What is a performant algorithm for implementing APL grade up?

Question

For APLers: I am only concerned with monadic grade up over vectors.

For non-APLers: Grade up is a function that takes a numeric vector V of size n and returns an integer vector R of equal size. The index of the smallest element of V is placed in R[0], the index of the next smallest element in R[1], ..., the largest element of V in R[n-1]. The value of V must be unchanged.

Grading is obviously related to sorting: R provides the indices into V to access V in sorted order. Grade up must be stable: that is, if two elements of V with indices i, j where i < j are equal, then i, j will appear consecutively in R and in that order. An O(n^2) implementation is easy, but I don't see how to adapt a standard stable O(n log n) sorting implementation for this purpose. An algorithm using only constant space would be desirable.

Answer 1

For this particular task, you can use any in-place sorting algorithm: Fill R with the indexes from 0 to n-1, and then just sort it.

For the sort, instead of comparing two indexes i and j by value, you compare V[i] to V[j]. If they are equal, then break the tie by comparing i and j.

Since no two indexes will compare equal, even an unstable algorithm like Quicksort will produce stable results. Many languages include a default sort that takes a comparison function as input.