Bucket sort is an algorithm that works by distributing elements of an array into buckets based on their values. It then sorts each bucket using insertion sort and concatenates the buckets to produce the final sorted array. The time complexity is O(n) to distribute elements into buckets, O(n^2) to sort each bucket using insertion sort, and O(n) to concatenate buckets, resulting in an overall average case time complexity of O(n+k) where n is the number of elements and k is the number of buckets.

1. BUCKET SORT
Design and Analysis Algorithm

2. BUCKET SORT
93 42 69 90 52 51 22 29
Array A
Array B0
1
2
3
4
5
6
7
8
9
42
52 51
69
93 90
Now we will sort each bucket using
insertion sort
22 29

3. 0
1
2
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5
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7
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9
42
51 52
69
22 29
90 93
29 42 51 52 69 90 93
[0] [1] [2] [3] [4] [5] [6] [7]
After sorting each
bucket using
insertion sort
Sorted Array

4. TIME COMPLEXITY
n=length[A] 1 time
for i=1 to n n times
do insert A[i] into bucket B[n A[i]] O(n)
for i=0 to n-1 n times
do sort bucket B[i] with insertion sort ∑O(n^2)
Concatenate bucket O(n)

5. BEST CASE
 In the best case every element belongs to
different buckets so the complexity is O(n+k)
 Where n=number of elements and k is the
number of buckets

6. AVERAGE CASE
 In average case running time of bucket sort
is:
T(n)=Θ(n)+∑O(n^2)
 So the average complexity is O(n+k)

7. WORST CASE
 In worst case complexity is O(n^2), as every
element belongs to one bucket so we have to
use insertion sort n elements.
 Instead of insertion sort we could merge or
heap sort because their worst case running
time is O(n log n).
 But we use insertion sort because we expect
the buckets to be small and insertion sort
works much faster for small array.