I just learned about Gibbs Sampling which is an MCMC method. Given a distribution $\pi$, we want to sample an item according to $\pi$.
Maybe my alternative suggestion would sound somewhat naive (even stupid) but why can't we just draw a number in random from $[0,M]$ for some sufficiently large enough $M$. Then, we divide the range to buckets with appropriate sizes according to the distribution.
This will be a true sampling of $\pi$.
One could argue that my suggestion demands a PRNG, but Gibbs Sampling uses randomness too when deciding the next state from the neighbors of the current state.
So for a reasonable distribution, wouldn't my suggestion work way better? It's essentially $O(1)$ and accurate.