Data extraction via histogram and arithmetic mean queries: Fundamental limits and algorithms

The problems of extracting information from a data set via histogram queries or arithmetic mean queries are considered. We first show that the fundamental limit on the number of histogram queries, m, so that the entire data set of size n can be extracted losslessly, is m = Θ(n/log n), sub-linear in the size of the data set. For proving the lower bound (converse), we use standard arguments based on simple counting. For proving the upper bound (achievability), we proposed two query mechanisms. The first mechanism is random sampling, where in each query, the items to be included in the queried subset are uniformly randomly selected. With random sampling, it is shown that the entire data set can be extracted with vanishing error probability using Ω(n/log n) queries. The second one is a non-adaptive deterministic algorithm. With this algorithm, it is shown that the entire data set can be extracted exactly (no error) using Ω(n/log n) queries. We then extend the results to arithmetic mean queries, and show that for data sets taking values in a real-valued finite arithmetic progression, the fundamental limit on the number of arithmetic mean queries to extract the entire data set is also Θ(n/log n).