Question

(Bucket Sort) A bucket sort begins with a one-dimensional array of positive integers to be sorted and a two-dimensional array of integers with rows subscripted from 0 to 9 and columns subscripted from 0 to \(n-1\), where \(n\) is the number of values in the array to be sorted. Each row of the two- dimensional array is referred to as a bucket. Write a function bucketsort that takes an integer array and the array size as arguments and performs as follows: a) Place each value of the one-dimensional array into a row of the bucket array based on the value's ones digit. For example, 97 is placed in row 7,3 is placed in row 3 and 100 is placed in row \(0 .\) This is called a "distribution pass." b) Loop through the bucket array row by row, and copy the values back to the original array. This is called a "gathering pass." The new order of the preceding values in the onedimensional array is 100,3 and 97 c) Repeat this process for cach subsequent digit position (tens, hundreds, thousands, etc.). On the second 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 pass, 100 is placed in row 1,3 is placed in row zero and 97 is placed in row zero (after the 3). After the last gathering pass, the original array is now in sorted order. Note that the two-dimensional array of buckets is 10 times the size of the integer array being sorted. This sorting technique provides better performance than an insertion sort, but requires much more memory. The insertion sort requires space for only one additional element of data. This is an example of the space-time trade-off: The bucket sort uses more memory than the insertion 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

To perform a bucket sort: 1) Initialize a 10x(n) two-dimensional bucket array. 2) Distribute array elements into buckets based on their current digit. 3) Perform a gathering pass and copy back to the original array. 4) Repeat for the next higher digit. 5) When all digits have been processed, the array is sorted.

Step-by-step solution

Initialize Buckets

Create a two-dimensional array (buckets) with 10 rows, each corresponding to a digit from 0 to 9. Each row should have the capacity to hold up to n elements, where n is the size of the input array.

Distribution Pass

For each element in the input array, determine the relevant digit (starting with the ones digit). Place the element in the corresponding row of the buckets. For example, if the element is 97, place it in row 7, if the element is 3, place it in row 3, and if the element is 100, place it in row 0.

Gathering Pass

Iterate through the bucket rows in order from 0 to 9. Copy the elements from the buckets back into the original array, preserving the order they appear within each bucket. This rearranges the elements in the original array according to their ones digit.

Repeat for Other Digits

Repeat the distribution and gathering passes for each subsequent digit place (tens, hundreds, thousands, etc.). Make sure to reset buckets before each new distribution pass. During each pass, ignore non-existing higher digits (place them in row 0).

Sort Complete

After the last gathering pass has been completed for the highest digit place present in the numbers, the original array will be sorted. No further action is required if all digits have been processed.

Key concepts

Sorting algorithms

Sorting algorithms are fundamental techniques used in computer science to reorder items in collections—such as arrays—so that their elements are in a specific sequence, often numerical or lexicographical. There are numerous sorting algorithms, each with unique mechanisms, advantages, and drawbacks.

Some popular sorting algorithms include:

• Bubble Sort: Repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order.
• Quick Sort: Selects a 'pivot' element and partitions the array around the pivot, then recursively applies the same logic to the subarrays.
• Merge Sort: Divides the array into halves, sorts each half, and then merges them back together.
• Bucket Sort: As described in our example, distributes elements into buckets and then sorts these buckets individually, often using another sorting algorithm.

The choice of sorting algorithm depends on the context of the problem, such as the size and distribution of data, as well as desired trade-offs between efficiency and complexity.

C++ programming

C++ is a powerful, high-performance programming language that is widely used for system/software development and game programming, among other applications. In C++ programming, sorting algorithms are implemented to manage data sets efficiently.

For example, the Bucket Sort algorithm exercise would require knowledge of arrays, loops, and functions in C++. Specifically, one must initialize and manage a two-dimensional array, use loops to distribute elements into buckets, and implement a function to sort the array.

Here is a simplified representation of steps in C++:

• Declare the buckets as a 2D vector of integers.
• Use a loop to iterate over elements and determine the appropriate bucket.
• Sort contents of each bucket, if necessary (in our exercise, this is implicit).
• Flatten the buckets back into the original array in a sorted order.

Understanding C++ data structures and control flow is essential in tackling such sorting problems.

Distributive sorting techniques

Distributive sorting techniques, such as Bucket Sort, Counting Sort, and Radix Sort, involve distributing the items to be sorted into a number of categories or 'buckets.' The distribution is based on certain attributes or key values, like the individual digits in Bucket Sort.

The main steps in a distributive sort include:

• Determining the category or bucket for each item based on a part of its key value.
• Placing the item into the corresponding bucket.
• Sorting items within those buckets if necessary.
• Concatenating the buckets back in order to achieve the sorted list.

Distributive sorts often avoid comparison by categorizing elements directly, hence can achieve performance better than O(n log n) under certain conditions. These algorithms are particularly effective for sorting integers, strings, or other data where the key values are discrete and can be distributed over a finite range of categories.

Space-time trade-off in sorting

The space-time trade-off in sorting algorithms is a concept wherein a balance is struck between the amount of memory (space) used and the speed of execution (time). Different sorting algorithms demonstrate different trade-offs.

For instance:

• Bubble Sort: Has a minimal space overhead as it operates in-place, but it tends to be slow with a time complexity of O(n^2).
• Quick Sort: Also operates mainly in-place but can require O(log n) additional space for the call stack and has an average time complexity of O(n log n).
• Merge Sort: Requires additional space proportional to the size of the array to hold the merged result, but ensures a stable time complexity of O(n log n).
• Bucket Sort: Can be very fast with a time complexity close to O(n); however, as discussed in the exercise, it usually requires more space as additional structures (buckets) are used to distribute the elements.

This trade-off is vital when selecting an appropriate sorting algorithm, as it is often necessary to consider limitations on memory and time requirements based on the application or environment in which the sort is being conducted.