Run generation revisited: What goes up may or may not come down

We revisit the classic problem of run generation. Run generation is the first phase of external-memory sorting, where the objective is to scan through the data, reorder elements using a small buffer of size M, and output runs (contiguously sorted chunks of elements) that are as long as possible. We develop algorithms for minimizing the total number of runs (or equivalently, maximizing the average run length) when the runs are allowed to be sorted or reverse sorted. We study the problem in the online setting, both with and without resource augmentation, and in the offline setting. First, we analyze alternating-up-down replacement selection (runs alternate between sorted and reverse sorted), which was studied by Knuth as far back as 1963. We show that this simple policy is asymptotically optimal. Next, we give online algorithms having smaller competitive ratios with resource augmentation. We demonstrate that performance can also be improved with a small amount of foresight. Lastly, we present algorithms tailored for “nearly sorted” inputs which are guaranteed to have sufficiently long optimal runs.