The frequent items problem, under polynomial decay, in the streaming model

We consider the problem of estimating the frequency count of data stream elements under polynomial decay functions. In these settings every element in the stream is assigned with a time-decreasing weight, using a non-increasing polynomial function. Decay functions are used in applications where older data is less significant, less interesting or even less reliable than recent data. Consider a data stream of N elements drawn from a universe U. We propose three poly-logarithmic algorithms for the problem. The first one, deterministic, uses O([email protected]^2logN(loglogN+logU)) bits, where @[email protected]?(0,1) is the approximation parameter. The second one, probabilistic, uses O([email protected]^[email protected]@e) bits or O([email protected]^[email protected]) bits, depending on the decay function parameter, where @[email protected]?(0,1) is the probability of failure. The third one, deterministic in the stochastic model, uses O([email protected]) bits or O([email protected]^2logN) bits, also depending on the decay parameter as will be described in this paper. This variant of the problem is important and has many applications. To our knowledge, it has never been studied before.