Michael Mitzenmacher

Network Applications of Bloom Filters: A Survey

A Bloom filter is a simple space-efficient randomized data structure for representing a set in order to support membership queries. Bloom filters allow false positives but the space savings often outweigh this drawback when the probability of an error is controlled. Bloom filters have been used in database applications since the 1970s, but only in recent years have they become popular in the networking literature. The aim of this paper is to survey the ways in which Bloom filters have been used and modified in a variety of network problems, with the aim of providing a unified mathematical and practical framework for understanding them and stimulating their use in future applications.

A digital fountain approach to reliable distribution of bulk data

The proliferation of applications that must reliably distribute bulk data to a large number of autonomous clients motivates the design of new multicast and broadcast protocols. We describe an ideal, fully scalable protocol for these applications that we call a digital fountain. A digital fountain allows any number of heterogeneous clients to acquire bulk data with optimal efficiency at times of their choosing. Moreover, no feedback channels are needed to ensure reliable delivery, even in the face of high loss rates.We develop a protocol that closely approximates a digital fountain using a new class of erasure codes that for large block sizes are orders of magnitude faster than standard erasure codes. We provide performance measurements that demonstrate the feasibility of our approach and discuss the design, implementation and performance of an experimental system.

A Brief History of Generative Models for Power Law and Lognormal Distributions

Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.

Efficient erasure correcting codes

We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate R and any given real number /spl epsiv/ a family of linear codes of rate R which can be encoded in time proportional to ln(1//spl epsiv/) times their block length n. Furthermore, a codeword can be recovered with high probability from a portion of its entries of length (1+/spl epsiv/)Rn or more. The recovery algorithm also runs in time proportional to n ln(1//spl epsiv/). Our algorithms have been implemented and work well in practice; various implementation issues are discussed.

The power of two choices in randomized load balancing

We consider the following natural model: customers arrive as a Poisson stream of rate /spl lambda/n, /spl lambda/<1, at a collection of n servers. Each customer chooses some constant d servers independently and uniformly at random from the n servers and waits for service at the one with the fewest customers. Customers are served according to the first-in first-out (FIFO) protocol and the service time for a customer is exponentially distributed with mean 1. We call this problem the supermarket model. We wish to know how the system behaves and in particular we are interested in the effect that the parameter d has on the expected time a customer spends in the system in equilibrium. Our approach uses a limiting, deterministic model representing the behavior as n/spl rarr//spl infin/ to approximate the behavior of finite systems. The analysis of the deterministic model is interesting in its own right. Along with a theoretical justification of this approach, we provide simulations that demonstrate that the method accurately predicts system behavior, even for relatively small systems. Our analysis provides surprising implications. Having d=2 choices leads to exponential improvements in the expected time a customer spends in the system over d=1, whereas having d=3 choices is only a constant factor better than d=2. We discuss the possible implications for system design.

Improved low-density parity-check codes using irregular graphs

We construct new families of error-correcting codes based on Gallagers (1973) low-density parity-check codes. We improve on Gallagers results by introducing irregular parity-check matrices and a new rigorous analysis of hard-decision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binary-symmetric and Gaussian channels. For example, using belief propagation, for rate 1/4 codes on 16000 bits over a binary-symmetric channel, previous low-density parity-check codes can correct up to approximately 16% errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density parity-check codes may be able to match or beat turbo code performance.

Min-Wise Independent Permutations

We define and study the notion of min-wise independent families of permutations. We say that F?Sn (the symmetric group) is min-wise independent if for any set X?n and any x?X, when ? is chosen at random in F we havePr(min{?(X)}=?(x))=1|X| . In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under ?. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter near-duplicate documents. However, in the course of our investigation we have discovered interesting and challenging theoretical questions related to this concept?we present the solutions to some of them and we list the rest as open problems.

Privacy preserving keyword searches on remote encrypted data

We consider the following problem: a user

Practical loss-resilient codes

\mathcal{U}

A digital fountain approach to asynchronous reliable multicast

wants to store his files in an encrypted form on a remote file server