Pigeonhole sort

Pigeonhole sort
Class	Sorting algorithm
Data structure	Array
Worst-case performance	, where N is the range of key values and n is the input size
Worst-case space complexity
Optimal	If and only if

Pigeonhole sorting is a sorting algorithm that is suitable for sorting lists of elements where the number n of elements and the length N of the range of possible key values 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" (analogous to a pigeon-hole messagebox in an office or desk), one pigeonhole for each key in the range of the keys in 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 increasing order of keys, and for each pigeonhole, put its elements into the original array in increasing order.

Example

Suppose one 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")

The pigeonhole array is then iterated over in order, and the elements are moved 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

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.

[1] "NIST's Dictionary of Algorithms and Data Structures: pigeonhole sort".

[2] Black, Paul E. "Dictionary of Algorithms and Data Structures". NIST. Retrieved 6 November 2015.

[1]

[2]

v t e 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 Proportion extend sort Quicksort
Selection sorts	Selection sort Heapsort Smoothsort Cartesian tree sort Tournament sort Cycle sort Weak-heap 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 Interpolation sort Pigeonhole sort Proxmap sort Radix sort Flashsort
Concurrent sorts	Bitonic sorter Batcher odd–even mergesort Pairwise sorting network Samplesort
Hybrid sorts	Block merge sort Kirkpatrick–Reisch sort Timsort Introsort Spreadsort Merge-insertion sort
Other	Topological sorting Pre-topological order Pancake sorting Spaghetti sort
Impractical sorts	Stooge sort Slowsort Bogosort

Example

See also

References