Home > Articles > Programming > C/C++

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

Like this article? We recommend

The Merge-Sort Algorithm

The Merge-Sort Algorithm

One of the faster sorting techniques is the merge-sort algorithm, but it has one major drawback: It requires the allocation of a temporary array as large as the array to be sorted. If memory is scarce, the merge-sort algorithm might not be feasible.

Consider what it takes to merge two already sorted groups into a single sorted group. For example:

1  4  9          3  5  7

Once a merge is performed, the resulting array contains the following:

1  3  4  5  7  9

It's not too hard to come up with an algorithm that performs the merge. We keep track of an index (I or J) for each sub-range and then step through, comparing the current element in one group to the current element in the other. Only after an element is copied do we advance the index to the next element.

In this example, we begin by comparing 1 in the first list to 3 in the second list. Because 1 is the smaller number, we copy 1 to the temporary array and advance the first index to point to 4. Then we compare 4 in the first list to 3 in the second list. Now 3 is the smaller number, so we copy that number to the temporary array and advance the second index.

Here's the pseudo-code:

Initialize index I to the beginning of first sub-range.
Initialize index J to the beginning of second sub-range.
While neither sub-range's end point is reached,
      If A[I] < A[J]
            Copy A[I] to temporary array and increment I.
Else
            Copy A[J] to next temporary array and increment J.
While index I has not reached its end point,
       Copy A[I] to next temporary array and increment I.
While index J has not reached its end point,
      Copy A[J] to next temporary array and increment J.
Copy contents of the temporary array back to the first array.

Eventually, one of the two indexes is incremented to its end point. After that happens, the remainder of the other sub-range is copied. The result creates a temporary array that has all the elements in one continuous order. The final step is to copy these elements back to the first array.

Roughly N comparisons are performed, where N is the size of the groups put together. The algorithm also has to copy elements to and from the temporary array, but the total amount of work is still proportional to N, the number of elements involved.

Therefore a merge operation has a cost in time of O(n).

Now, how can we advance from this limited ability (merging two presorted sub-ranges) to the much more powerful ability of sorting any array whatsoever?

It's easy to see that if a sub-range of an array is of size 1 (that is, the range contains only one member), it's already sorted. That's a trivial observation, but it's going to be important.

What if a range is of size 2? For example:

7    3

We can break this range into two sub-ranges, each of size 1. Remember, sub-ranges of size 1 are already sorted! Merging these two "ranges" produces one sorted range, two elements in length:

3    7

Assume this has been done for two groups of 2. We now have two pre-sorted sub-ranges, and we can merge them to produce a single range, four elements in length. For example:

3    7         2    10

Merging the two sub-ranges (3, 7 and 2, 10) produces the following:

2    3    7    10

A range of four elements has been fully sorted. Next, assume that we've done this for two ranges of size 4.

2    3    7    10         1    8    9    11

We now do one last merge, to produce:

1    2    3    7    8    9    10   11

Voilà! A range eight members in length has been sorted.

We can repeat this process to sort ever-larger ranges, merging ranges of size 16, and then 32, 64, 128, and so on. The size of these ranges increases exponentially.

Look at the situation another way: As the size of the array (N) increases, how fast does the number of levels increase? The answer is an inverse-exponentiation function; that is, the logarithm of N, or log N. This figure increases relatively slowly. For example, it takes only one more level to sort up to 128 elements than to sort 64.

The total amount of work to be done at each level is always O(n). Therefore, the total time taken by this algorithm is as follows:

O(n log n)

For large N, this is less expensive than O(n2). Later in this article, I'll demonstrate how much difference this makes in practice. But first, let's code a merge sort.

Programming a Merge Sort

The previous section analyzed the merge-sort algorithm through a bottom-up approach. To code the algorithm, we need to take a top-down approach and use recursion. Recursion causes a function to call itself over and over until it reaches the terminal case, at which point the function starts returning, and things move from the bottom up. Here's the pseudo-code:

MergeSort(A[], iBegin, iEnd):
If the size of the range is less than 2
      Return immediately (we're already sorted!)
Find the midpoint index, M, by averaging indexes iBegin and iEnd.
MergeSort(A, iBegin, M). (Recursive call!)
MergeSort(A, M, iEnd). (Recursive call!)
Merge the sub-ranges (iBegin, M) and (M, iEnd).

These ranges are exclusive of their end points. For example, the range iBegin to M includes positions up to but not including M. The range M to iEnd includes positions up to but not including iEnd.

The C++ source code for a merge sort function is rather simple, although the code for the merge itself is a little more involved:

// Merge Ranges: Merge presorted sub-range iBegin to iMid
//  with presorted sub-range iMid to iEnd.
//  Ranges are exclusive of end points.
void merge_ranges(int A[], int TempArray[],
int iBegin, int iMid, int iEnd) {
    int i = iBegin;   // i will index first sub-range.
    int j = iMid;     // j will index second sub-range.
    int k = i;        // index into temporary array

    // While neither end point has been reached,
    //  Use A[i] < A[j] comparison to step through, always
    //  copying the lesser of the two indexed elements.
    while (i < iMid && j < iEnd) {
        if (A[i] < A[j])
            TempArray[k++] = A[i++];
        else
            TempArray[k++] = A[j++];
    }
    // Now, eat the "tail" remaining...
    while (i < iMid)
       TempArray[k++] = A[i++];
    while (j < iEnd)
       TempArray[k++] = A[j++];

    // Finally, copy back contents of temp. array.
    for (int i = iBegin; i < iEnd; i++)
        A[i] = TempArray[i];
}

void merge_sort(int A[], int TempArray[], int iBegin, int iEnd) {
    if (iEnd—iBegin < 2)  // If range is size 1, it is sorted!
        return;
    int iMid = (iBegin + iEnd) / 2;  // Calculate midpoint.
    // Recursively sort each half of range.
    merge_sort(A, TempArray, iBegin, iMid);
    merge_sort(A, TempArray, iMid, iEnd);
    // Now, merge these two sorted ranges.
    merge_ranges(A, TempArray, iBegin, iMid, iEnd);
}
  • + 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.

Overview


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.

Surveys

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.

Newsletters

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.

Security


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

Children


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

Marketing


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.

Choice/Opt-out


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.

Links


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