Fast and Simple Sorting Using Partial Information

We consider the problem of sorting n items, given the outcomes of m pre-existing comparisons. We present a simple and natural deterministic algorithm that runs in $\text{O}(m+logT)$ time and does $\text{O}(logT)$ comparisons, where T is the number of total orders consistent with the pre-existing comparisons. Our running time and comparison bounds are best possible up to constant factors, thus resolving a problem that has been studied intensely since 1976 (Fredman, Theoretical Computer Science). The best previous algorithm with a bound of $\text{O}(logT)$ on the number of comparisons has a time bound of $\text{O}\left(n^{2.5}\right)$ and is more complicated. Our algorithm combines three classic algorithms: topological sort, heapsort with the right kind of heap, and efficient search in a sorted list. It outputs the items in sorted order one by one. It can be modified to stop early, thereby solving the important and more general top-k sorting problem: Given k and the outcomes of some pre-existing comparisons, output the smallest k items in sorted order. The modified algorithm solves the top-k sorting problem in minimum time and comparisons, to within constant factors.