text: Quantiles, such as the median or percentiles, provide concise and useful information about the distribution of a collection of items, drawn from a linearly ordered universe. We study data structures, called quantile summaries, which keep track of all quantiles, up to an error of at most @math . That is, an @math -approximate quantile summary first processes a stream of items and then, given any quantile query @math , returns an item from the stream, which is a @math -quantile for some @math . We focus on comparison-based quantile summaries that can only compare two items and are otherwise completely oblivious of the universe. The best such deterministic quantile summary to date, by Greenwald and Khanna (ACM SIGMOD '01), stores at most @math items, where @math is the number of items in the stream. We prove that this space bound is optimal by providing a matching lower bound. Our result thus rules out the possibility of constructing a deterministic comparison-based quantile summary in space @math , for any function @math that does not depend on @math . A consequence of our results is also to show a lower bound for randomized algorithms.
  @cite_5: Wireless sensor networks offer the potential to span and monitor large geographical areas inexpensively. Sensors, however, have significant power constraint (battery life), making communication very expensive. Another important issue in the context of sensor-based information systems is that individual sensor readings are inherently unreliable. In order to address these two aspects, sensor database systems like TinyDB and Cougar enable in-network data aggregation to reduce the communication cost and improve reliability. The existing data aggregation techniques, however, are limited to relatively simple types of queries such as SUM, COUNT, AVG, and MIN MAX. In this paper we propose a data aggregation scheme that significantly extends the class of queries that can be answered using sensor networks. These queries include (approximate) quantiles, such as the median, the most frequent data values, such as the consensus value, a histogram of the data distribution, as well as range queries. In our scheme, each sensor aggregates the data it has received from other sensors into a fixed (user specified) size message. We provide strict theoretical guarantees on the approximation quality of the queries in terms of the message size. We evaluate the performance of our aggregation scheme by simulation and demonstrate its accuracy, scalability and low resource utilization for highly variable input data sets.
  @cite_6: An ∈-approximate quantile summary of a sequence of N elements is a data structure that can answer quantile queries about the sequence to within a precision of ∈N. We present a new online algorithm for computing∈-approximate quantile summaries of very large data sequences. The algorithm has a worst-case space requirement of O(1÷∈ log(∈N)). This improves upon the previous best result of O(1÷∈ log2(∈N)). Moreover, in contrast to earlier deterministic algorithms, our algorithm does not require a priori knowledge of the length of the input sequence. Finally, the actual space bounds obtained on experimental data are significantly better than the worst case guarantees of our algorithm as well as the observed space requirements of earlier algorithms.
summary: The Greenwald-Khanna algorithm @cite_6 is generally regarded as the best deterministic quantile summary. The space bound of @math follows from a somewhat involved proof, and it has been questioned whether this approach could be simplified or improved. Our work answers this second question in the negative. For a known universe @math of bounded size, Shrivastava al @cite_5 designed a quantile summary using @math words. Note that their algorithm is not comparison-based and so the result is incomparable to the upper bound of @math .