  We consider the problem of sorting $n$ items, given the outcomes of $m$
pre-existing comparisons. We present a simple, natural deterministic algorithm
that runs in $O(m+\log T)$ time and does $O(\log T)$ 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 $O(\lg T)$ on the number of comparisons has a time bound of
$O(n^{2.5})$ and is more complicated. (A recent independent and concurrent work
by Van der Hoog and Rutschmann implies an $O(n^{2.371552})$-time algorithm with
$O(\log T)$ comparisons.)
  Our algorithm combines three classic algorithms: topological sort, heapsort
with the right kind of heap, and efficient insertion into a sorted list. It
outputs the items in sorted order one by one. As a result, it can be modified
to solve 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.
