Question

Bubble sort can be improved. Smart bubble sort keeps track of how many exchanges are done within any single poss through the unsorted section of the list. If no ewchanges occur, then the list is sorted and the algorithm should stop. a. Write a pseudocode version of the smart bubble sort algorithm. b. Perform a smart bubble sort on the following list. How many comparisons are required? \(7,4,12,9,11\) c. Describe the best-case scenario for smart bubble sort on an n-element list. How many comparisons are required? How many exchanges are required? d. Under what circumstances does smart bubble sort do the same number of comparisons as regular bubble sort?

Short answer

Smart bubble sort requires 8 comparisons here. It performs optimally when the list is already sorted, requiring 0 exchanges. It matches regular bubble sort in a worst-case scenario with maximum swaps needed.

Step-by-step solution

Understanding Smart Bubble Sort

Smart bubble sort is an optimization on the regular bubble sort algorithm. While iterating through the list, if we encounter a pass with no exchanges, we stop as the list is already sorted. This helps in saving unnecessary passes through a sorted list.

Writing Pseudocode for Smart Bubble Sort

1. Initialize `n` as the length of the list. 2. Repeat for `i` from 0 to `n-1`: - Set `no_exchanges` to true. - Repeat for `j` from 0 to `n-i-2`: - If the element at `j` is greater than the element at `j+1`, swap them and set `no_exchanges` to false. - If `no_exchanges` is true, break out of the loop as the list is sorted.

Performing Smart Bubble Sort on the Given List

List: [7, 4, 12, 9, 11] - First pass: 7 swaps with 4 -> [4, 7, 12, 9, 11] then 12 swaps with 9 -> [4, 7, 9, 12, 11], finally 12 swaps with 11 -> [4, 7, 9, 11, 12]. Total comparisons = 4. - Second pass: No swapping needed as the list is already sorted. Total comparisons = 4. Total comparisons required = 8.

Best-case Scenario for Smart Bubble Sort

The best-case scenario occurs when the list is already sorted. In this case, after one pass through the list, no exchanges are needed, allowing the algorithm to stop after `n-1` comparisons and 0 exchanges.

Comparison with Regular Bubble Sort

Smart bubble sort does the same number of comparisons as regular bubble sort when each pass still requires at least one exchange to eventually sort the list completely. This usually happens in the worst case when the list is in reverse order.

Key concepts

Pseudocode

Pseudocode provides a high-level description of an algorithm, using easier to understand language which is closer to everyday English rather than strict coding syntax.
It outlines each step clearly, without going into details of actual code syntax, making it accessible for those who may not be experienced programmers.

Here's a simple pseudocode for a Smart Bubble Sort:

• Set `n` as the length of the list.
• For `i` from 0 to `n-1`, do the following:
• Initialize a variable, `no_exchanges`, set to true. This tracks if any swaps were made during the pass.
• For each `j` from 0 to `n-i-2`, check:
• If the item at position `j` is greater than the item at `j + 1`, swap them.
• If a swap occurs, set `no_exchanges` to false.
• If `no_exchanges` remains true after a full pass, it means the list is sorted, and you can break out of the loop early.

This method helps avoid unnecessary steps once the sorting is completed early, making it more efficient under certain conditions.

Best-case Scenario

In the best-case scenario, the list is already sorted from the beginning.
This unique situation allows the Smart Bubble Sort algorithm to exhibit its efficiency by recognizing that no swaps are needed.

The algorithm will confirm the sorted order in just one pass by:

• Comparing each adjacent pair through the list (hence, `n-1` comparisons for a list of length `n`).
• Making zero exchanges, since no items need to change positions.

After completing one pass with no exchanges, Smart Bubble Sort smoothly exits, reducing the number of operations greatly compared to its worst case.
This makes it noticeably faster in scenarios where data is often pre-sorted or nearly sorted.

Computational Efficiency

Computational efficiency determines how effectively an algorithm runs, which often depends on the input data and the structure of the algorithm itself.
Smart Bubble Sort incorporates a strategic break mechanism that aids greatly during sorting tasks, especially on pre-sorted or nearly-sorted lists.

By recognizing if a pass through the list involves zero exchanges, the algorithm can terminate early:

• Provides backward-efficiency, reducing unnecessary cycles.


• If the list is sorted or close to, the runtime drops significantly, often to linear time, that is, \(O(n)\).


• This approach caters to prerequisites often found in real-world scenarios, offering practical benefits in terms of time savings.

Thus, unlike regular bubble sort which runs in quadratic time \(O(n^2)\) for all cases, Smart Bubble Sort leverages its efficiency by minimizing unnecessary processing, especially beneficial in best-case scenarios.

Comparisons and Exchanges

In sorting algorithms, "comparisons" check the order between elements, while "exchanges" (or "swaps") modify the order by swapping positions of elements when necessary.
Smart Bubble Sort improves on regular bubble sort in how it uses these operations efficiently.

The number of comparisons and exchanges directly affects sorting speed and total processing time:

• Comparisons assess if an element is greater than its neighbor, critical for determining if swaps are needed.
• In the best case where no exchanges occur, Smart Bubble Sort only requires \(n-1\) comparisons, with zero exchanges.
• In less ideal cases where the list is not sorted, Smart Bubble Sort manages the same number of comparisons and can still reduce exchanges compared to regular bubble sort.

When the list is sorted in reverse order (the worst-case scenario), every element still needs to be compared and swapped, equaling the traditional bubble sort in both comparisons and exchanges.