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Abstract
In the setting of streaming interactive proofs (SIPs), a client (verifier) needs to compute a given function on a massive stream of data, arriving online, but is unable to store even a small fraction of the data. It outsources the processing to a third party service (prover) but is unwilling to blindly trust answers returned by this service. Thus, the service cannot simply supply the desired answer; it must convince the verifier of its correctness via a short interaction after the stream has been seen. In this work we study “barely interactive” SIPs. Specifically, we show that one or two rounds of interaction suffice to solve several query problems---including index, median, nearest neighbor search, pattern matching, and range counting---with polylogarithmic space and communication costs. Such efficiency with $O(1)$ rounds of interaction was thought to be impossible based on previous work. On the other hand, we initiate a formal study of the limitations of constant-round SIPs by introducing a new hierarchy of communication models called online interactive proofs (OIPs). The online nature of these models is analogous to the streaming restriction placed upon the verifier in a SIP. We give upper and lower bounds that (1) characterize, up to quadratic blowups, every finite level of the OIP hierarchy in terms of other well-known communication complexity classes, (2) separate the first four levels of the hierarchy, and (3) reveal that the hierarchy collapses to the fourth level. Our study of OIPs reveals marked contrasts and some parallels with the classic Turing machine theory of interactive proofs, establishes limits on the power of existing techniques for developing constant-round SIPs, and provides a new characterization of (nononline) Arthur--Merlin communication in terms of an online model.
Highlights
The surging popularity of commercial cloud computing services, and more generally outsourced computations, has revealed compelling new applications for the study of interactive proofs with highly restricted verifiers
We identify an implicit assumption in the Klauck–Prakash lower bound argument: it applies only to protocols in which the verifier’s messages to the prover are independent of the input. This happened to hold in all previous streaming interactive proofs (SIPs), which are descended from the sum-check protocol of Lund et al [32]
We introduced new “online” communication hierarchies, Online Interactive Proofs (OIPs)+ and OIP, which can be seen as restricted variants of the standard Arthur-Merlin communication model
Summary
The surging popularity of commercial cloud computing services, and more generally outsourced computations, has revealed compelling new applications for the study of interactive proofs with highly restricted verifiers. E.g., a retailer (verifier) who lacks the resources to locally process a massive input (say, the set of all its transactions), but can access a powerful but untrusted cloud service provider (prover), who processes the input on the retailer’s behalf. The verifier must work within the confines of the restrictive data streaming paradigm, using only a small amount of working memory. The prover must both answer queries about the input (say, “how many pairs of blue jeans have I ever sold?”), and prove that the answer is correct. We refer to this general scenario as verifiable data stream computation. It is useful to look at this computational scenario as “data stream algorithms with access to a powerful (space-unlimited) prover.” As is well known, most interesting data streaming
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