LIBRARIES Efficient and Private Distance Approximation in the Communication and Streaming Models

This thesis studies distance approximation in two closely related models- the streaming model and the two-party communication model. In the streaming model, a massive data stream is presented in an arbitrary order to a randomized algorithm that tries to approximate certain statistics of tile data with only a few (usually one) passes over the data. For instance, the data may be a flow of packets on the internet or a set of records in a large database. The size of the data necessitates the use of extremely efficient randomized approximation algorithms. Problems of interest include approximating the number of distinct elements, approximating the surprise index of a stream, or more generally, approximating the norm of a dynamically-changing vector in which coordinates are updated multiple times in an arbitrary order. In the two-party communication model, there are two parties who wish to efficiently compute a relation of their inputs. We consider the problem of approximating Lp distances for any p> 0. It turns out that lower bounds on the communication complexity of these re-lations yield lower bounds on the memory required of streaming algorithms for the problem