Zero-One Laws for Sliding Windows and Universal Sketches

Given a stream of data, a typical approach in streaming algorithms is to design a sophisticated algorithm with small memory that computes a specific statistic over the streaming data. Usually, if one wants to compute a different statistic after the stream is gone, it is impossible. But what if we want to compute a different statistic after the fact? In this paper, we consider the following fascinating possibility: can we collect some small amount of specific data during the stream that is "universal," i.e., where we do not know anything about the statistics we will want to later compute, other than the guarantee that had we known the statistic ahead of time, it would have been possible to do so with small memory? This is indeed what we introduce (and show) in this paper with matching upper and lower bounds: we show that it is possible to collect universal statistics of polylogarithmic size, and prove that these universal statistics allow us after the fact to compute all other statistics that are computable with similar amounts of memory. We show that this is indeed possible, both for the standard unbounded streaming model and the sliding window streaming model

Taming Big Data By Streaming

Data streams have emerged as a natural computational model for numerous applications of big data processing. In this model, algorithms are assumed to have access to a limited amount of memory and can only make a single pass (or a few passes) over the data, but need to produce sufficiently accurate answers for some objective functions on the dataset. This model captures various real-world applications and stimulates new scalable tools for solving important problems in the big data era. This dissertation focuses on the following two aspects of the streaming model. 1. Understanding the capability of the streaming model. For a vector aggregation stream, i.e., when the stream is a sequence of updates to an underlying $n$-dimensional vector $v$ (for very large $n$), we establish nearly tight space bounds on streaming algorithms of approximating functions of the form $\sum_{i=1}^n g(v_i)$ for nearly all functions $g$ of one-variable and $l(v)$ for all symmetric norms $l$. These results provide a deeper understanding of the streaming computation model. 2. Tighter upper bounds. We provide better streaming $k$-median clustering algorithms in a dynamic points stream, i.e., a stream of insertion and deletion of points on a discrete Euclidean space ($[\Delta]^d$ for sufficiently large $\Delta$ and $d$). Our algorithms use k\cdot\poly(d \log \Delta) space/update time and maintain with high probability an approximate $k$-median solution to the streaming dataset. All previous algorithms for computing an approximation for the $k$-median problem over dynamic data streams required space and update time exponential in $d$

Sketching as a Tool for Efficient Networked Systems

Today, computer systems need to cope with the explosive growth of data in the world. For instance, in data-center networks, monitoring systems are used to measure traffic statistics at high speed; and in financial technology companies, distributed processing systems are deployed to support graph analytics. To fulfill the requirements of handling such large datasets, we build efficient networked systems in a distributed manner most of the time. Ideally, we expect the systems to meet service-level objectives (SLOs) using the least amount of resource. However, existing systems constructed with conventional in-memory algorithms face the following challenges: (1) excessive resource requirements (e.g., CPU, ASIC, and memory) with high cost; (2) infeasibility in a larger scale; (3) processing the data too slowly to meet the objectives. To address these challenges, we propose sketching techniques as a tool to build more efficient networked systems. Sketching algorithms aim to process the data with one or several passes in an online, streaming fashion (e.g., a stream of network packets), and compute highly accurate results. With sketching, we only maintain a compact summary of the entire data and provide theoretical guarantees on error bounds. This dissertation argues for a sketching based design for large-scale networked systems, and demonstrates the benefits in three application contexts: (i) Network monitoring: we build generic monitoring frameworks that support a range of applications on both software and hardware with universal sketches. (ii) Graph pattern mining: we develop a swift, approximate graph pattern miner that scales to very large graphs by leveraging graph sketching techniques. (iii) Halo finding in N-body simulations: we design scalable halo finders on CPU and GPU by leveraging sketch-based heavy hitter algorithms