Chernoff--Hoeffding 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 Chernoff--Hoeffding bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routi...

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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

Micah Adler # Department of Computer Science, University of Massachusetts, Amherst, MA 01003-4610, USA micah@cs.umass.edu Eran Halperin + Science Institute and Computer Science Division University of California Berkeley, CA 94720.

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 non-bipartite 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 A|E|, for all A between 0.845 and 1. This extends a result of Karloff. 1.

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

Contents 1 Chernoff--Hoeffding 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 CH-bounds 23 2.1 Data Structures: Skip Lists and Treaps . . . . . . . . . .

Abstract. It is well-known 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 multi-party computation with t < n/2 after the first detection of such a time slot.

We consider a synchronous model of distributed computation in which n nodes communicate via point-topoint 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 hashing-based 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.

Proofs of classical Chernoff-Hoeffding bounds has been used to obtain polynomial-time 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 polynomial-time 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...