Derive k-dependent performance equations for top-k E2LSH/E2LSHoS

Derive explicit analytical equations characterizing how both the in-memory E2LSH query time and the number of hash-bucket reads (I/O operations) in E2LSH-on-Storage depend on the top-k parameter k for Euclidean c-approximate top-k nearest neighbor search with E2LSH parameters (m, L, S), to enable precise prediction of storage IOPS requirements for top-k queries.

Background

In analyzing storage requirements to match in-memory E2LSH speed, the authors model query time via computational and I/O components. They note that both in-memory E2LSH time and E2LSHoS I/O counts grow sublinearly with database size n, but they explicitly state the absence of equations for the dependence on k in top-k ANNS. Despite empirical observations, this analytic gap limits formal prediction of IOPS needs for larger k.

Filling this gap would allow the model to determine how increasing k impacts both computation and I/O, strengthening the generality of the analysis framework used throughout the paper.