On the algebra of data sketches

We consider the problem of designing a distributed data sketch for scenario in which data stream is observed by many independent network nodes. We require that a sketch apart from being computationally and memory efficient should also be mergeable in a way that mimics set theory operations on related data sets. For example, when monitoring network traffic, one may consider how many distinct packets passed through a given node (sum of sets for different time windows) or passed through two given nodes (intersection of sets from two locations) and what is their total size (intersection of weighted sets). In this paper we propose a sketch that allows to efficiently summarize sets constructed from a sequence of set theory operations. We also provide an analytical control over the trade-off between the accuracy and storage/computational requirements. In comparison to the previous works the proposed solution 1) allows the weights of elements, 2) allows performing set theory operations simultaneous on a large number of sketches, 3) does not require computationally expensive numerical calculations and guarantees low overheads.