Streaming Algorithms Measured in Terms of the Computed Quantity

The last decade witnessed the extensive studies of algorithms for data streams. In this model, the input is given as a sequence of items passing only once or a few times, and we are required to compute (often approximately) some statistical quantity using a small amount of space. While many lower bounds on the space complexity have been proved for various tasks, almost all of them were done by reducing the problems to the cases where the desired statistical quantity is at one extreme end. For example, the lower bound of triangle-approximating was showed by reducing the problem to distinguishing between graphs without triangle and graphs with only one triangle. However, data in many practical applications are not in the extreme, and/or usually we are interested in computing the statistical quantity only if it is in some range (and otherwise reporting "too large" or "too small"). This paper takes this practical relaxation into account by putting the computed quantity itself into the measure of space complexity. It turns out that all three possible types of dependence of the space complexity on the computed quantity exist: as the quantity goes from one end to the other, the space complexity can goes from max to min, remains at max, or goes to somewhere between.