# New Verification Schemes for Frequency-Based Functions on Data Streams

We study the general problem of computing frequency-based functions, i.e., the sum of any given function of data stream frequencies. Special cases include fundamental data stream problems such as computing the number of distinct elements (F_0), frequency moments (F_k), and heavy-hitters. It can also be applied to calculate the maximum frequency of an element (F_∞). Given that exact computation of most of these special cases provably do not admit any sublinear space algorithm, a natural approach is to consider them in an enhanced data streaming model, where we have a computationally unbounded but untrusted prover sending proofs or help messages to ease the computation. Think of a memory-restricted client delegating the computation to a stronger cloud service whom it doesn't want to trust blindly. Using its limited memory, it wants to verify the proof that the cloud sends. Chakrabarti et al. (ICALP '09) introduced this setting as the "annotated data streaming model" and showed that multiple problems including exact computation of frequency-based functions—that have no sublinear algorithms in basic streaming—do have annotated streaming algorithms, also called "schemes", with both space and proof-length sublinear in the input size. We give a general scheme for computing any frequency-based function with both space usage and proof-size of O(n^2/3log n) bits, where n is the size of the universe. This improves upon the best known bound of O(n^2/3log^4/3 n) given by the seminal paper of Chakrabarti et al. and as a result, also improves upon the best known bounds for the important special cases of computing F_0 and F_∞. We emphasize that while being quantitatively better, our scheme is also qualitatively better in the sense that it is simpler than the previously best scheme that uses intricate data structures and elaborate subroutines.

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