Counting Triangles under Updates in Worst-Case Optimal Time

TL;DR

This paper presents a novel incremental algorithm for counting triangles in a database that achieves worst-case optimal update times under certain conjectures, improving over classical methods.

Contribution

It introduces a space-time tradeoff framework that optimally maintains triangle counts under updates, unifying and surpassing previous approaches.

Findings

Achieves worst-case optimal update time conditioned on the OMv conjecture.

Unifies classical and factorized view maintenance as special cases.

Recovers optimal complexity for static triangle counting.

Abstract

We consider the problem of incrementally maintaining the triangle count query under single-tuple updates to the input relations. We introduce an approach that exhibits a space-time tradeoff such that the space-time product is quadratic in the size of the input database and the update time can be as low as the square root of this size. This lowest update time is worst-case optimal conditioned on the Online Matrix-Vector Multiplication conjecture. The classical and factorized incremental view maintenance approaches are recovered as special cases of our approach within the space-time tradeoff. In particular, they require linear-time update maintenance, which is suboptimal. Our approach also recovers the worst-case optimal time complexity for computing the triangle count in the non-incremental setting.