Question

(Bucket Sort) A bucket sort begins with a one-dimensional vector of positive integers to be sorted and a two-dimensional vector 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 vector is referred to as a bucket. Write a class named Bucketsort containing a function called sort that operates as follows: a. Place each value of the one-dimensional vector into a row of the bucket vector, 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 vector row by row, and copy the values back to the original vector. This procedure is called a gathering pass. The new order of the preceding values in the one-dimensional vector 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 vector 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 vector is in sorted order. Note that the two-dimensional vector of buckets is 10 times the length of the integer vector being sorted. This sorting technique provides better performance than a bubble sort, but requires much more memorythe bubble sort requires space for only one additional element of data. This comparison is an example of the spacetime 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 vector on each pass. Another possibility is to create a second two-dimensional bucket vector and repeatedly swap the data between the two bucket vectors.

Short answer

Implement a distribution and gathering pass for each digit position in the numbers to perform bucket sort.

Step-by-step solution

Initializing Data

Start by identifying the one-dimensional array of positive integers that needs to be sorted. Also, prepare a two-dimensional array (a vector of buckets) with 10 rows, one for each digit (0-9). The number of columns should be equal to the length of the input array since, in the worst case, all numbers could end up in the same bucket.

Distribution Pass for Ones Digit

For each number in the input array, determine the value of its ones (rightmost) digit. Place each number into the corresponding row (bucket) in the two-dimensional array based on this digit. For example, the number 97 goes into row 7, and the number 3 goes into row 3.

Gathering Pass

Once all numbers are placed in their respective buckets based on the ones digit, gather all numbers from the buckets and copy them back to the original array starting from the lowest index. This will result in a partially sorted array based on the ones digit.

Repeat Process for Tens Digit

Perform the same distribution and gathering process for the tens digit. Determine each number’s tens digit and place it into the appropriate row in the bucket array. After placing all numbers by their tens digits, gather them back into the original array.

Repeat Process for Further Digits

Continue the distribution and gathering process for higher-place digits (hundreds, thousands, etc.) until all digits have been processed. For each pass, extract and sort numbers by the respective digit value and copy them back to the original array.

Conclusion of Sorting

After completing the passes for each digit in the largest number, the sorting of the initial array is complete. The output will be the sorted array, now in ascending order, based on all the digit places.

Key concepts

Sorting Algorithms

Sorting algorithms are methods used to arrange data in a specific order, most commonly in ascending or descending order. They are fundamental in computer science as they optimize the process of accessing or arranging data in a meaningful way.
There are several types of sorting algorithms, each with its own mechanisms and efficiencies. For instance, Quick Sort and Merge Sort are known for their speed, but they involve more complex operations compared to simpler algorithms such as Bubble Sort or Selection Sort.
Bucket Sort, the algorithm of focus here, is distinctive because it distributes elements into different "buckets" or sections based on their values for a particular digit. This makes Bucket Sort particularly efficient with uniformly distributed data over a fixed range. Essentially, the concept of Bucket Sort follows a distributive approach, grouping values before sorting them within each group and then combining them.
In summary, sorting algorithms serve as essential tools for organizing data, and choosing the right one depends on the specific data characteristics and performance requirements.

Time Complexity

Understanding time complexity is key to evaluating how efficient an algorithm is as it scales with larger input sizes. It helps us predict how an algorithm's runtime increases as the size of the input data grows.
Bucket Sort has an average time complexity of \( O(n + k) \), where \( n \) is the number of elements in the array and \( k \) is the number of buckets. This assumes that the input distribution is uniform and that the time to sort within a bucket is sufficiently small.
However, if data is not perfectly distributed or if each bucket contains many elements, additional sorting within the buckets can degrade overall performance. This could push Bucket Sort towards a worse-case scenario resembling \( O(n^2) \).
Thus, while Bucket Sort can perform efficiently under optimal conditions, it's important to consider the nature of the dataset when evaluating this algorithm's time complexity.

Space Complexity

Space complexity refers to the amount of working storage an algorithm needs. For sorting algorithms, understanding space complexity is essential for assessing their practicality, especially when dealing with large datasets.
Bucket Sort is known to have a relatively high space complexity because it requires additional memory storage in the form of buckets. Specifically, it uses \( O(n + k) \) space, where \( n \) is the number of elements and \( k \) is the number of buckets.
This space requirement implies a need for larger memory allocation compared to more space-efficient algorithms such as Insertion Sort or Bubble Sort, which generally use \( O(1) \) auxiliary space.
Despite its good average performance, the high space complexity of Bucket Sort can be a drawback for applications with memory constraints, underscoring the trade-off between speed and space efficiency.

Radix Sort

Radix Sort is a fascinating sorting algorithm that's often compared to Bucket Sort. Like Bucket Sort, Radix Sort handles numbers through distribution based on their digit places, starting from the least significant digit to the most significant digit.
One of the key distinctions is that Radix Sort operates efficiently with a fixed-sized base (usually ten for decimal digits), and it typically utilizes a stable sorting algorithm like Counting Sort for internally sorting the values at each level of digit significance.
The efficiency of Radix Sort lies in its ability to process numbers in multiple passes, sorting one digit at a time without comparing total values directly. It works particularly well on integers and can be highly efficient when the maximum digit length is known, making it a good choice for datasets with known limits.
With a time complexity of \( O(nk) \), where \( n \) is the number of keys, and \( k \) is the maximum number of digits in any key, Radix Sort can outperform other sorts under certain conditions, though it does require additional space due to the auxiliary sorting process.