Semi-Streaming Algorithms for Annotated Graph Streams

Abstract

Considerable effort has been devoted to the development of streaming algorithms for analyzing massive graphs. Unfortunately, many results have been negative, establishing that a wide variety of problems require $\Omega(n^2)$ space to solve. One of the few bright spots has been the development of semi-streaming algorithms for a handful of graph problems -- these algorithms use space $O(n\cdot\text{polylog}(n))$. In the annotated data streaming model of Chakrabarti et al., a computationally limited client wants to compute some property of a massive input, but lacks the resources to store even a small fraction of the input, and hence cannot perform the desired computation locally. The client therefore accesses a powerful but untrusted service provider, who not only performs the requested computation, but also proves that the answer is correct. We put forth the notion of semi-streaming algorithms for annotated graph streams (semi-streaming annotation schemes for short). These are protocols in which both the client's space usage and the length of the proof are $O(n \cdot \text{polylog}(n))$. We give evidence that semi-streaming annotation schemes represent a substantially more robust solution concept than does the standard semi-streaming model. On the positive side, we give semi-streaming annotation schemes for two dynamic graph problems that are intractable in the standard model: (exactly) counting triangles, and (exactly) computing maximum matchings. The former scheme answers a question of Cormode. On the negative side, we identify for the first time two natural graph problems (connectivity and bipartiteness in a certain edge update model) that can be solved in the standard semi-streaming model, but cannot be solved by annotation schemes of "sub-semi-streaming" cost. That is, these problems are just as hard in the annotations model as they are in the standard model.Comment: This update includes some additional discussion of the results proven. The result on counting triangles was previously included in an ECCC technical report by Chakrabarti et al. available at http://eccc.hpi-web.de/report/2013/180/. That report has been superseded by this manuscript, and the CCC 2015 paper "Verifiable Stream Computation and Arthur-Merlin Communication" by Chakrabarti et a