Programming Praxis

When I made the list of heap algorithms in the previous exercise, I realized that we don’t have an implementation of what is probably the most common form of heaps, the mutable heaps of imperative languages implemented in an array x. The basic idea is to arrange the items in the array so the largest is at index 1, and each item x_i is greater than x_2i and x_2i+1 (that’s a max-heap—a min-heap with the smallest item at the root has the sense of the comparison reversed).

A heap stored sub-array x_1..n−1 is unlikely to remain a heap if a new item is placed at x_n. The siftup function restores the heap property to x_1..n by repeatedly swapping the item at x_n with its parent at x_n/2 until it reaches its proper place in the heap.

The siftdown operation takes a heap stored in a sub-array x_1..n in which the item at x₁ is out of place and restores the heap property to x_1..n by repeatedly swapping the out-of-order item with its lesser child until it reaches the proper place in the array.

Given the siftup and siftdown operations, sorting an array is easy. First, for each item from the second to the last, sift it up to its proper position, forming a heap in x_1..n. Then, swap each item from the last to the second with the first item and sift the new first item down to its proper position in the heap.

Your task is to write functions that implement the siftup, siftdown, and heapsort procedures. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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