An Improved Interactive Streaming Algorithm for the Distinct Elements Problem.

The exact computation of the number of distinct elements (frequency moment F-0) is a fundamental problem in the study of data streaming algorithms. We denote the length of the stream by n where each symbol is drawn from a universe of size m. While it is well known that the moments F-0, F-1, F-2 can be approximated by efficient streaming algorithms [1], it is easy to see that exact computation of F0, F2 requires space Omega(m). In previous work, Cormode et al. [9] therefore considered a model where the data stream is also processed by a powerful helper, who provides an interactive proof of the result. They gave such protocols with a polylogarithmic number of rounds of communication between helper and verifier for all functions in NC. This number of rounds (O(log(2) m) in the case of F-0) can quickly make such protocols impractical. Cormode et al. also gave a protocol with log m + 1 rounds for the exact computation of F-0 where the space complexity is O (logmlog n + log(2) m) but the total communication O(root n logm(log n + logm)). They managed to give logm round protocols with polylog(m, n) complexity for many other interesting problems including F-2, Inner product and Range-sum, but computing F-0 exactly with polylogarithmic space and communication and O(logm) rounds remained open. In this work, we give a streaming interactive protocol with logm rounds for exact computation of F-0 using O(logm(log n + logmlog logm)) bits of space and the communication is O (logm (log n + log(3) m(log logm)(2) )). The update time of the verifier per symbol received is O(log(2) m).