Bucket sort works as follows: Let array denote the array to be sorted and k denote the number of buckets to use.

The floor function must be used to convert a floating number to an integer ( and possibly casting of datatypes too ).

Using bucketSort itself as nextSort produces a relative of radix sort; in particular, the case n = 2 corresponds to quicksort (although potentially with poor pivot choices).

When the input contains several keys that are close to each other (clustering), those elements are likely to be placed in the same bucket, which results in some buckets containing more elements than average.

The worst-case scenario occurs when all the elements are placed in a single bucket.

The overall performance would then be dominated by the algorithm used to sort each bucket, for example

Consider the case that the input is uniformly distributed.

The first step, which is initialize the buckets and find the maximum key value in the array, can be done in

If division and multiplication can be done in constant time, then scattering each element to its bucket also costs

average time, given a uniformly distributed input.

[1] A common optimization is to put the unsorted elements of the buckets back in the original array first, then run insertion sort over the complete array; because insertion sort's runtime is based on how far each element is from its final position, the number of comparisons remains relatively small, and the memory hierarchy is better exploited by storing the list contiguously in memory.

average time complexity even without uniformly distributed input.

The most common variant of bucket sort operates on a list of n numeric inputs between zero and some maximum value M and divides the value range into b buckets each of size M/b.

[3] However, the performance of this sort degrades with clustering; if many values occur close together, they will all fall into a single bucket and be sorted slowly.

This performance degradation is avoided in the original bucket sort algorithm by assuming that the input is generated by a random process that distributes elements uniformly over the interval [0,1).

[1] Similar to generic bucket sort as described above, ProxmapSort works by dividing an array of keys into subarrays via the use of a "map key" function that preserves a partial ordering on the keys; as each key is added to its subarray, insertion sort is used to keep that subarray sorted, resulting in the entire array being in sorted order when ProxmapSort completes.

ProxmapSort differs from bucket sorts in its use of the map key to place the data approximately where it belongs in sorted order, producing a "proxmap" — a proximity mapping — of the keys.

Another variant of bucket sort known as histogram sort or counting sort adds an initial pass that counts the number of elements that will fall into each bucket using a count array.

[4] Using this information, the array values can be arranged into a sequence of buckets in-place by a sequence of exchanges, leaving no space overhead for bucket storage.

[failed verification] The Postman's sort is a variant of bucket sort that takes advantage of a hierarchical structure of elements, typically described by a set of attributes.

This is the algorithm used by letter-sorting machines in post offices: mail is sorted first between domestic and international; then by state, province or territory; then by destination post office; then by routes, etc.

This is similar to a radix sort that works "top down," or "most significant digit first.

This creates n/8 "buckets" to which the remaining 7/8 of the items are distributed.

The variable bucket size of bucket sort allows it to use O(n) memory instead of O(M) memory, where M is the number of distinct values; in exchange, it gives up counting sort's O(n + M) worst-case behavior.

While this choice is effective for uniformly distributed inputs, other means of choosing the pivot in quicksort such as randomly selected pivots make it more resistant to clustering in the input distribution.

The n-way mergesort algorithm also begins by distributing the list into n sublists and sorting each one; however, the sublists created by mergesort have overlapping value ranges and so cannot be recombined by simple concatenation as in bucket sort.

However, this added expense is counterbalanced by the simpler scatter phase and the ability to ensure that each sublist is the same size, providing a good worst-case time bound.

Top-down radix sort can be seen as a special case of bucket sort where both the range of values and the number of buckets is constrained to be a power of two.

Consequently, each bucket's size is also a power of two, and the procedure can be applied recursively.

This approach can accelerate the scatter phase, since we only need to examine a prefix of the bit representation of each element to determine its bucket.