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Bucket Sort

This page provides a brief introduction to bucket sort.

Definition

Bucket sort is a sorting algorithm suitable for cases where the data to be sorted has a large value range but is relatively uniformly distributed.

Process

Bucket sort proceeds as follows:

  1. Set up a fixed-size array as empty buckets;
  2. Traverse the sequence and place each element into its corresponding bucket;
  3. Sort each non-empty bucket;
  4. Gather the elements from the non-empty buckets back into the original sequence.

Properties

Stability

If a stable inner sorting algorithm is used and the relative order of elements is preserved when inserting them into buckets, then bucket sort is a stable sorting algorithm.

Since each bucket contains few elements, insertion sort is typically used. In this case, bucket sort is a stable sorting algorithm.

Time Complexity

The average time complexity of bucket sort is \(O(n + n^2/k + k)\) (dividing the value range into \(n\) blocks + sorting + recombining elements), which becomes \(O(n)\) when \(k\approx n\).1

The worst-case time complexity of bucket sort is \(O(n^2)\).

Implementation

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constexpr int N = 100010;

int n, w, a[N];
vector<int> bucket[N];

void insertion_sort(vector<int>& A) {
  for (int i = 1; i < A.size(); ++i) {
    int key = A[i];
    int j = i - 1;
    while (j >= 0 && A[j] > key) {
      A[j + 1] = A[j];
      --j;
    }
    A[j + 1] = key;
  }
}

void bucket_sort() {
  int bucket_size = w / n + 1;
  for (int i = 0; i < n; ++i) {
    bucket[i].clear();
  }
  for (int i = 1; i <= n; ++i) {
    bucket[a[i] / bucket_size].push_back(a[i]);
  }
  int p = 0;
  for (int i = 0; i < n; ++i) {
    insertion_sort(bucket[i]);
    for (int j = 0; j < bucket[i].size(); ++j) {
      a[++p] = bucket[i][j];
    }
  }
}

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N = 100010
w = n = 0
a = [0] * N
bucket = [[] for i in range(N)]


def insertion_sort(A):
    for i in range(1, len(A)):
        key = A[i]
        j = i - 1
        while j >= 0 and A[j] > key:
            A[j + 1] = A[j]
            j -= 1
        A[j + 1] = key


def bucket_sort():
    bucket_size = int(w / n + 1)
    for i in range(0, n):
        bucket[i].clear()
    for i in range(1, n + 1):
        bucket[int(a[i] / bucket_size)].append(a[i])
    p = 0
    for i in range(0, n):
        insertion_sort(bucket[i])
        for j in range(0, len(bucket[i])):
            a[p] = bucket[i][j]
            p += 1

References and Notes

  1. (English) Bucket sort - Wikipedia ↩

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