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The Pigeonhole Sort algorithm functions by placing each item into a 'pigeonhole' corresponding to its key value.

The Pigeonhole Sort algorithm is a sorting algorithm that is suitable for sorting lists of elements where the number of elements and the number of possible key values are approximately the same. It requires O(n + Range) time where n is the number of elements in input array and ‘Range’ is the number of possible values in array.

The algorithm works by creating an array of 'pigeonholes' equal to the size of the range of input values. Each pigeonhole initially contains no 'pigeons' (i.e., they are empty). The algorithm then goes through the input list and for each element, it places it into the pigeonhole corresponding to its key value. This is done by subtracting the minimum value of the key range from the key value of the element, to calculate the index of the pigeonhole.

Once all the elements have been placed into pigeonholes, the algorithm then goes through the pigeonholes in order and for each non-empty pigeonhole, it removes the elements and places them back into the input list. This results in a sorted list, as the pigeonholes are visited in order of their key values.

The Pigeonhole Sort algorithm is a type of bucket sort. It is efficient when the number of elements and the range of possible key values are approximately the same. However, it is not suitable for sorting large lists of elements, or lists where the range of key values is much larger than the number of elements. This is because the size of the array of pigeonholes is equal to the range of key values, so the algorithm can use a large amount of memory if the key range is large.

In summary, the Pigeonhole Sort algorithm is a simple and efficient sorting algorithm for certain types of lists. It works by placing each element into a pigeonhole corresponding to its key value, and then collecting the elements from the pigeonholes in order.

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