How does the bucket sort algorithm sort elements?

Bucket sort algorithm sorts elements by distributing them into different buckets, then sorting these buckets individually.

Bucket sort, also known as bin sort, is a distribution sort algorithm that works by distributing the elements of an array into a number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm or by recursively applying the bucket sort algorithm. The sorted values from the buckets are then combined to get the sorted array.

The process begins by determining a 'bucket range', which is the range of values that each bucket can hold. This is typically done by dividing the maximum value in the array by the number of buckets. For example, if the maximum value is 100 and there are 10 buckets, each bucket would hold a range of 10 values.

Next, the algorithm iterates over the array, placing each element into the appropriate bucket. This is done by comparing the element's value to the bucket range. For instance, if the bucket range is 10, an element with a value of 35 would be placed into the fourth bucket (since 35 falls between 30 and 40).

Once all elements have been distributed into buckets, each bucket is sorted individually. This can be done using any sorting algorithm, but often the same bucket sort algorithm is applied recursively. This is particularly effective if the elements are uniformly distributed over the range, as it can significantly reduce the sorting time.

Finally, the sorted values from each bucket are concatenated in order to produce the final sorted array. This is done by iterating over the buckets in order, and appending each bucket's elements to the end of the output array.

In terms of complexity, bucket sort performs best when the input is uniformly distributed over the range. In this case, it can achieve a time complexity of O(n), making it one of the most efficient sorting algorithms. However, if the input is not uniformly distributed, the performance can degrade to O(n²), which is similar to the worst-case performance of simpler comparison-based sorting algorithms.