Graham Cormode

An improved data stream summary: the count-min sketch and its applications

We introduce a new sublinear space data structure--the count-min sketch--for summarizing data streams. Our sketch allows fundamental queries in data stream summarization such as point, range, and inner product queries to be approximately answered very quickly; in addition, it can be applied to solve several important problems in data streams such as finding quantiles, frequent items, etc. The time and space bounds we show for using the CM sketch to solve these problems significantly improve those previously known--typically from 1/e2 to 1/e in factor.

What's hot and what's not: tracking most frequent items dynamically

Most database management systems maintain statistics on the underlying relation. One of the important statistics is that of the “hot items” in the relation: those that appear many times (most frequently, or more than some threshold). For example, end-biased histograms keep the hot items as part of the histogram and are used in selectivity estimation. Hot items are used as simple outliers in data mining, and in anomaly detection in many applications.We present new methods for dynamically determining the hot items at any time in a relation which is undergoing deletion operations as well as inserts. Our methods maintain small space data structures that monitor the transactions on the relation, and, when required, quickly output all hot items without rescanning the relation in the database. With user-specified probability, all hot items are correctly reported. Our methods rely on ideas from “group testing.” They are simple to implement, and have provable quality, space, and time guarantees. Previously known algorithms for this problem that make similar quality and performance guarantees cannot handle deletions, and those that handle deletions cannot make similar guarantees without rescanning the database. Our experiments with real and synthetic data show that our algorithms are accurate in dynamically tracking the hot items independent of the rate of insertions and deletions.

An Improved Data Stream Summary: The Count-Min Sketch and Its Applications

We introduce a new sublinear space data structure—the Count-Min Sketch— for summarizing data streams. Our sketch allows fundamental queries in data stream summarization such as point, range, and inner product queries to be approximately answered very quickly; in addition, it can be applied to solve several important problems in data streams such as finding quantiles, frequent items, etc. The time and space bounds we show for using the CM sketch to solve these problems significantly improve those previously known — typically from 1/e 2 to 1/e in factor.

Finding frequent items in data streams

The frequent items problem is to process a stream of items and find all items occurring more than a given fraction of the time. It is one of the most heavily studied problems in data stream mining, dating back to the 1980s. Many applications rely directly or indirectly on finding the frequent items, and implementations are in use in large scale industrial systems. However, there has not been much comparison of the different methods under uniform experimental conditions. It is common to find papers touching on this topic in which important related work is mischaracterized, overlooked, or reinvented. In this paper, we aim to present the most important algorithms for this problem in a common framework. We have created baseline implementations of the algorithms, and used these to perform a thorough experimental study of their properties. We give empirical evidence that there is considerable variation in the performance of frequent items algorithms. The best methods can be implemented to find frequent items with high accuracy using only tens of kilobytes of memory, at rates of millions of items per second on cheap modern hardware.

Combinatorial Algorithms for Compressed Sensing

In sparse approximation theory, the fundamental problem is to reconstruct a signal AisinRn from linear measurements (A,psii) with respect to a dictionary of psiis. Recently, there is focus on the novel direction of Compressed Sensing where the reconstruction can be done with very few-O(klogn)-linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, the results prove that there exists a single O(klogn)timesn measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying mathematics and because of its potential applications. In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1/epsiv, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1+epsiv and improves the reconstruction time from poly(n) to poly (klogn). Our second result is a randomized construction of O(kpolylog(n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on compressed sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including learning theory, streaming algorithms and complexity theory for this case. Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement.

Semantics of Ranking Queries for Probabilistic Data and Expected Ranks

When dealing with massive quantities of data, top-k queries are a powerful technique for returning only the k most relevant tuples for inspection, based on a scoring function. The problem of efficiently answering such ranking queries has been studied and analyzed extensively within traditional database settings. The importance of the top-k is perhaps even greater in probabilistic databases, where a relation can encode exponentially many possible worlds. There have been several recent attempts to propose definitions and algorithms for ranking queries over probabilistic data. However, these all lack many of the intuitive properties of a top-k over deterministic data. Specifically, we define a number of fundamental properties, including exact-k, containment, unique-rank, value-invariance, and stability, which are all satisfied by ranking queries on certain data. We argue that all these conditions should also be fulfilled by any reasonable definition for ranking uncertain data. Unfortunately, none of the existing definitions is able to achieve this. To remedy this shortcoming, this work proposes an intuitive new approach of expected rank. This uses the well-founded notion of the expected rank of each tuple across all possible worlds as the basis of the ranking. We are able to prove that, in contrast to all existing approaches, the expected rank satisfies all the required properties for a ranking query. We provide efficient solutions to compute this ranking across the major models of uncertain data, such as attribute-level and tuple-level uncertainty. For an uncertain relation of N tuples, the processing cost is O(N logN)—no worse than simply sorting the relation. In settings where there is a high cost for generating each tuple in turn, we provide pruning techniques based on probabilistic tail bounds that can terminate the search early and guarantee that the top-k has been found. Finally, a comprehensive experimental study confirms the effectiveness of our approach.

Approximation algorithms for clustering uncertain data

There is an increasing quantity of data with uncertainty arising from applications such as sensor network measurements, record linkage, and as output of mining algorithms. This uncertainty is typically formalized as probability density functions over tuple values. Beyond storing and processing such data in a DBMS, it is necessary to perform other data analysis tasks such as data mining. We study the core mining problem of clustering on uncertain data, and define appropriate natural generalizations of standard clustering optimization criteria. Two variations arise, depending on whether a point is automatically associated with its optimal center, or whether it must be assigned to a fixed cluster no matter where it is actually located. For uncertain versions of k-means and k-median, we show reductions to their corresponding weighted versions on data with no uncertainties. These are simple in the unassigned case, but require some care for the assigned version. Our most interesting results are for uncertain k-center, which generalizes both traditional k-center and k-median objectives. We show a variety of bicriteria approximation algorithms. One picks O(kε--1log2n) centers and achieves a (1 + ε) approximation to the best uncertain k-centers. Another picks 2k centers and achieves a constant factor approximation. Collectively, these results are the first known guaranteed approximation algorithms for the problems of clustering uncertain data.

Holistic UDAFs at streaming speeds

Many algorithms have been proposed to approximate holistic aggregates, such as quantiles and heavy hitters, over data streams. However, little work has been done to explore what techniques are required to incorporate these algorithms in a data stream query processor, and to make them useful in practice.In this paper, we study the performance implications of using user-defined aggregate functions (UDAFs) to incorporate selection-based and sketch-based algorithms for holistic aggregates into a data stream management systems query processing architecture. We identify key performance bottlenecks and tradeoffs, and propose novel techniques to make these holistic UDAFs fast and space-efficient for use in high-speed data stream applications. We evaluate performance using generated and actual IP packet data, focusing on approximating quantiles and heavy hitters. The best of our current implementations can process streaming queries at OC48 speeds (2x 2.4Gbps).

Space efficient mining of multigraph streams

The challenge of monitoring massive amounts of data generated by communication networks has led to the interest in data stream processing. We study streams of edges in massive communication multigraphs, defined by (source, destination) pairs. The goal is to compute properties of the underlying graph while using small space (much smaller than the number of communicants), and to avoid bias introduced because some edges may appear many times, while others are seen only once. We give results for three fundamental problems on multigraph degree sequences: estimating frequency moments of degrees, finding the heavy hitter degrees, and computing range sums of degree values. In all cases we are able to show space bounds for our summarizing algorithms that are significantly smaller than storing complete information. We use a variety of data stream methods: sketches, sampling, hashing and distinct counting, but a common feature is that we use cascaded summaries: nesting multiple estimation techniques within one another. In our experimental study, we see that such summaries are highly effective, enabling massive multigraph streams to be effectively summarized to answer queries of interest with high accuracy using only a small amount of space.

Conquering the Divide: Continuous Clustering of Distributed Data Streams

Data is often collected over a distributed network, but in many cases, is so voluminous that it is impractical and undesirable to collect it in a central location. Instead, we must perform distributed computations over the data, guaranteeing high quality answers even as new data arrives. In this paper, we formalize and study the problem of maintaining a clustering of such distributed data that is continuously evolving. In particular, our goal is to minimize the communication and computational cost, still providing guaranteed accuracy of the clustering. We focus on the k-center clustering, and provide a suite of algorithms that vary based on which centralized algorithm they derive from, and whether they maintain a single global clustering or many local clusterings that can be merged together. We show that these algorithms can be designed to give accuracy guarantees that are close to the best possible even in the centralized case. In our experiments, we see clear trends among these algorithms, showing that the choice of algorithm is crucial, and that we can achieve a clustering that is as good as the best centralized clustering, with only a small fraction of the communication required to collect all the data in a single location.