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This answer gives exact formulas for the expected number of steps, with and without replacement. To be clear, we interpret OP's problem as detailed in OP's Python gist: each step of the process makes …

Your bound (at least when $t\le E[S]$) seems to follow from the standard multiplicative Chernoff bound, as follows. Lemma 1. (a standard multiplicative Chernoff bound) Let $S$ be the sum of independen …

$E[m]$ equals $n(H_n-1)$. Here's a more complete proof sketch, following the argument suggested in my comment. We start by showing that each permutation of the first $k-1$ elements is equally like …

For notational convenience define r.v.s $T_S = \min\{i : |S_i| = 1\}$ (recalling $S_i = S \cap H_1 \cap \cdots \cap H_i$), and $T_H = \min\big\{i : H_1 \cap H_2 \cap \cdots \cap H_i = \{0^n\}\big\}$. …

Here is a proof that this algorithm runs in $O(n\,m)$ time in expectation and with high probability. First consider the algorithm modified so that $k$ is chosen in $\{2,3,..,\min(c,n)\}$ instead of r …