Question

Describe an in-place version of the quick-select algorithm in pseudo-code.

Short answer

Select pivot, partition elements, and recursively apply quick-select to the appropriate partition until the element is found.

Step-by-step solution

- Choose a Pivot

Select a pivot element from the array. This can be done by choosing the last element, the first element, or a random element.

- Partitioning

Rearrange the elements in the array such that elements less than the pivot come before it and elements greater than the pivot come after it. This is done in-place, using two pointers.

- Recurrence

If the pivot is in the k-th position, return it. Otherwise, if the k-th element is in the left partition, recursively apply quick-select on the left partition. Otherwise, apply quick-select on the right partition.

- Base Case

The base case occurs when there is only one element in the partition, which happens when the recursion reaches a single element or the search is complete.

Key concepts

pivot selection

In the quick-select algorithm, the first crucial step is to choose a pivot element. The pivot is pivotal (pun intended) because it determines how the array will be divided for further processing.
There are several common strategies for selecting a pivot:

• Choosing the first element
• Choosing the last element
• Choosing a random element
• Choosing the median of the first, middle, and last elements

Each strategy has its pros and cons. For simplicity, many implementations pick the last element. However, choosing a random element can help ensure more balanced partitions, especially for non-random input data.
Note, the choice of the pivot can affect the efficiency of the algorithm.

partitioning

Partitioning is the process of rearranging the elements in the array around the chosen pivot. Elements less than the pivot go to its left, and elements greater go to its right.
This step is performed in-place with the help of two pointers. One pointer starts at the beginning of the array and the other starts just before the pivot. These pointers move towards each other, swapping elements when necessary to ensure all elements on the left are lesser and all on the right are greater:

pointer1 = start
pointer2 = end - 1
while pointer1 <= pointer2:
    if array[pointer1] < pivot:
        pointer1 += 1
    elif array[pointer2] > pivot:
        pointer2 -= 1
    else:
        swap(array[pointer1], array[pointer2])
        pointer1 += 1
        pointer2 -= 1

This logic ensures the pivot ends up in its correct position, ready for the next step.

recurrence

Recurrence refers to how the algorithm proceeds after partitioning. If after partitioning, the pivot is in the k-th position (where k is the index of the element we're looking for), the algorithm terminates and returns the pivot.
If the pivot is not in the k-th position, we determine which half of the array (left or right of the pivot) contains the k-th element. The algorithm is then recursively applied to that half:

• If k is less than the index of pivot, apply the algorithm to the left partition.
• If k is greater, apply it to the right partition.

Recurrence allows quick-select to zoom into the correct half of the array, significantly reducing the problem size with each recursive call.
Thus, this smart reduction ensures the algorithm is much more efficient than a brute-force search.

base case

The base case in quick-select is straightforward. It occurs when the partition to be processed contains only one element. In such a case, that single element must be the k-th smallest element.
Another base case is reached if the element found during partitioning is already in its final position corresponding to k. In either situation, quick-select terminates.
The simplicity of the base case helps in ensuring that the recursion depth is minimized, guaranteeing that the algorithm doesn't run indefinitely. Below is a pseudo-code snippet for the base case:

if start == end:
    return array[start]
elif k == pivotIndex:
    return array[pivotIndex]

Reaching the base case ensures that the search is complete, giving us the desired k-th smallest element.