The Pigeonhole Sort algorithm organizes elements by placing each item into a “pigeonhole” that corresponds to its key value.

Pigeonhole Sort operates by creating an array of “pigeonholes” or “buckets” for each potential key value found in the input data. The algorithm begins by iterating through the input, placing each item into the appropriate pigeonhole based on its key value. After all items have been allocated to their respective pigeonholes, the algorithm proceeds to traverse these pigeonholes in order, collecting and outputting the items contained in each. This process ultimately results in a sorted list of items.

The first step in the Pigeonhole Sort algorithm involves determining the range of key values present in the input. This is achieved by identifying the minimum and maximum key values. The total number of pigeonholes required is then calculated by subtracting the minimum key value from the maximum, and adding one. This approach ensures that there is a dedicated pigeonhole for every possible key value.

Subsequently, an array of pigeonholes is initialized, with each one starting off empty. The algorithm then goes through the input once more, placing each item into the appropriate pigeonhole. This placement is performed by subtracting the minimum key value from the item’s key value, which provides the index of the corresponding pigeonhole.

Once all items have been sorted into their designated pigeonholes, the algorithm iterates through the pigeonholes in sequential order. For each pigeonhole, it outputs the items it contains in the order they were placed. This sequence results in a fully sorted list of items.

Pigeonhole Sort is a straightforward and efficient sorting algorithm, particularly when the range of key values is limited. However, its performance can degrade significantly when the range of key values is extensive, as it necessitates considerable memory to accommodate the pigeonholes. Additionally, this algorithm is not well-suited for sorting items with non-integer key values, since it relies on these key values as indices within an array.