We define and study the notion of min-wise independent families of permutations. We say that F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(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 π. 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.

A locality sensitive hashing scheme is a distribution on a family F of hash functions operating on a collection of objects, such that for two objects x, y, Prh∈F[h(x) = h(y)] = sim(x,y), where sim(x,y) ∈ [0, 1] is some similarity function defined on the collection of objects. Such a scheme leads to a compact representation of objects so that similarity of objects can be estimated from their compact sketches, and also leads to efficient algorithms for approximate nearest neighbor search and clustering. Min-wise independent permutations provide an elegant construction of such a locality sensitive hashing scheme for a collection of subsets with the set similarity measure sim(A, B) = |A∩B| |A∪B |. We show that rounding algorithms for LPs and SDPs used in the context of approximation algorithms can be viewed as locality sensitive hashing schemes for several interesting collections of objects. Based on this insight, we construct new locality sensitive hashing schemes for: 1. A collection of vectors with the distance between ⃗u and ⃗v measured by θ(⃗u,⃗v)/π, where θ(⃗u,⃗v) is the angle between ⃗u and ⃗v. This yields a sketching scheme for estimating the cosine similarity measure between two vectors, as well as a simple alternative to minwise independent permutations for estimating set similarity. 2. A collection of distributions on n points in a metric space, with distance between distributions measured by the Earth Mover Distance (EMD), (a popular distance measure in graphics and vision). Our hash functions map distributions to points in the metric space such that, for distributions P and Q,

In this chapter we consider the following problem: given a set P of points in a high-dimensional space, construct a data structure which given any query point q nds the point in P closest to q. This problem, called nearest neighbor search is of significant importance to several areas of computer science, including pattern recognition, searching in multimedial data, vector compression [GG91], computational statistics [DW82], and data mining. Many of these applications involve data sets which are very large (e.g., a database containing Web documents could contain over one billion documents). Moreover, the dimensionality of the points is usually large as well (e.g., in the order of a few hundred). Therefore, it is crucial to design algorithms which scale well with the database size as well as with the dimension. The nearest-neighbor problem is an example of a large class of proximity problems, which, roughly speaking, are problems whose definitions involve the notion of...

We compare several algorithms for identifying mirrored hosts on the World Wide Web. The algorithms operate on the basis of URL strings and linkage data: the type of information easily available from web proxies and crawlers. Identification of mirrored hosts can improve web-based information retrieval in several ways: First, by identifying mirrored hosts, search engines can avoid storing and returning duplicate documents. Second, several new information retrieval techniques for the Web make inferences based on the explicit links among hypertext documents -- mirroring perturbs their graph model and degrades performance. Third, mirroring information can be used to redirect users to alternate mirror sites to compensate for various failures, and can thus improve the performance of web browsers and proxies. # This work was presented at the Workshop on Organizing Web Space at the Fourth ACM Conference on Digital Libraries 1999. We evaluated 4 classes of "top-down" algorithms for detecting ...

Ongoing advancements in technology lead to everincreasing storage capacities. In spite of this, optimizing storage usage can still provide rich dividends. Several techniques based on delta-encoding and duplicate block suppression have been shown to reduce storage overheads, with varying requirements for resources such as computation and memory. We propose a new scheme for storage reduction that reduces data sizes with an effectiveness comparable to the more expensive techniques, but at a cost comparable to the faster but less effective ones. The scheme, called Redundancy Elimination at the Block Level (REBL), leverages the benefits of compression, duplicate block suppression, and delta-encoding to eliminate a broad spectrum of redundant data in a scalable and efficient manner. REBL generally encodes more compactly than compression (up to a factor of 14) and a combination of compression and duplicate suppression (up to a factor of 6.7). REBL also encodes similarly to a technique based on delta-encoding, reducing overall space significantly in one case. Furthermore, REBL uses super-fingerprints, a technique that reduces the data needed to identify similar blocks while dramatically reducing the computational requirements of matching the blocks: it turns comparisons into hash table lookups. As a result, using super-fingerprints to avoid enumerating matching data objects decreases computation in the resemblance detection phase of REBL by up to a couple orders of magnitude.

Virtual machine monitors (VMMs) are a popular platform for Internet hosting centers and cloud-based compute services. By multiplexing hardware resources among virtual machines (VMs) running commodity operating systems, VMMs decrease both the capital outlay and management overhead of hosting centers. Appropriate placement and migration policies can take advantage of statistical multiplexing to effectively utilize available processors. However, main memory is not amenable to such multiplexing and is often the primary bottleneck in achieving higher degrees of consolidation. Previous efforts have shown that content-based page sharing provides modest decreases in the memory footprint of VMs running similar operating systems and applications. Our studies show that significant additional gains can be had by leveraging both sub-page level sharing (through page patching) and in-core memory compression. We build Difference Engine, an extension to the Xen virtual machine monitor, to support each of these—in addition to standard copy-on-write full page sharing—and demonstrate substantial savings not only between VMs running similar applications and operating systems (up to 90%), but even across VMs running disparate workloads (up to 65%). In head-to-head memory-savings comparisons, Difference Engine outperforms VMware ESX server by a factor of 1.5 for homogeneous workloads and by a factor of 1.6–2.5 for heterogeneous workloads. In all cases, the performance overhead of Difference Engine is less than 7%. 1

Many objects, such as les, electronic messages, and web pages, contain overlapping content. Numerous past research projects have observed that one can compress one object relative to another one by computing the differences between the two, but these delta-encoding systems have almost invariably required knowledge of a specific relationship between them most commonly, two versions using the same name at different points in time. We consider cases in which this relationship is determined dynamically, by efficiently determining when a sufficient resemblance exists between two objects in a relatively large collection. We look at specific examples of this technique, namely web pages, email, and files in a file system, and evaluate the potential data reduction and the factors that influence this reduction. We find that delta-encoding using this resemblance detection technique can improve on simple compression by up to a factor of two, depending on workload, and that a small fraction of objects can potentially account for a large portion of these savings.

Clustering is one of the most crucial techniques for dealing with the massive amount of information present on the web. Clustering can either be performed once offline, independent of search queries, or performed online on the results of search queries. Our offline approach aims to efficiently cluster similar pages on the web, using the technique of Locality-Sensitive Hashing (LSH), in which web pages are hashed in such a way that similar pages have a much higher probability of collision than dissimilar pages. Our preliminary experiments on the Stanford WebBase have shown that the hash-based scheme can be scaled to millions of urls. 1.

In the windowed data stream model, we observe items coming in over time. At any time t, we consider the window of the last N observations a t\Gamma(N \Gamma1) ; a t\Gamma(N \Gamma2) ; : : : ; a t , each a i 2 f1; : : : ; ug; we are allowed to ask queries about the data in the window, say, we wish to compute the minimum or the median of the items in the window. A crucial restriction is that we are only allowed o(N) (often polylogarithmic in N) storage space, that is, space smaller than the window size, so the items within the window can not be archived. Window data stream model arose out of the need to formally reason about the underlying data analyses problems in applications like inter-networking and transactions processing.

We present TAPER, a scalable data replication protocol that synchronizes a large collection of data across multiple geographically distributed replica locations. TAPER can be applied to a broad range of systems, such as software distribution mirrors, content distribution networks, backup and recovery, and federated file systems. TA-PER is designed to be bandwidth efficient, scalable and content-based, and it does not require prior knowledge of the replica state. To achieve these properties, TA-PER provides: i) four pluggable redundancy elimination phases that balance the trade-off between bandwidth savings and computation overheads, ii) a hierarchical hash tree based directory pruning phase that quickly matches identical data from the granularity of directory trees to individual files, iii) a content-based similarity detection technique using Bloom filters to identify similar files, and iv) a combination of coarse-grained chunk matching with finer-grained block matches to achieve bandwidth efficiency. Through extensive experiments on various datasets, we observe that in comparison with rsync, a widely-used directory synchronization tool, TAPER reduces bandwidth by 15 % to 71%, performs faster matching, and scales to a larger number of replicas. 1