Question

(Bubble Sort Enbancements) The bubble sort described in Exercise 7.11 is inefficient for large arrays. Make the following simple modifications to improve the performance of the bubble sort: a) After the first pass, the largest number is guaranteed to be in the highest-numbered element of the array; after the second pass, the two highest numbers are "in place," and so on. Instead of making nine comparisons on every pass, modify the bubble sort to make eight comparisons on the second pass, seven on the third pass, and so on. b) The data in the array may already be in the proper order or near-proper order, so why make nine passes if fewer will suffice? Modify the sort to check at the end of each pass if any swaps have been made. If none have been made, then the data must already be in the proper order, so the program should terminate. If swaps have been made, then at least one more pass is needed.

Short answer

Enhance bubble sort by reducing comparisons each pass and adding an early termination condition if no swaps are made during a pass.

Step-by-step solution

Adjust the Number of Comparisons per Pass

Modify the original bubble sort algorithm so that the number of comparisons decreases by one with each pass. After the first pass, the largest number is in the right position, so the next pass should do one less comparison, and so on. This means for an array of length n, the first pass does n-1 comparisons, the second pass does n-2, continuing until only one comparison is made.

Implement Early Termination

Introduce a flag variable to help determine if any swaps have been made in the current pass. At the start of each pass, set the flag to false. Whenever a swap occurs, set the flag to true. At the end of each pass, if the flag is still false (meaning no swaps occurred), the array is sorted, and you can terminate the algorithm early.

Combine Adjustments into the Bubble Sort Algorithm

Integrate both optimizations into the bubble sort. Loop over the array, and in each iteration, reduce the number of comparisons based on the current pass number, and monitor the flag for any swaps. The algorithm stops when a complete pass occurs without any swaps, indicating that the array is already sorted.

Key concepts

Algorithm Efficiency

In the world of computer science, algorithm efficiency is a measure of how effectively an algorithm performs in terms of time and space usage. The goal is to minimize the resources required to solve a given problem. With sorting algorithms like bubble sort, efficiency can be dramatically affected by the number of elements being sorted. In the context of the Bubble Sort Optimization exercise, boosting efficiency involves reducing the total number of comparisons and incorporating an early termination condition. This makes sure the algorithm doesn't spend unnecessary cycles on an array that is already sorted or perform redundant comparisons when the largest elements have been placed in their final position.

Comparison Reduction

The principle of comparison reduction is critical in optimizing sorting algorithms. By default, bubble sort compares adjacent elements and performs swaps to sort the array, requiring potentially many comparisons.

In the optimized version, the number of comparisons per pass decreases since following each pass, the n-1 elements are sorted already. For example, in a 10-element array, after the first pass, we only need to make 8 comparisons on the next pass, reducing the total number from 45 (-1 comparisons for an array of 9 in a naive bubble sort) to 36, significantly cutting down the sorting time.

Early Termination Condition

An early termination condition is a smart way to improve the performance of a sorting algorithm. In the case of bubble sort, after completing a pass without any swaps, we can conclude that the array is already sorted.

This optimization uses a boolean flag that is set to true if a swap occurs. If, by the end of the pass, the flag remains false, the algorithm can terminate early, skipping the remaining unnecessary passes. This condition is particularly effective for nearly sorted or fully sorted arrays, where a naive bubble sort would still perform a full set of checks.

Sorting Algorithms

Sorting is a foundational task in programming, and sorting algorithms are designed to rearrange a sequence of elements into a particular order—typically numerical or lexicographical. Common sorting algorithms include bubble sort, quick sort, merge sort, and insertion sort, each with its own advantages in different scenarios.

Bubble sort, the subject of our exercise, is known for its simplicity and includes comparing pairs of adjacent elements and swapping them if they are in the wrong order. Despite being less efficient compared to more complex algorithms, its ease of understanding and implementation makes it a valuable educational tool.

Programming Optimization Techniques

There are various programming optimization techniques utilized to refine code and improve efficiency. These can range from simple logic improvements to complex algorithmic changes.

In our bubble sort optimizations, two main techniques are employed: reducing the number of comparisons by considering the elements already sorted in previous passes, and introducing an early termination condition to stop the algorithm if the array is sorted before all passes are completed. These small yet effective tweaks underscore how incremental changes in a program's logic can lead to significant performance enhancements.