Which of the following sorting algorithms are stable: insertion sort, merge sort, heapsort, and quicksort? Give a simple scheme that makes any sorting algorithm stable. How much additional time and space does your scheme entail?
Stable: Insertion sort, merge sort
Not stable: Heapsort, quicksort
We can make any algorithm stable by mapping the array to an array of pairs, where the first element in each pair is the original element and the second is its index. Then we sort lexicographically. This scheme takes additional $\Theta(n)$ space.