Parallel Algorithms for Select and Partition with Noisy Comparisons

We consider the problem of finding the $k^{th}$ highest element in a totally ordered set of $n$ elements (select), and partitioning a totally ordered set into the top $k$ and bottom $n-k$ elements (partition) using pairwise comparisons. Motivated by settings like peer grading or crowdsourcing, where multiple rounds of interaction are costly and queried comparisons may be inconsistent with the ground truth, we evaluate algorithms based both on their total runtime and the number of interactive rounds in three comparison models: noiseless (where the comparisons are correct), erasure (where comparisons are erased with probability $1-\gamma$), and noisy (where comparisons are correct with probability $1/2+\gamma/2$ and incorrect otherwise). We provide numerous matching upper and lower bounds in all three models. Even our results in the noiseless model, which is quite well-studied in the TCS literature on parallel algorithms, are novel.