoded very efficiently. That makes them particularly suitable for devices which must decode cheaply, such as compact disk players and remote satellite receivers. We hope that the connection we draw between expander graphs and error correcting codes will stimulate research in both fields. 1.1. Error correcting codes An error correcting code is a mapping from messages to codewords such that the mappi...
Concentration Of Measure And Isoperimetric Inequalities In Product Spaces
, 1995
"... . The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product# N of probability spaces has measure at least one half, "most" of the points of# N are "close" to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word "most" ..."
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Cited by 271 (3 self)
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. The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product# N of probability spaces has measure at least one half, "most" of the points of# N are "close" to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word "most" is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work "close" is defined in three main ways, each of them giving rise to related, but di#erent inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools. AMS Classification numbers: Primary 60E15, 28A35, 60G99; Secondary 60G15, 68C15. Typeset by A M ST E X 1 2 M. TALAGRAND Table of Contents I.
Priority Encoding Transmission
 IEEE Transactions on Information Theory
, 1994
"... We introduce a new method, called Priority Encoding Transmission, for sending messages over lossy packetbased networks. When a message is to be transmitted, the user specifies a priority value for each part of the message. Based on the priorities, the system encodes the message into packets for tra ..."
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Cited by 264 (11 self)
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We introduce a new method, called Priority Encoding Transmission, for sending messages over lossy packetbased networks. When a message is to be transmitted, the user specifies a priority value for each part of the message. Based on the priorities, the system encodes the message into packets for transmission and sends them to (possibly multiple) receivers. The priority value of each part of the message determines the fraction of encoding packets sufficient to recover that part. Thus, even if some of the encoding packets are lost enroute, each receiver is still able to recover the parts of the message for which a sufficient fraction of the encoding packets are received. International Computer Science Institute, Berkeley, California. Research supported in part by National Science Foundation operating grant NCR941610 y Computer Science Department, Swiss Federal Institute of Technology, Zurich, Switzerland. Research done while a postdoc at the International Computer Science Institute...
SmallBias Probability Spaces: Efficient Constructions and Applications
 SIAM J. Comput
, 1993
"... We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is ..."
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Cited by 258 (15 self)
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We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are fflbiased can be used to construct "almost" kwise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using fflbiased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...
Broadcast Encryption
, 1994
"... We introduce new theoretical measures for the qualitative and quantitative assessment of encryption schemes designed for broadcast transmissions. The goal is to allow a central broadcast site to broadcast secure transmissions to an arbitrary set of recipients while minimizing key management related ..."
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Cited by 249 (10 self)
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We introduce new theoretical measures for the qualitative and quantitative assessment of encryption schemes designed for broadcast transmissions. The goal is to allow a central broadcast site to broadcast secure transmissions to an arbitrary set of recipients while minimizing key management related transmissions. We present several schemes that allow a center to broadcast a secret to any subset of privileged users out of a universe of size n so that coalitions of k users not in the privileged set cannot learn the secret. The most interesting scheme requires every user to store O(k log k log n) keys and the center to broadcast O(k 2 log 2 k log n) messages regardless of the size of the privileged set. This scheme is resilient to any coalition of k users. We also present a scheme that is resilient with probability p against a random subset of k users. This scheme requires every user to store O(log k log(1=p)) keys and the center to broadcast O(k log 2 k log(1=p)) messages. Prel...
Correlation Clustering
 MACHINE LEARNING
, 2002
"... We consider the following clustering problem: we have a complete graph on # vertices (items), where each edge ### ## is labeled either # or depending on whether # and # have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as mu ..."
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Cited by 222 (4 self)
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We consider the following clustering problem: we have a complete graph on # vertices (items), where each edge ### ## is labeled either # or depending on whether # and # have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as much as possible with the edge labels. That is, we want a clustering that maximizes the number of # edges within clusters, plus the number of edges between clusters (equivalently, minimizes the number of disagreements: the number of edges inside clusters plus the number of # edges between clusters). This formulation is motivated from a document clustering problem in which one has a pairwise similarity function # learned from past data, and the goal is to partition the current set of documents in a way that correlates with # as much as possible; it can also be viewed as a kind of "agnostic learning" problem. An interesting
Approximate distance oracles
 J. ACM
"... Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
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Cited by 210 (8 self)
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Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs. 1
Free Bits, PCPs and NonApproximability  Towards Tight Results
, 1996
"... This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems. ..."
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Cited by 208 (40 self)
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This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.
Efficient Algorithms for Discovering Association Rules
, 1994
"... Association rules are statements of the form "for 90 % of the rows of the relation, if the row has value 1 in the columns in set W , then it has 1 also in column B". Agrawal, Imielinski, and Swami introduced the problem of mining association rules from large collections of data, and gave a method ba ..."
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Cited by 204 (11 self)
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Association rules are statements of the form "for 90 % of the rows of the relation, if the row has value 1 in the columns in set W , then it has 1 also in column B". Agrawal, Imielinski, and Swami introduced the problem of mining association rules from large collections of data, and gave a method based on successive passes over the database. We give an improved algorithm for the problem. The method is based on careful combinatorial analysis of the information obtained in previous passes; this makes it possible to eliminate unnecessary candidate rules. Experiments on a university course enrollment database indicate that the method outperforms the previous one by a factor of 5. We also show that sampling is in general a very efficient way of finding such rules. Keywords: association rules, covering sets, algorithms, sampling. 1 Introduction Data mining (database mining, knowledge discovery in databases) has recently been recognized as a promising new field in the intersection of databa...
The Power of Two Choices in Randomized Load Balancing
 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS
, 1996
"... Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d ..."
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Cited by 201 (23 self)
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Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d bins chosen independently and uniformly at random. It has recently been shown that the maximum load is then only log log n log d +O(1) with high probability. Thus giving each ball two choices instead of just one leads to an exponential improvement in the maximum load. This result demonstrates the power of two choices, and it has several applications to load balancing in distributed systems. In this thesis, we expand upon this result by examining related models and by developing techniques for stu...