/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Bubble sort can be improved. Sma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 14

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.
05

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A tennis tournarnsnt has 342 players. A singlo match imolves 2 players. The winner of a match will play the winner of a match in the next round, wheress losers are eliminated from the toumament. The 2 players who have won all previous rounds play in the final game, and the wirner wins the tournament. What is the total number of matches needed to determine the winner? a. Here is one algorithm to answer this question. Compute \(342 / 2=171\) to get the number of pairs (matches) in the first round, which results in 171 winners to go on to the second round. Compute \(171 / 2=85\) with 1 left over, which results in 85 matches in the second round and 85 winners, plus the 1 left over, to go on to the third round. For the third round compute \(86 / 2=43,50\) the third round has 43 matches, and so on. The total number of matches is \(171+85+43+\ldots\) Finish this process to find the total number of matches. b. Here is another algorithm to solve this problem, Each match results in exactly one loser, so there must be the same number of matches as losers in the tournament. Compute the total number of losers in the entire tournament. (Hint: This isf't really a computation; it is a one-sentence argument.) c. What is your opinion on the relative clarity, elegance, and efficiency of the two algorithms?

a. An algorithm that is \(\Theta(n)\) takes 10 seconds to execute on a particular computer when \(n=100\). How long would you expect it to take when \(n=500 ?\) b. An algorithm that is \(\Theta\left(n^{2}\right)\) takes 10 seconds to execute on a particular computer when \(n=100\). How long would you expect it to take when \(n=500 ?\)

An English Christmas carol, "The Twelve Days of Christmas, " dates from the late \(1700 \mathrm{~s}\). The 12 verses in the song are cumulative, each verse adding an additional gift given by "my true love." The twelfth verse says "On the twelfth day of Christmas, my true love gave to me ..." 12 Drummers Drumming 11 Pipers Piping 10 Lords-a-Leaping \(\ldots\) and so forth down to ... 3 French Hens 2 Turtle Doves And a Partridge in a Pear Tree. a. Use Gauss's formula to find the total number of gifts given on Day 12 . b. How many total gifts are given over all 12 days? Hint: $$ 1(2)+2(3)+3(4)+\ldots+n(n+1)=\frac{n(n+1)(n+2)}{3} $$

The selection sort algorithm could be modified to stop when the unscrted section of the list contains only one number, because that one number must be in the correct position. Show that this modification wrould have no effect on the number of comparisons required to sort an neelement list.

Suppose the pattern-matching problem is changed to require locating only the first instance, if any, of the pattern within the text. a. Describe the worst case, give an example, and give the exact number of comparisons (of a pattern character with a text character) required. b. Describe the best case, give an example, and give the exact number of comparisons required.
See all solutions

Recommended explanations on Computer Science Textbooks

Blockchain Technology

Read Explanation

Functional Programming

Read Explanation

Computer Programming

Read Explanation

Problem Solving Techniques

Read Explanation

Computer Organisation and Architecture

Read Explanation

Databases

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.