Arthur-Merlin streaming complexity

Tom Gur, Ran Raz

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical AM streaming algorithm for a class of data stream problems. The algorithm offers a tradeoff between the length of the proof and the space complexity that is needed to verify it. As an application, we give an AM streaming algorithm for the Distinct Elements problem. Given a data stream of length m over alphabet of size n, the algorithm uses Õ(s) space and a proof of size Õ(w), for every s,w such that s•w>n (where Õ hides a polylog(m,n) factor). We also prove a lower bound, showing that every MA streaming algorithm for the Distinct Elements problem that uses s bits of space and a proof of size w, satisfies s•w=Ω(n). Furthermore, the lower bound also holds for approximating the number of distinct elements within a multiplicative factor of 1±1/n. As a part of the proof of the lower bound for the Distinct Elements problem, we show a new lower bound of Ω(n) on the MA communication complexity of the Gap Hamming Distance problem, and prove its tightness.

Original languageEnglish (US)
Pages (from-to)145-165
Number of pages21
JournalInformation and Computation
StatePublished - Jul 28 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Information Systems
  • Computer Science Applications
  • Computational Theory and Mathematics


  • Communication complexity
  • Data streams
  • Probabilistic proof systems


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