Stream sketches, sampling, and sabotage

Exact solutions are unattainable for important problems. The calculations are limited by the memory of our computers and the length of time that we can wait for a solution. The field of approximation algorithms has grown to address this problem; it is practically important and theoretically fascinating. We address three questions along these lines. What are the limits of streaming computation? Can we efficiently compute the likelihood of a given network of relationships? How robust are the solutions to combinatorial optimization problems? High speed network monitoring and rapid acquisition of scientific data require the development of space efficient algorithms. In these settings it is impractical or impossible to store all of the data, nonetheless the need for analyzing it persists. Typically, the goal is to compute some simple statistics on the input using sublinear, or even polylogarithmic, space. Our main contributions here are the complete classification of the space necessary for several types of statistics. Our sharpest results characterize the complexity in terms of the domain size and stream length