# Problem 47: Annotated Streaming

Suggested by | Graham Cormode |
---|---|

Source | Bertinoro 2011 |

Short link | https://sublinear.info/47 |

In the annotated stream model [ChakrabartiCM-09], a stream is augmented with ‘annotation’, which takes the form of a proof of a property of the stream. In its simplest form, the annotation may just be a reordering of the stream to make it easy to compute a function of interest. The key parameters in this model are $H$, the size of the annotation, and $V$, the space needed by the streaming party to view the stream and verify the proof. The annotation proposed should be such that an honest annotation will always be accepted, while a mistaken annotation will be detected and rejected with high probability.

We consider the problem of counting the number of triangles in a graph described by a stream of edges (where each edge is promised to occur at most once). Partial results from the above reference are that $H = O(n^2)$ and $V = \tilde{O}(1)$ is possible, as is $H = O(n^{3/2}), V= O(n^{3/2})$.

**Question:** Can one achieve $H = V = \tilde{O}(n)$?

## Update[edit]

This question was answered affirmatively by Thaler [Thaler-16].