Home > Articles > Programming > C/C++

  • Print
  • + Share This
Like this article? We recommend

Like this article? We recommend

Other Algorithms

Other Algorithms

Sorting belongs to a class of problems with multiple solutions; that is, there are more than one or two ways to sort an array. Let's look at some other sorting techniques.

Bubble Sort

The bubble sort algorithm is only slightly more sophisticated than selection sort. This algorithm makes a pass through the array, comparing each element to its immediate neighbor. When two neighboring elements are out of order relative to each other, they're swapped. This action causes the largest element to "bubble up" to the last position during the first pass. Therefore, the next pass operates on only the first N-1 elements. The process is repeated on the first N-2 elements, first N-3 elements, and so on. The process continues until no swaps needed to be done during one full pass. When that happens, the algorithm quits.

The pseudo-code is fairly simple, and so is bubble-sort C++ code if you care to write it:

For I = N-1 down to 1
     For J = 0 up to but not including I
          If A[J] > A[J+1]
               Swap(A[J], A[J+1])
     If no swaps were done during this pass, exit.

In the typical case, the duration will be O(n2). But in the best case, with an array that is presorted, the duration is O(n), because the bubble sort quits after just one pass. This makes the bubble sort potentially faster than other algorithms. In general, though, a bubble sort takes O(n2), which makes it a poor algorithm for large N.


The quicksort algorithm is in some ways the most sophisticated sorting algorithm of all. It starts by selecting a "pivot" point within a range. It then places every element that's less than the pivot to its left, and every element that's greater than the pivot to its right. Each pass has a duration of O(n). When this is done recursively for smaller and smaller ranges, the entire array is sorted.

The algorithm can be summarized as follows, in which the ranges are defined to be exclusive of end points:

Quicksort(A[], iBegin, iEnd):
If iBegin—iEnd < 2
Select pivot P within the range iBegin to iEnd.
Partition the range so that all values less than P are to its left and all values greater than P are to its right. As a result, P is placed at a new position, NP.
Quicksort(A, iBegin, NP)
Quicksort(A, NP, iEnd)

In the typical case, the time duration is O(n log n), matching the speed of a merge sort. However, in the worst case, the time duration is O(n2), which is much less desirable.

The degenerate case can happen because it's not always easy to choose a good pivot point. The most naïve quicksort implementations just select the first or last element in the range as the pivot. But if the array is already sorted, this approach results in a degenerate case in which every range is split into ranges of size 1 and size N-1, thereby making a quicksort as slow as a selection sort! To avoid that problem, quicksort algorithms often use the midpoint (at index iMid) or take the median value of iBegin, iMid, and iEnd. But even that strategy causes poor results if all values in a sub-range are equal.

Despite this worst-case drawback, a quicksort has some advantages over a merge sort. In the typical case, quicksort is usually somewhat faster (typically in the range of 40–50%). But it has another, more substantial advantage. A merge sort requires significant extra space, of size O(n): The size of the temporary array required is equal to that of the array being sorted. This requirement is acceptable only when you have large amounts of memory available. In contrast, the quicksort algorithm needs relatively little extra space. Specifically, it needs only O(log n), which is required for the stack as a result of recursion.

Assuming that the degenerate case doesn't happen, it should be easy to see why the quicksort algorithm makes far fewer comparisons than the selection sort. Again, consider that a selection sort always does N(N-1)/2 comparisons, enough to compare each and every combination of two elements. (A bubble sort makes the same number of comparisons, unless it quits early.) But a quicksort divides a range into two smaller ranges. Once an element is grouped into a sub-range, that element is never compared to anything outside its range. This technique greatly reduces the number of comparisons.

Each level of the algorithm—during which every element is grouped into a smaller left or right partition—takes a total duration of O(n). As long as reasonable pivot points are selected, the sort has log n levels of recursion. This is why quicksort, like merge sort, has a duration of O(n log n).

Lessons Learned

Lessons Learned

What information can we take away from comparing all these algorithms? Basically, we've observed two rules.

  • Opt for the shortest duration. When designing algorithms to be used repeatedly (or in very large projects), consider how the complexity and time duration increase with the size of the data. The smallest durations, in order, are as follows:
    1. constant, O(1)
    2. logarithmic, O(log n)
    3. linear, O(n)

    But these durations may not always be achievable. If at all possible, avoid exponential time durations such as O(n2). If you can achieve times of O(n log n) instead of O(n2) , that's a major victory for speed and efficiency.

  • Accept that tradeoffs are part of the deal. The other point we've seen in this article is that old tradeoff: speed versus compactness (or, put another way, time versus space). The merge-sort algorithm guarantees a duration that's never more than O(n log n), which is both its best and worst case. But the tradeoff for this speed is that this algorithm requires more space. You can't have everything.
  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020