Lars Hagen describes in his blog post a strategy to solve the problem of a fast incremental sort.

From the article

I recently came across the need for an incremental sorting algorithm, and started to wonder how to do it optimally.

The incremental sorting problem is described here, and is an “online” version of partial sort. That is, if you have already sorted kk elements, you should be able to quickly sort k+1k+1 elements, and so on.

Incremental sorts can be useful for a number of cases: