Space-optimal single-pass counting of cliques and other substructures

Design space-optimal single-pass streaming algorithms for counting cliques and other graph structures in repeated-edge arrival and Right-to-be-Forgotten Graph Streaming models, extending the established space-optimal multi-pass clique-counting algorithms and the single-pass triangle-counting results.

Background

The paper establishes space-optimal algorithms for counting triangles in repeated-edge arrival streams and in the Right-to-be-Forgotten Graph Streaming (RFGS) model. It also develops constant-pass algorithms for counting cliques in repeated-edge arrival streams, but does not provide corresponding single-pass algorithms for cliques or more general subgraph structures in these generalized streaming models.

The unresolved problem is therefore to obtain single-pass algorithms with optimal space complexity for clique counting and related graph-substructure counting tasks under repeated edge occurrences and, where applicable, forget operations. The paper explicitly identifies this as an open question without specifying a particular clique size or subgraph family beyond these categories.

References

Designing space optimal single-pass algorithm for counting cliques and other structures in these new models is an open question.

— Counting Triangles in Graph Streams with Repeatable and Forgettable Edges  (2609.19943 - Chakraborty et al., 17 Sep 2026) in Section 5, Conclusions

What are the optimal classical and quantum space bounds for triangle counting under the same exact-$R$ random-order promise and a common space measure? In particular, can one prove a quantum--classical separation in this model?

— Random Order in Quantum Streaming: Replenishment and Robust Lower Bounds  (2610.06727 - Voronova, 5 Oct 2026) in Section 6, paragraph "Open questions"