Bucket sort is a sort algorithm that works by partitioning an array into a finite number of buckets and then sorting each bucket. It is a generalization of pigeonhole sort. Bucket sort runs in linear time (Θ(n)) when input is drawn from a uniform distribution. Not being a comparison sort , it is not subject to the Ω(n log n) lower bound.

It works as follows:

1. Set up an array of initially empty "buckets" the size of the range.
2. Go over the original array, putting each object in its bucket.
3. Sort each non-empty bucket.
4. Put elements from non-empty buckets back into the original array.

Pseudocode

' A is the array
' n is the number of buckets
' MSBITS(x) returns the most significant bits of x.
'    This could be floor(x/2^k) (where k is a
'    nonnegative integer) for sorting numbers, or
'    the first character of x for sorting strings.
' NEXT-SORT is a sort algorithm
BUCKET-SORT(A, n, MSBITS, NEXT-SORT):
  make array B of n lists
  for i = 1 to n:
    insert A[i] into list B[MSBITS(A[i])]
    for i = 0 to n - 1:
      NEXT-SORT(B[i])
  concatenate the lists B[0]...B[n-1] in order

Relationships to other sorting algorithms

Using BUCKET-SORT itself as the NEXT-SORT produces a relative of the radix sort. Using BUCKET-SORT with n == 2 and itself as the NEXT-SORT produces quicksort.