Pigeonhole sort

Pigeonhole sort
Class	Sorting algorithm
Data structure	Array
Worst-case performance	$O(N+n)$ , where N is the range of key values and n is the input size
Worst-case space complexity	$O(N+n)$

Pigeonhole sorting is a sorting algorithm that is suitable for sorting lists of elements where the number of elements (n) and the length of the range of possible key values (N) are approximately the same.^[1] It requires O(n + N) time. It is similar to counting sort, but differs in that it "moves items twice: once to the bucket array and again to the final destination [whereas] counting sort builds an auxiliary array then uses the array to compute each item's final destination and move the item there."^[2]

The pigeonhole algorithm works as follows:

Given an array of values to be sorted, set up an auxiliary array of initially empty "pigeonholes," one pigeonhole for each key through the range of the original array.
Going over the original array, put each value into the pigeonhole corresponding to its key, such that each pigeonhole eventually contains a list of all values with that key.
Iterate over the pigeonhole array in order, and put elements from non-empty pigeonholes back into the original array.

Example

Suppose we were sorting these value pairs by their first element:

(5, "hello")
(3, "pie")
(8, "apple")
(5, "king")

For each value between 3 and 8 we set up a pigeonhole, then move each element to its pigeonhole:

3: (3, "pie")
4:
5: (5, "hello"), (5, "king")
6:
7:
8: (8, "apple")

We then iterate over the pigeonhole array in order and move them back to the original list.

The difference between pigeonhole sort and counting sort is that in counting sort, the auxiliary array does not contain lists of input elements, only counts:

3: 1
4: 0
5: 2
6: 0
7: 0
8: 1

Using this information we can perform a series of exchanges on the input array that puts it in order, moving items only once.

For arrays where N is much larger than n, bucket sort is a generalization that is more efficient in space and time.

References

↑ NIST's Dictionary of Algorithms and Data Structures: pigeonhole sort
↑ Black, Paul E. "Dictionary of Algorithms and Data Structures". NIST. Retrieved 6 November 2015.

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Sorting algorithms

Theory	Computational complexity theory Big O notation Total order Lists Inplacement Stability Comparison sort Adaptive sort Sorting network Integer sorting X + Y sorting Transdichotomous model Quantum sort

Exchange sorts	Bubble sort Cocktail shaker sort Odd–even sort Comb sort Gnome sort Quicksort Slowsort Stooge sort Bogosort

Selection sorts	Selection sort Heapsort Smoothsort Cartesian tree sort Tournament sort Cycle sort

Insertion sorts	Insertion sort Shellsort Splaysort Tree sort Library sort Patience sorting

Merge sorts	Merge sort Cascade merge sort Oscillating merge sort Polyphase merge sort

Distribution sorts	American flag sort Bead sort Bucket sort Burstsort Counting sort Pigeonhole sort Proxmap sort Radix sort Flashsort

Concurrent sorts	Bitonic sorter Batcher odd–even mergesort Pairwise sorting network

Hybrid sorts	Block merge sort Timsort Introsort Spreadsort

Other	Topological sorting Pancake sorting Spaghetti sort

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Pigeonhole sort

Example

See also

References