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37
ChernoffHoeffding Bounds for Applications with Limited Independence
 SIAM J. Discrete Math
, 1993
"... ChernoffHoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these, and which more importantly requires only limited independence among the rando ..."
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Cited by 104 (10 self)
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ChernoffHoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these, and which more importantly requires only limited independence among the random variables, thereby importing a variety of standard results to the case of limited independence for free. Additional methods are also presented, and the aggregate results are sharp and provide a better understanding of the proof techniques behind these bounds. They also yield improved bounds for various tail probability distributions and enable improved approximation algorithms for jobshop scheduling. The "limited independence" result implies that a reduced amount of randomness and weaker sources of randomness are sufficient for randomized algorithms whose analyses use the ChernoffHoeffding bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routi...
A stochastic process on the hypercube with applications to peertopeer networks
 Proc. STOC 2003
"... Consider the following stochastic process executed on a graph G = (V, E) whose nodes are initially uncovered. In each step, pick a node at random and if it is uncovered, cover it. Otherwise, if it has an uncovered neighbor, cover a random uncovered neighbor. Else, do nothing. This can be viewed as a ..."
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Cited by 57 (2 self)
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Consider the following stochastic process executed on a graph G = (V, E) whose nodes are initially uncovered. In each step, pick a node at random and if it is uncovered, cover it. Otherwise, if it has an uncovered neighbor, cover a random uncovered neighbor. Else, do nothing. This can be viewed as a structured coupon collector process. We show that for a large family of graphs, O(n) steps suffice to cover all nodes of the graph with high probability, where n is the number of vertices. Among these graphs are dregular graphs with d = Ω(log n log log n), random dregular graphs with d = Ω(log n) and the kdimensional hypercube where n = 2 k. This process arises naturally in answering a question on load balancing in peertopeer networks. We consider a distributed hash table in which keys are partitioned across a set of processors, and we assume that the number of processors
Are bitvectors optimal?
"... ... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must u ..."
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Cited by 57 (7 self)
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... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must use \Omega ( n2ffl2 log(n=ffl) log m) bits of storage. We also
Bounds For Dispersers, Extractors, And DepthTwo Superconcentrators
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
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Bipartite subgraphs and the smallest eigenvalue
 Combinatorics, Probability & Computing
, 2000
"... Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue µ of any nonbipartite graph on n vertices with diameter D and maximum degree ∆ satisfies µ � − ∆ + 1 (D+1)n. This improves prev ..."
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Cited by 26 (1 self)
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Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue µ of any nonbipartite graph on n vertices with diameter D and maximum degree ∆ satisfies µ � − ∆ + 1 (D+1)n. This improves previous estimates and is tight up to a constant factor. The second result is the determination of the precise approximation guarantee of the max cut algorithm of Goemans and Williamson for graphs G =(V,E) in which the size of the max cut is at least AE, for all A between 0.845 and 1. This extends a result of Karloff. 1.
Concentration
, 1998
"... Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impac ..."
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Cited by 17 (2 self)
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Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impact in probabilistic methods in discrete mathematics and in the mathematics of operational research and theoretical computer science. Recently Talagrand introduced an exciting new method for bounding probabilities of large deviations, which often proves superior to the bounded differences approach. In this paper we
Unconditional Byzantine Agreement and MultiParty Computation Secure Against Dishonest Minorities from Scratch
 In Advances in Cryptology  EUROCRYPT 2002, Lecture Notes in Computer Science
, 2002
"... Abstract. It is wellknown that n players, connected only by pairwise secure channels, can achieve unconditional broadcast if and only if the number t of cheaters satisfies t < n/3. In this paper, we show that this bound can be improved — at the sole price that the adversary can prevent successful c ..."
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Cited by 16 (6 self)
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Abstract. It is wellknown that n players, connected only by pairwise secure channels, can achieve unconditional broadcast if and only if the number t of cheaters satisfies t < n/3. In this paper, we show that this bound can be improved — at the sole price that the adversary can prevent successful completion of the protocol, but in which case all players will have agreement about this fact. Moreover, a first time slot during which the adversary forgets to cheat can be reliably detected and exploited in order to allow for future broadcasts with t < n/2. This even allows for secure multiparty computation with t < n/2 after the first detection of such a time slot.
Concentration of Measure for the Analysis of Randomised Algorithms
, 1998
"... Contents 1 ChernoffHoeffding Bounds 7 1.1 What is "Concentration of Measure"? . . . . . . . . . . . . . . . . 7 1.2 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 8 1.3 The Chernoff Bound . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Heterogeneous Variables . . . ..."
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Cited by 15 (2 self)
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Contents 1 ChernoffHoeffding Bounds 7 1.1 What is "Concentration of Measure"? . . . . . . . . . . . . . . . . 7 1.2 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 8 1.3 The Chernoff Bound . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Heterogeneous Variables . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 The Hoeffding Extension . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Useful Forms of the Bound . . . . . . . . . . . . . . . . . . . . . . 12 1.7 A Variance Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 The Normal Approximation . . . . . . . . . . . . . . . . . . . . . 14 1.9 CH Bounds with Limited Independence . . . . . . . . . . . . . . . 14 1.10 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Applying the CHbounds 23 2.1 Data Structures: Skip Lists and Treaps . . . . . . . . . .
Fast FaultTolerant Concurrent Access to Shared Objects
, 1996
"... We consider a synchronous model of distributed computation in which n nodes communicate via pointtopoint messages, subject to the following constraints: (i) in a single "step", a node can only send or receive O(log n) words, and (ii) communication is unreliable in that a constant fraction of all me ..."
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Cited by 10 (1 self)
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We consider a synchronous model of distributed computation in which n nodes communicate via pointtopoint messages, subject to the following constraints: (i) in a single "step", a node can only send or receive O(log n) words, and (ii) communication is unreliable in that a constant fraction of all messages are lost at each step due to node and/or link failures. We design and analyze a simple local protocol for providing fast concurrent access to shared objects in this faulty network environment. In our protocol, clients use a hashingbased method to access shared objects. When a large number of clients attempt to read a given object at the same time, the object is rapidly replicated to an appropriate number of servers. Once the necessary level of replication has been achieved, each remaining request for the object is serviced within O(1) expected steps. Our protocol has practical potential for supporting high levels of concurrency in distributed file systems over wide area networks.
Algorithmic ChernoffHoeffding Inequalities in Integer Programming
, 1993
"... Proofs of classical ChernoffHoeffding bounds has been used to obtain polynomialtime implementations of Spencer's derandomization method of conditional probabilities on usual finite machine models: given m events whose complements are large deviations corresponding to weighted sums of n mutually in ..."
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Cited by 9 (3 self)
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Proofs of classical ChernoffHoeffding bounds has been used to obtain polynomialtime implementations of Spencer's derandomization method of conditional probabilities on usual finite machine models: given m events whose complements are large deviations corresponding to weighted sums of n mutually independent Bernoulli trials, Raghavan's lattice approximation algorithm constructs for 0 \Gamma 1 weights and integer deviation terms in O(mn)time a point for which all events hold. For rational weighted sums of Bernoulli trials the lattice approximation algorithm or Spencer's hyperbolic cosine algorithm are deterministic precedures, but a polynomialtime implementation was not known. We resolve this problem with an O(mn 2 log mn ffl )time algorithm, whenever the probability that all events hold is at least ffl ? 0. Since such algorithms simulate the proof of the underlying large deviation inequality in a constructive way, we call it the algorithmic version of the inequality. Applicati...