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 arrives 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 \ interesting \ reliable than recent data. We propose 3 poly-logarithmic algorithms for the problem. The first one, deterministic, uses $ O (\frac{1}{\epsilon ^{2}} \log N ( \log \log N + \log U) ) $ bits. The second one, probabilistic, uses $O ( \frac{1}{\epsilon ^{2}} \log \frac{1}{\epsilon \delta} \log N )$ bits and the third one, deterministic in the stochastic model, uses $O(\frac{1}{\epsilon ^{2}} \log N)$ bits. In addition we show that using additional additive error can improve, in some cases, the space bounds. This variant of the problem is important and has many applications. To our knowledge it was never studied before.