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The $h$-index is a metric used to measure the impact of a user in apublication setting, such as a member of a social network with many highlyliked posts or a researcher in an academic domain with many highly citedpublications. Specifically, the $h$-index of a user is the largest integer $h$such that at least $h$ publications of the user have at least $h$ units ofpositive feedback.  We design an algorithm that, given query access to the $n$ publications of auser and each publication's corresponding positive feedback number, outputs a$(1\pm \varepsilon)$-approximation of the $h$-index of this user withprobability at least $1-\delta$ in time \[  O(\frac{n \cdot \ln{(1/\delta)}}{\varepsilon^2 \cdot h}),  \] where $h$ is the actual $h$-index which is unknown to the algorithma-priori. We then design a novel lower bound technique that allows us to provethat this bound is in fact asymptotically optimal for this problem in allparameters $n,h,\varepsilon,$ and $\delta$.  Our work is one of the first in sublinear time algorithms that addressesobtaining asymptotically optimal bounds, especially in terms of the error andconfidence parameters. As such, we focus on designing novel techniques for thistask. In particular, our lower bound technique seems quite general -- toshowcase this, we also use our approach to prove an asymptotically optimallower bound for the problem of estimating the number of triangles in a graph insublinear time, which now is also optimal in the error and confidenceparameters. This result improves upon prior lower bounds of Eden, Levi, Ron,and Seshadhri (FOCS'15) for this problem, as well as multiple follow-ups thatextended this lower bound to other subgraph counting problems.

Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting