Home > Articles > Programming > Algorithms

This chapter is from the book

1.3. Analysis of Algorithms

Though the analysis of sorting and merge-sort that we considered in §1.2 demonstrates the intrinsic “difficulty” of the sorting problem, there are many important questions related to sorting (and to mergesort) that it does not address at all. How long might an implementation of mergesort be expected to run on a particular computer? How might its running time compare to other O(NlogN) methods? (There are many.) How does it compare to sorting methods that are fast on average, but perhaps not in the worst case? How does it compare to sorting methods that are not based on compares among elements? To answer such questions, a more detailed analysis is required. In this section we briefly describe the process of doing such an analysis.

To analyze an algorithm, we must first identify the resources of primary interest so that the detailed analysis may be properly focused. We describe the process in terms of studying the running time since it is the resource most relevant here. A complete analysis of the running time of an algorithm involves the following steps:

  • Implement the algorithm completely.
  • Determine the time required for each basic operation.
  • Identify unknown quantities that can be used to describe the frequency of execution of the basic operations.
  • Develop a realistic model for the input to the program.
  • Analyze the unknown quantities, assuming the modeled input.
  • Calculate the total running time by multiplying the time by the frequency for each operation, then adding all the products.

The first step in the analysis is to carefully implement the algorithm on a particular computer. We reserve the term program to describe such an implementation. One algorithm corresponds to many programs. A particular implementation not only provides a concrete object to study, but also can give useful empirical data to aid in or to check the analysis. Presumably the implementation is designed to make efficient use of resources, but it is a mistake to overemphasize efficiency too early in the process. Indeed, a primary application for the analysis is to provide informed guidance toward better implementations.

The next step is to estimate the time required by each component instruction of the program. In principle and in practice, we can often do so with great precision, but the process is very dependent on the characteristics of the computer system being studied. Another approach is to simply run the program for small input sizes to “estimate” the values of the constants, or to do so indirectly in the aggregate, as described in Exercise 1.7. We do not consider this process in detail; rather we focus on the “machine-independent” parts of the analysis in this book.

Indeed, to determine the total running time of the program, it is necessary to study the branching structure of the program in order to express the frequency of execution of the component instructions in terms of unknown mathematical quantities. If the values of these quantities are known, then we can derive the running time of the entire program simply by multiplying the frequency and time requirements of each component instruction and adding these products. Many programming environments have tools that can simplify this task. At the first level of analysis, we concentrate on quantities that have large frequency values or that correspond to large costs; in principle the analysis can be refined to produce a fully detailed answer. We often refer to the “cost” of an algorithm as shorthand for the “value of the quantity in question” when the context allows.

The next step is to model the input to the program, to form a basis for the mathematical analysis of the instruction frequencies. The values of the unknown frequencies are dependent on the input to the algorithm: the problem size (usually we name that N) is normally the primary parameter used to express our results, but the order or value of input data items ordinarily affects the running time as well. By “model,” we mean a precise description of typical inputs to the algorithm. For example, for sorting algorithms, it is normally convenient to assume that the inputs are randomly ordered and distinct, though the programs normally work even when the inputs are not distinct. Another possibility for sorting algorithms is to assume that the inputs are random numbers taken from a relatively large range. These two models can be shown to be nearly equivalent. Most often, we use the simplest available model of “random” inputs, which is often realistic. Several different models can be used for the same algorithm: one model might be chosen to make the analysis as simple as possible; another model might better reflect the actual situation in which the program is to be used.

The last step is to analyze the unknown quantities, assuming the modeled input. For average-case analysis, we analyze the quantities individually, then multiply the averages by instruction times and add them to find the running time of the whole program. For worst-case analysis, it is usually difficult to get an exact result for the whole program, so we can only derive an upper bound, by multiplying worst-case values of the individual quantities by instruction times and summing the results.

This general scenario can successfully provide exact models in many situations. Knuth’s books [17][18][19][20] are based on this precept. Unfortunately, the details in such an exact analysis are often daunting. Accordingly, we typically seek approximate models that we can use to estimate costs.

The first reason to approximate is that determining the cost details of all individual operations can be daunting in the context of the complex architectures and operating systems on modern computers. Accordingly, we typically study just a few quantities in the “inner loop” of our programs, implicitly hypothesizing that total cost is well estimated by analyzing just those quantities. Experienced programmers regularly “profile” their implementations to identify “bottlenecks,” which is a systematic way to identify such quantities. For example, we typically analyze compare-based sorting algorithms by just counting compares. Such an approach has the important side benefit that it is machine independent. Carefully analyzing the number of compares used by a sorting algorithm can enable us to predict performance on many different computers. Associated hypotheses are easily tested by experimentation, and we can refine them, in principle, when appropriate. For example, we might refine comparison-based models for sorting to include data movement, which may require taking caching effects into account.

Exercise 1.11 Run experiments on two different computers to test the hypothesis that the running time of mergesort divided by the number of compares that it uses approaches a constant as the problem size increases.

Approximation is also effective for mathematical models. The second reason to approximate is to avoid unnecessary complications in the mathematical formulae that we develop to describe the performance of algorithms. A major theme of this book is the development of classical approximation methods for this purpose, and we shall consider many examples. Beyond these, a major thrust of modern research in the analysis of algorithms is methods of developing mathematical analyses that are simple, sufficiently precise that they can be used to accurately predict performance and to compare algorithms, and able to be refined, in principle, to the precision needed for the application at hand. Such techniques primarily involve complex analysis and are fully developed in our book [10].

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