In this paper, we present an abstract framework for online approximation of time-series data that yields a unified set of algorithms for several popular models: data streams, amnesic approximation, and out-of-order stream approximation. Our framework essentially develops a popular greedy method of bucket-merging into a more generic form, for which we can prove space-quality approximation bounds. When specialized to piecewise linear bucket approximations and commonly used error metrics, such as L2 or L∞, our framework leads to provable error bounds where none were known before, offers new results, or yields simpler and unified algorithms. The conceptual simplicity of our scheme translates into highly practical implementations, as borne out in our simulation studies: the algorithms produce nearoptimal approximations, require very small memory footprints, and run extremely fast.

We consider negotiations between publishers and advertisers in a marketplace for ads. Motivated by Google’s online PrintAds system which is such a marketplace, we focus on the role of the market runner in improving market efficiency. We abstract the problem of pricing guidance where the market runner provides an initial price-point for negotiations based on data analysis. The problem is nuanced because the market runner can not fully reveal the price data for any of the publishers. We introduce two solutions for pricing guidance, the first using clustering and the second using support vector machines, and present experimental evaluation of our methods. Pricing guidance by the market runner is a novel direction, and we expect more research in the future. 1.

The Lattice Histogram is a recently proposed data summarization technique that achieves approximation quality preferable to that of an optimal plain histogram. Like other hierarchical synopsis methods, a lattice histogram (LH) aims to approximate data using a hierarchical structure. Still, this structure is not defined a priori; it consists an unknown, not a given, of the problem. Past work has defined the properties that an LH needs to obey and developed general-purpose approximation algorithms for the construction thereof. Still, two major issues remain unaddressed: First, the construction of an optimal LH for a given error metric is a problem unsolved to date. Second, the proposed algorithms suffer from too high space and time complexities that render their application in real-world settings problematic. In this paper, we address both these questions, focusing on the case that the target error metric is a maximum error metric. Our algorithms treat both the error-bounded LH construction problem, in which the space occupied by an LH is minimized under an error constraint, as well as the classic space-bounded problem. First, we develop a dynamicprogramming scheme that detects an optimal LH under a given maximum-error bound. Second, we propose an efficient, practical, greedy algorithm that solves the same problem with much lower time and space requirements. Then, we show how both our algorithms can be applied to the classic space-bounded problem, aiming at minimizing error under a bound on space. Our experimental study with real-world data sets shows the effectiveness of our methods compared to competing summarization techniques. Moreover, our findings show that our greedy heuristic performs almost as well as the optimal solution in terms of accuracy. 1.