Question

A bucket sort begins with a one-dimensional array of positive integers to be sorted and a two-dimensional array of integers with rows indexed from 0 to 9 and columns indexed from 0 to \(n-1,\) where \(n\) is the number of values to be sorted. Each row of the two-dimensional array is referred to as a bucket. Write a class named BucketSort containing a method called sort that operates as follows: a) Place each value of the one-dimensional array into a row of the bucket array, based on the value's "ones" (rightmost) digit. For example, 97 is placed in row 7,3 is placed in row 3 and 100 is placed in row \(0 .\) This procedure is called a distribution pass. b) Loop through the bucket array row by row, and copy the values back to the original array. This procedure is called a gathering pass. The new order of the preceding values in the one-dimensional array is 100,3 and 97 c) Repeat this process for each subsequent digit position (tens, hundreds, thousands, etc.). On the second (tens digit) pass, 100 is placed in row 0,3 is placed in row 0 (because 3 has no tens digit) and 97 is placed in row \(9 .\) After the gathering pass, the order of the values in the one-dimensional array is 100,3 and \(97 .\) On the third (hundreds digit) pass, 100 is placed in row 1,3 is placed in row 0 and 97 is placed in row 0 (after the 3 ). After this last gathering pass, the original array is in sorted order. Note that the two-dimensional array of buckets is 10 times the length of the integer array being sorted. This sorting technique provides better performance than a bubble sort, but requires much more memory - the bubble sort requires space for only one additional element of data. This comparison is an example of the space-time trade-off: The bucket sort uses more memory than the bubble sort, but performs better. This version of the bucket sort requires copying all the data back to the original array on each pass. Another possibility is to create a second two- dimensional bucket array and repeatedly swap the data between the two bucket arrays.

Short answer

Use the bucket sort steps: distribute by ones, tens, hundreds digits, gathering each time.

Step-by-step solution

Initialize Buckets

Create a 2D array, or a list of lists, where each sub-list corresponds to a bucket for digits 0 to 9. The number of columns in each bucket should match the length of the input array.

Distribution Pass by Ones Digit

For each integer in the input array, determine the ones digit (i.e., the last digit of the integer) and place the integer into the corresponding bucket. For example, place 97 in bucket 7 and 3 in bucket 3.

Gathering Pass for Ones Digit

Traverse each bucket in order (from 0 to 9) and copy back the integers into the original array sequentially. This will reorder the array based on the ones digit.

Distribution Pass by Tens Digit

Repeat the distribution pass, but this time use the tens digit. For numbers with no tens digit, consider it as 0. Place each number in the corresponding bucket (e.g., 97 goes to bucket 9, 3 goes to bucket 0, 100 goes to bucket 0).

Gathering Pass for Tens Digit

Again, traverse each bucket in order and copy back the integers to the original array. This rearranges the array based on the tens digit.

Distribution Pass by Hundreds Digit

Perform the distribution pass using the hundreds digit this time. For numbers with fewer than hundreds places, treat the digit as 0. Sort numbers into their respective buckets (e.g., place 100 in bucket 1, 3 in bucket 0, and 97 in bucket 0).

Final Gathering Pass

Traverse all buckets, in order from 0 to 9, and copy the numbers back to the original array. Now, the array should be fully sorted.

Key concepts

Sorting Algorithms

When it comes to arranging data in a specific order, sorting algorithms play a crucial role. Bucket Sort is a type of sorting algorithm that divides elements into categories, or 'buckets'. This method capitalizes on the values' digit positions to organize the numbers effectively. Each bucket sorts numbers based on individual digits, such as ones, tens, and hundreds, in successive rounds. After sorting within each bucket, the algorithm assembles them back into the main array. This process continues until the digits in every place have been utilized to sort the collection. Thus, offering an efficient way to get data sorted by progressively handling digit groupings rather than sorting numbers via comparison directly.

Space-Time Trade-Off

In the world of computing, space-time trade-off is a concept where more memory usage might lead to faster execution times. With Bucket Sort, more space is occupied due to the creation of additional buckets for sorting operations. This additional memory helps in achieving faster sorting times compared to more traditional comparison-based sorts. Unlike a simple bubble sort that only needs a small amount of extra space, bucket sort requires space equal to the input array size upon each pass. As more memory is consumed by the buckets, the trade-off results in a more rapid sorting process despite the larger memory footprint. Students must understand how these trade-offs balance between memory use and execution speed to choose the best sorting algorithm for their needs.

Digital Distribution

Digital distribution in Bucket Sort is all about distributing the elements into buckets. This process is done by considering the value's individual digits, starting from the least significant digit, for sorting. Each integer's digit determines the specific bucket in which it will be placed.

• This step is often called a 'distribution pass', where elements are sorted based on a specific digit.
• The process repeats for each subsequent digit position to further refine the order.
• The concept of digital distribution simplifies data handling by breaking down successive digit comparisons and sorting iteratively.

Through this systematic approach, Bucket Sort handles even large datasets digitally distributed in manageable amounts.

Algorithm Efficiency

Algorithm efficiency refers to how well an algorithm performs in terms of time complexity and resource usage. Bucket Sort is efficient for sorting data with uniformly distributed input values. It utilizes linear time complexity, especially effective when data spans a limited range of integer values.

• The sorting is completed in linear time \( O(n+k) \), where \( n \) is the number of elements and \( k \) is the number of buckets.
• Its performance is superior to many other sorts when working with input suited to its partition-based strategy.
• Despite the additional memory, the algorithm can significantly reduce the number of comparisons needed.

Understanding the efficiency of Bucket Sort involves recognizing the suitability of the data and conditions under which it outperforms other algorithms, making it an essential choice for particular sorting tasks.