Zhao 1
Fangyi Zhao
LLD 100A
September 10, 2017
Professor Alkire
Rhetorical Analysis of Professional Writing
Introduction
In a society, which resources and temptations increase frequently, a public life (such as
political movement, social movements, etc.) and personal life (such as a job interview, daily
shopping, etc.) requires more and more communicating power. Communication’s effect
depends on the size of the rhetoric skills, the amount of rhetorical resources rather than just
the amount of knowledge. Therefore, most of the professional journals or articles include
rhetoric. The most commonly used of the writers are rhetorical appeals and rhetorical
strategies. There are four important elements which are logos, ethos, pathos and kairos when
writer want to appeal readers. Logos are rational and logical appeals, pathos are emotional
appeals, ethos are ethical appeals, and kairos are timing, place and culture appeals. Writers
may also use some rhetorical strategies like description, narration, exemplification, and
definitions to help prove the argument.
To be a computer science major’s student, coding seems like the biggest part for us.
However, algorithm is the soul of the computer, and it can’t be separate with any software.
When complete a software, different programmers will use different way to achieve it.
Nevertheless, how to find the best software in these different versions? Most of people may
consider interface, user interaction, and functionality as conditions to estimate software. But,
for a professional way, the software which spend the less time is the best software. In order to
achieve this purpose, algorithm make a great contribution. The article is from the textbook of
my CS-146 class, Introduction to Algorithms, which is published in 2009 and written by
Cormen, Leiserson, Rivest, and Stein. Section 1.1 The Role of Algorithm in Computer
introduces what is algorithm, and what is the role of algorithm in computer by using some
rhetorical appeal and rhetorical strategies. This article introduces the basic concept of
algorithm in computer and use some examples of the application to show how the algorithms
apply in different case.
The purpose of Cormen, Leiserson, Rivest, and Stein’s article is to give common a basic
idea and concept of algorithm. Because it is a professional article, authors use Logos and
Ethos which are logic and authority in this section rather than pathos. The main rhetorical
strategies that authors use are definition, exemplification, and compare and contrast. Some
definitions and examples make the article more creditability and logically. I will examine
more detail of appeals and strategies in my following article.
Zhao 2
Analysis of Rhetorical Appeals
Logos
Logos, which means logic, is the main rhetorical appeal using in this article because it helps
readers use logic to understand each step or element in the whole concept. Cormen,
Leiserson, Rivest, and Stein use some facts to show how algorithm solve the problem. For
example, Human Genome Project identified all the 100,000 genes in human DNA,
determined the sequences, sorted the information, and analyzed the data. In order to achieve,
they must use sophisticated algorithms. That helps readers could clearly know how huge the
project is, how complicated it is, and how important the algorithm is. Furthermore, authors
use other applications to show the important of algorithm. Oil companies want to know how
to get maximum profit, internet service providers want to serve customers more effectively,
and political candidates want to buy the most useful advertisement to get the chance to win
the election. Authors show that all of the problem that in the above can be solve by linear
programming which is also included in algorithm. These facts helps author persuade readers
algorithm play big role in computing.
Ethos
Ethos, which means reliability and creditability is another rhetorical appeal using in the
article. Cormen, Leiserson, Rivest, and Stein ever mentioned algorithm should correct in
order to get a correct answer. However, they also stated the incorrect condition. Even though
common think incorrect answer will produce by the wrong algorithm, they still wrote down
the special case, which is answer could be right when people can control their error rate.
These statements made them reliability because they are not arbitrary to say algorithm need
always be right, and it remain some special cases sometimes.
Authors also spend lots of time on writing the appendix from A to D, which include example,
graph, statistic, probability, and data, to further explain the concepts mentioned in the book.
Furthermore, authors are all professional like “Thomas H. Cormen is a professor of Computer
Science and former Director of the Institute for Writing and Rhetoric at Dartmouth College,
Charles E. Leiserson is Professor of Computer Science and Engineering at the Massachusetts
Institute of Technology, Ronald L. Rivest is Andrew and Erna Viterbi Professor of Electrical
Engineering and Computer Science at the Massachusetts Institute of Technology, and Clifford
Stein is Professor of Industrial Engineering and Operations Research at Columbia
University” (Web MIT 2017) . For another example, they use a large amount of reference,
which is wrote down in the last of the book, to help strong their creditability. Therefore, ethos
are another main rhetoric appeal in this article.
Zhao 3
Analysis of Rhetorical Strategies
Definition
Definition is a common rhetoric strategy in the academic essay to help writers define an idea
or explain an academic term. In the article “Section 1.1 The Role of Algorithm in Computer”,
authors introduced the definition of the algorithm in the beginning: “Informally, an algorithm
is any well-defined computational procedure that takes some value, or set of values, as input
and produces some value, or set of values, as output” (Introduction to Algorithms 5). Authors
used definition to explain what is algorithm, which takes the input and use some special
procedures to get the output. Algorithm is a key word for this article. Therefore, using
definition to introduce the concept of algorithm helps authors illustrated more details about
their thesis and also helps readers establish some senses about the algorithm.
Exemplification
Exemplification is another effective rhetoric strategy. In this article, authors use
exemplification, which means using examples to help describe or analysis the thesis, to show
how algorithm works. For example, authors gave the readers an array which is input
sequence, {31, 41, 59, 26, 41, 58}. After a sorting algorithm worked, system will give an
output sequence like {26, 31, 41, 41, 58, 59}. From the example, readers could easily find the
input is unordered array which includes some numbers. Sorting algorism make sequence
change from unordered to ordered. Authors used numbers and professional phrases to make
readers know what sorting algorithm is. For another example, authors gave us a condition that
a mechanical design the library with different parts. However, some of the parts may include
other parts. Therefore, it needs to list an order in order to make each part can appear before
other parts use it. If the mechanical design includes n parts, so, there exists n! possible orders.
From function we could know n! is a factorial function which grows faster than other
function. Therefore, in order to calculate which one is the possible way to solve the problem
need a complicated algorithm which called instance of topological sorting. Even though the
example include some of the professional concept, reader can also easily understand the
importance of algorithm. Similarly, these kind of questions need computer to compute. That
also shows how important the algorithm is in the computing. Overall, exemplification is an
effective way to help author to present their idea in this article.
Zhao 4
Comparison and contrast
Comparison and Contrast is a rhetorical strategy to help writers develop and analysis the
article’s ideas. “Comparison examine similarities; contrasts examine differences” (23 Alkire).
Authors use compare and contrast to help analysis some complicated definition or vividly
describe some ideas. In this article, authors use lots of example to show some of the problems
like the transport company want to find the shortest path to decrease the cost, Internet routing
node want find the shortest way to pass message in order to save time, and someone want to
find the nearest way as early as possible to arrive at the destination, have some candidate
solutions and find “best” in the solutions when using the algorithm. Furthermore, authors use
contrast to give another example to show not each problem has sat of candidate solution
when using the algorithm. “For example, suppose we are given a set of numerical values
representing samples of a signal, and we want to compute the discrete Fourier transform of
these samples. The discrete Fourier transform converts the time domain to the frequency
domain, producing a set of numerical coefficients, so that we can determine the strength of
various frequencies in the sampled signal.” (Introduction to Algorithms 9). From the
example, author use another example which want to compute the discrete Fourier transform
which can’t have a set of solution but still can be solved by another algorithm which called
the ‘fast Fourier transform’. From the contrast, authors show that algorithm not only can
solve ‘find best question’, but also can solve ‘discrete question’. That also shows there exists
lots of different kinds of questions can be solved by algorithm. This rhetorical strategy helps
authors enhance their thesis that algorithm is really important in computing.
Conclusion
In the article, authors use logos to introduce the concept of algorithm step by step and use
some facts to emphasis the role of algorithm. They also use ethos to supplement the definition
method of application of algorithm to show their profession and creditability. Definition of
rhetorical strategy helps readers understand the concept of algorithm and how it related to
algorithm in computing. Exemplification bring lots of example to help writers show how
algorithm works in actual life and how important of the role in computing. Finally, writers
use contrast to extend the application of the algorithm and reinforce the ability of algorithm
in computing in order to show their thesis.
Zhao 5
https://mitpress.mit.edu/books/introduction-algorithms
T H O M A S H. C O R M E N
C H A R L E S E. L E I S E R S O N
R O N A L D L. R I V E S T
C L I F F O R D STEIN
INTRODUCTION TO
ALGORITHMS
T H I R D
E D I T I O N
Introduction to Algorithms
Third Edition
Thomas H. Cormen
Charles E. Leiserson
Ronald L. Rivest
Clifford Stein
Introduction to Algorithms
Third Edition
The MIT Press
Cambridge, Massachusetts
London, England
c 2009 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means
(including photocopying, recording, or information storage and retrieval) without permission in writing from the
publisher.
For information about special quantity discounts, please email special sales@mitpress.mit.edu.
This book was set in Times Roman and Mathtime Pro 2 by the authors.
Printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Introduction to algorithms / Thomas H. Cormen . . . [et al.].—3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-262-03384-8 (hardcover : alk. paper)—ISBN 978-0-262-53305-8 (pbk. : alk. paper)
1. Computer programming. 2. Computer algorithms. I. Cormen, Thomas H.
QA76.6.I5858 2009
005.1—dc22
2009008593
10 9 8 7 6 5 4 3 2
Contents
Preface
xiii
I Foundations
Introduction
3
1
The Role of Algorithms in Computing 5
1.1 Algorithms 5
1.2 Algorithms as a technology 11
2
Getting Started 16
2.1 Insertion sort 16
2.2 Analyzing algorithms 23
2.3 Designing algorithms 29
3
Growth of Functions 43
3.1 Asymptotic notation 43
3.2 Standard notations and common functions
4
?
5
?
53
Divide-and-Conquer 65
4.1 The maximum-subarray problem 68
4.2 Strassen’s algorithm for matrix multiplication 75
4.3 The substitution method for solving recurrences 83
4.4 The recursion-tree method for solving recurrences 88
4.5 The master method for solving recurrences 93
4.6 Proof of the master theorem 97
Probabilistic Analysis and Randomized Algorithms 114
5.1 The hiring problem 114
5.2 Indicator random variables 118
5.3 Randomized algorithms 122
5.4 Probabilistic analysis and further uses of indicator random variables
130
vi
Contents
II Sorting and Order Statistics
Introduction
6
7
8
9
147
Heapsort 151
6.1 Heaps 151
6.2 Maintaining the heap property
6.3 Building a heap 156
6.4 The heapsort algorithm 159
6.5 Priority queues 162
154
Quicksort 170
7.1 Description of quicksort 170
7.2 Performance of quicksort 174
7.3 A randomized version of quicksort
7.4 Analysis of quicksort 180
Sorting in Linear Time 191
8.1 Lower bounds for sorting
8.2 Counting sort 194
8.3 Radix sort 197
8.4 Bucket sort 200
179
191
Medians and Order Statistics 213
9.1 Minimum and maximum 214
9.2 Selection in expected linear time 215
9.3 Selection in worst-case linear time 220
III Data Structures
Introduction
10
11
?
229
Elementary Data Structures 232
10.1 Stacks and queues 232
10.2 Linked lists 236
10.3 Implementing pointers and objects
10.4 Representing rooted trees 246
Hash Tables 253
11.1 Direct-address tables 254
11.2 Hash tables 256
11.3 Hash functions 262
11.4 Open addressing 269
11.5 Perfect hashing 277
241
Contents
12
?
13
14
vii
Binary Search Trees 286
12.1 What is a binary search tree? 286
12.2 Querying a binary search tree 289
12.3 Insertion and deletion 294
12.4 Randomly built binary search trees 299
Red-Black Trees 308
13.1 Properties of red-black trees
13.2 Rotations 312
13.3 Insertion 315
13.4 Deletion 323
308
Augmenting Data Structures 339
14.1 Dynamic order statistics 339
14.2 How to augment a data structure
14.3 Interval trees 348
345
IV Advanced Design and Analysis Techniques
Introduction
357
15
Dynamic Programming 359
15.1 Rod cutting 360
15.2 Matrix-chain multiplication 370
15.3 Elements of dynamic programming 378
15.4 Longest common subsequence 390
15.5 Optimal binary search trees 397
16
Greedy Algorithms 414
16.1 An activity-selection problem 415
16.2 Elements of the greedy strategy 423
16.3 Huffman codes 428
16.4 Matroids and greedy methods 437
16.5 A task-scheduling problem as a matroid
?
?
17
Amortized Analysis 451
17.1 Aggregate analysis 452
17.2 The accounting method 456
17.3 The potential method 459
17.4 Dynamic tables 463
443
viii
Contents
V Advanced Data Structures
Introduction
18
B-Trees 484
18.1 Definition of B-trees 488
18.2 Basic operations on B-trees 491
18.3 Deleting a key from a B-tree 499
19
Fibonacci Heaps 505
19.1 Structure of Fibonacci heaps 507
19.2 Mergeable-heap operations 510
19.3 Decreasing a key and deleting a node 518
19.4 Bounding the maximum degree 523
20
van Emde Boas Trees 531
20.1 Preliminary approaches 532
20.2 A recursive structure 536
20.3 The van Emde Boas tree 545
21
Data Structures for Disjoint Sets 561
21.1 Disjoint-set operations 561
21.2 Linked-list representation of disjoint sets 564
21.3 Disjoint-set forests 568
21.4 Analysis of union by rank with path compression
?
VI
481
Graph Algorithms
Introduction
587
22
Elementary Graph Algorithms 589
22.1 Representations of graphs 589
22.2 Breadth-first search 594
22.3 Depth-first search 603
22.4 Topological sort 612
22.5 Strongly connected components 615
23
Minimum Spanning Trees 624
23.1 Growing a minimum spanning tree 625
23.2 The algorithms of Kruskal and Prim 631
573
Contents
24
ix
Single-Source Shortest Paths 643
24.1 The Bellman-Ford algorithm 651
24.2 Single-source shortest paths in directed acyclic graphs
24.3 Dijkstra’s algorithm 658
24.4 Difference constraints and shortest paths 664
24.5 Proofs of shortest-paths properties 671
25
All-Pairs Shortest Paths 684
25.1 Shortest paths and matrix multiplication 686
25.2 The Floyd-Warshall algorithm 693
25.3 Johnson’s algorithm for sparse graphs 700
26
Maximum Flow 708
26.1 Flow networks 709
26.2 The Ford-Fulkerson method 714
26.3 Maximum bipartite matching 732
26.4 Push-relabel algorithms 736
26.5 The relabel-to-front algorithm 748
?
?
655
VII Selected Topics
Introduction
769
27
Multithreaded Algorithms 772
27.1 The basics of dynamic multithreading 774
27.2 Multithreaded matrix multiplication 792
27.3 Multithreaded merge sort 797
28
Matrix Operations 813
28.1 Solving systems of linear equations 813
28.2 Inverting matrices 827
28.3 Symmetric positive-definite matrices and least-squares approximation
832
29
Linear Programming 843
29.1 Standard and slack forms 850
29.2 Formulating problems as linear programs
29.3 The simplex algorithm 864
29.4 Duality 879
29.5 The initial basic feasible solution 886
859
x
Contents
30
Polynomials and the FFT 898
30.1 Representing polynomials 900
30.2 The DFT and FFT 906
30.3 Efficient FFT implementations 915
31
Number-Theoretic Algorithms 926
31.1 Elementary number-theoretic notions 927
31.2 Greatest common divisor 933
31.3 Modular arithmetic 939
31.4 Solving modular linear equations 946
31.5 The Chinese remainder theorem 950
31.6 Powers of an element 954
31.7 The RSA public-key cryptosystem 958
31.8 Primality testing 965
31.9 Integer factorization 975
?
?
32
?
33
String Matching 985
32.1 The naive string-matching algorithm 988
32.2 The Rabin-Karp algorithm 990
32.3 String matching with finite automata 995
32.4 The Knuth-Morris-Pratt algorithm 1002
Computational Geometry 1014
33.1 Line-segment properties 1015
33.2 Determining whether any pair of segments intersects
33.3 Finding the convex hull 1029
33.4 Finding the closest pair of points 1039
34
NP-Completeness 1048
34.1 Polynomial time 1053
34.2 Polynomial-time verification 1061
34.3 NP-completeness and reducibility 1067
34.4 NP-completeness proofs 1078
34.5 NP-complete problems 1086
35
Approximation Algorithms 1106
35.1 The vertex-cover problem 1108
35.2 The traveling-salesman problem 1111
35.3 The set-covering problem 1117
35.4 Randomization and linear programming
35.5 The subset-sum problem 1128
1123
1021
Contents
xi
VIII Appendix: Mathematical Background
Introduction
A
1143
Summations 1145
A.1 Summation formulas and properties
A.2 Bounding summations 1149
1145
B
Sets, Etc. 1158
B.1 Sets 1158
B.2 Relations 1163
B.3 Functions 1166
B.4 Graphs 1168
B.5 Trees 1173
C
Counting and Probability 1183
C.1 Counting 1183
C.2 Probability 1189
C.3 Discrete random variables 1196
C.4 The geometric and binomial distributions 1201
C.5 The tails of the binomial distribution 1208
?
D
Matrices 1217
D.1 Matrices and matrix operations
D.2 Basic matrix properties 1222
Bibliography
Index
1251
1231
1217
Preface
Before there were computers, there were algorithms. But now that there are computers, there are even more algorithms, and algorithms lie at the heart of computing.
This book provides a comprehensive introduction to the modern study of computer algorithms. It presents many algorithms and covers them in considerable
depth, yet makes their design and analysis accessible to all levels of readers. We
have tried to keep explanations elementary without sacrificing depth of coverage
or mathematical rigor.
Each chapter presents an algorithm, a design technique, an application area, or a
related topic. Algorithms are described in English and in a pseudocode designed to
be readable by anyone who has done a little programming. The book contains 244
figures—many with multiple parts—illustrating how the algorithms work. Since
we emphasize efficiency as a design criterion, we include careful analyses of the
running times of all our algorithms.
The text is intended primarily for use in undergraduate or graduate courses in
algorithms or data structures. Because it discusses engineering issues in algorithm
design, as well as mathematical aspects, it is equally well suited for self-study by
technical professionals.
In this, the third edition, we have once again updated the entire book. The
changes cover a broad spectrum, including new chapters, revised pseudocode, and
a more active writing style.
To the teacher
We have designed this book to be both versatile and complete. You should find it
useful for a variety of courses, from an undergraduate course in data structures up
through a graduate course in algorithms. Because we have provided considerably
more material than can fit in a typical one-term course, you can consider this book
to be a “buffet” or “smorgasbord” from which you can pick and choose the material
that best supports the course you wish to teach.
xiv
Preface
You should find it easy to organize your course around just the chapters you
need. We have made chapters relatively self-contained, so that you need not worry
about an unexpected and unnecessary dependence of one chapter on another. Each
chapter presents the easier material first and the more difficult material later, with
section boundaries marking natural stopping points. In an undergraduate course,
you might use only the earlier sections from a chapter; in a graduate course, you
might cover the entire chapter.
We have included 957 exercises and 158 problems. Each section ends with exercises, and each chapter ends with problems. The exercises are generally short questions that test basic mastery of the material. Some are simple self-check thought
exercises, whereas others are more substantial and are suitable as assigned homework. The problems are more elaborate case studies that often introduce new material; they often consist of several questions that lead the student through the steps
required to arrive at a solution.
Departing from our practice in previous editions of this book, we have made
publicly available solutions to some, but by no means all, of the problems and exercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to these solutions.
You will want to check this site to make sure that it does not contain the solution to
an exercise or problem that you plan to assign. We expect the set of solutions that
we post to grow slowly over time, so you will need to check it each time you teach
the course.
We have starred (?) the sections and exercises that are more suitable for graduate
students than for undergraduates. A starred section is not necessarily more difficult than an unstarred one, but it may require an understanding of more advanced
mathematics. Likewise, starred exercises may require an advanced background or
more than average creativity.
To the student
We hope that this textbook provides you with an enjoyable introduction to the
field of algorithms. We have attempted to make every algorithm accessible and
interesting. To help you when you encounter unfamiliar or difficult algorithms, we
describe each one in a step-by-step manner. We also provide careful explanations
of the mathematics needed to understand the analysis of the algorithms. If you
already have some familiarity with a topic, you will find the chapters organized so
that you can skim introductory sections and proceed quickly to the more advanced
material.
This is a large book, and your class will probably cover only a portion of its
material. We have tried, however, to make this a book that will be useful to you
now as a course textbook and also later in your career as a mathematical desk
reference or an engineering handbook.
Preface
xv
What are the prerequisites for reading this book?
You should have some programming experience. In particular, you should understand recursive procedures and simple data structures such as arrays and
linked lists.
You should have some facility with mathematical proofs, and especially proofs
by mathematical induction. A few portions of the book rely on some knowledge
of elementary calculus. Beyond that, Parts I and VIII of this book teach you all
the mathematical techniques you will need.
We have heard, loud and clear, the call to supply solutions to problems and
exercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to solutions for
a few of the problems and exercises. Feel free to check your solutions against ours.
We ask, however, that you do not send your solutions to us.
To the professional
The wide range of topics in this book makes it an excellent handbook on algorithms. Because each chapter is relatively self-contained, you can focus in on the
topics that most interest you.
Most of the algorithms we discuss have great practical utility. We therefore
address implementation concerns and other engineering issues. We often provide
practical alternatives to the few algorithms that are primarily of theoretical interest.
If you wish to implement any of the algorithms, you should find the translation of our pseudocode into your favorite programming language to be a fairly
straightforward task. We have designed the pseudocode to present each algorithm
clearly and succinctly. Consequently, we do not address error-handling and other
software-engineering issues that require specific assumptions about your programming environment. We attempt to present each algorithm simply and directly without allowing the idiosyncrasies of a particular programming language to obscure
its essence.
We understand that if you are using this book outside of a course, then you
might be unable to check your solutions to problems and exercises against solutions
provided by an instructor. Our Web site, http://mitpress.mit.edu/algorithms/, links
to solutions for some of the problems and exercises so that you can check your
work. Please do not send your solutions to us.
To our colleagues
We have supplied an extensive bibliography and pointers to the current literature.
Each chapter ends with a set of chapter notes that give historical details and references. The chapter notes do not provide a complete reference to the whole field
xvi
Preface
of algorithms, however. Though it may be hard to believe for a book of this size,
space constraints prevented us from including many interesting algorithms.
Despite myriad requests from students for solutions to problems and exercises,
we have chosen as a matter of policy not to supply references for problems and
exercises, to remove the temptation for students to look up a solution rather than to
find it themselves.
Changes for the third edition
What has changed between the second and third editions of this book? The magnitude of the changes is on a par with the changes between the first and second
editions. As we said about the second-edition changes, depending on how you
look at it, the book changed either not much or quite a bit.
A quick look at the table of contents shows that most of the second-edition chapters and sections appear in the third edition. We removed two chapters and one
section, but we have added three new chapters and two new sections apart from
these new chapters.
We kept the hybrid organization from the first two editions. Rather than organizing chapters by only problem domains or according only to techniques, this book
has elements of both. It contains technique-based chapters on divide-and-conquer,
dynamic programming, greedy algorithms, amortized analysis, NP-Completeness,
and approximation algorithms. But it also has entire parts on sorting, on data
structures for dynamic sets, and on algorithms for graph problems. We find that
although you need to know how to apply techniques for designing and analyzing algorithms, problems seldom announce to you which techniques are most amenable
to solving them.
Here is a summary of the most significant changes for the third edition:
We added new chapters on van Emde Boas trees and multithreaded algorithms,
and we have broken out material on matrix basics into its own appendix chapter.
We revised the chapter on recurrences to more broadly cover the divide-andconquer technique, and its first two sections apply divide-and-conquer to solve
two problems. The second section of this chapter presents Strassen’s algorithm
for matrix multiplication, which we have moved from the chapter on matrix
operations.
We removed two chapters that were rarely taught: binomial heaps and sorting
networks. One key idea in the sorting networks chapter, the 0-1 principle, appears in this edition within Problem 8-7 as the 0-1 sorting lemma for compareexchange algorithms. The treatment of Fibonacci heaps no longer relies on
binomial heaps as a precursor.
Preface
xvii
We revised our treatment of dynamic programming and greedy algorithms. Dynamic programming now leads off with a more interesting problem, rod cutting,
than the assembly-line scheduling problem from the second edition. Furthermore, we emphasize memoization a bit more than we did in the second edition,
and we introduce the notion of the subproblem graph as a way to understand
the running time of a dynamic-programming algorithm. In our opening example of greedy algorithms, the activity-selection problem, we get to the greedy
algorithm more directly than we did in the second edition.
The way we delete a node from binary search trees (which includes red-black
trees) now guarantees that the node requested for deletion is the node that is
actually deleted. In the first two editions, in certain cases, some other node
would be deleted, with its contents moving into the node passed to the deletion
procedure. With our new way to delete nodes, if other components of a program
maintain pointers to nodes in the tree, they will not mistakenly end up with stale
pointers to nodes that have been deleted.
The material on flow networks now bases flows entirely on edges. This approach is more intuitive than the net flow used in the first two editions.
With the material on matrix basics and Strassen’s algorithm moved to other
chapters, the chapter on matrix operations is smaller than in the second edition.
We have modified our treatment of the Knuth-Morris-Pratt string-matching algorithm.
We corrected several errors. Most of these errors were posted on our Web site
of second-edition errata, but a few were not.
Based on many requests, we changed the syntax (as it were) of our pseudocode.
We now use “D” to indicate assignment and “==” to test for equality, just as C,
C++, Java, and Python do. Likewise, we have eliminated the keywords do and
then and adopted “//” as our comment-to-end-of-line symbol. We also now use
dot-notation to indicate object attributes. Our pseudocode remains procedural,
rather than object-oriented. In other words, rather than running methods on
objects, we simply call procedures, passing objects as parameters.
We added 100 new exercises and 28 new problems. We also updated many
bibliography entries and added several new ones.
Finally, we went through the entire book and rewrote sentences, paragraphs,
and sections to make the writing clearer and more active.
xviii
Preface
Web site
You can use our Web site, http://mitpress.mit.edu/algorithms/, to obtain supplementary information and to communicate with us. The Web site links to a list of
known errors, solutions to selected exercises and problems, and (of course) a list
explaining the corny professor jokes, as well as other content that we might add.
The Web site also tells you how to report errors or make suggestions.
How we produced this book
Like the second edition, the third edition was produced in LATEX 2" . We used the
Times font with mathematics typeset using the MathTime Pro 2 fonts. We thank
Michael Spivak from Publish or Perish, Inc., Lance Carnes from Personal TeX,
Inc., and Tim Tregubov from Dartmouth College for technical support. As in the
previous two editions, we compiled the index using Windex, a C program that we
wrote, and the bibliography was produced with B IBTEX. The PDF files for this
book were created on a MacBook running OS 10.5.
We drew the illustrations for the third edition using MacDraw Pro, with some
of the mathematical expressions in illustrations laid in with the psfrag package
for LATEX 2" . Unfortunately, MacDraw Pro is legacy software, having not been
marketed for over a decade now. Happily, we still have a couple of Macintoshes
that can run the Classic environment under OS 10.4, and hence they can run MacDraw Pro—mostly. Even under the Classic environment, we find MacDraw Pro to
be far easier to use than any other drawing software for the types of illustrations
that accompany computer-science text, and it produces beautiful output.1 Who
knows how long our pre-Intel Macs will continue to run, so if anyone from Apple
is listening: Please create an OS X-compatible version of MacDraw Pro!
Acknowledgments for the third edition
We have been working with the MIT Press for over two decades now, and what a
terrific relationship it has been! We thank Ellen Faran, Bob Prior, Ada Brunstein,
and Mary Reilly for their help and support.
We were geographically distributed while producing the third edition, working
in the Dartmouth College Department of Computer Science, the MIT Computer
1 We investigated several drawing programs that run under Mac OS X, but all had significant shortcomings compared with MacDraw Pro. We briefly attempted to produce the illustrations for this
book with a different, well known drawing program. We found that it took at least five times as long
to produce each illustration as it took with MacDraw Pro, and the resulting illustrations did not look
as good. Hence the decision to revert to MacDraw Pro running on older Macintoshes.
Preface
xix
Science and Artificial Intelligence Laboratory, and the Columbia University Department of Industrial Engineering and Operations Research. We thank our respective universities and colleagues for providing such supportive and stimulating
environments.
Julie Sussman, P.P.A., once again bailed us out as the technical copyeditor. Time
and again, we were amazed at the errors that eluded us, but that Julie caught. She
also helped us improve our presentation in several places. If there is a Hall of Fame
for technical copyeditors, Julie is a sure-fire, first-ballot inductee. She is nothing
short of phenomenal. Thank you, thank you, thank you, Julie! Priya Natarajan also
found some errors that we were able to correct before this book went to press. Any
errors that remain (and undoubtedly, some do) are the responsibility of the authors
(and probably were inserted after Julie read the material).
The treatment for van Emde Boas trees derives from Erik Demaine’s notes,
which were in turn influenced by Michael Bender. We also incorporated ideas
from Javed Aslam, Bradley Kuszmaul, and Hui Zha into this edition.
The chapter on multithreading was based on notes originally written jointly with
Harald Prokop. The material was influenced by several others working on the Cilk
project at MIT, including Bradley Kuszmaul and Matteo Frigo. The design of the
multithreaded pseudocode took its inspiration from the MIT Cilk extensions to C
and by Cilk Arts’s Cilk++ extensions to C++.
We also thank the many readers of the first and second editions who reported
errors or submitted suggestions for how to improve this book. We corrected all the
bona fide errors that were reported, and we incorporated as many suggestions as
we could. We rejoice that the number of such contributors has grown so great that
we must regret that it has become impractical to list them all.
Finally, we thank our wives—Nicole Cormen, Wendy Leiserson, Gail Rivest,
and Rebecca Ivry—and our children—Ricky, Will, Debby, and Katie Leiserson;
Alex and Christopher Rivest; and Molly, Noah, and Benjamin Stein—for their love
and support while we prepared this book. The patience and encouragement of our
families made this project possible. We affectionately dedicate this book to them.
T HOMAS H. C ORMEN
C HARLES E. L EISERSON
RONALD L. R IVEST
C LIFFORD S TEIN
February 2009
Lebanon, New Hampshire
Cambridge, Massachusetts
Cambridge, Massachusetts
New York, New York
Introduction to Algorithms
Third Edition
I
Foundations
Introduction
This part will start you thinking about designing and analyzing algorithms. It is
intended to be a gentle introduction to how we specify algorithms, some of the
design strategies we will use throughout this book, and many of the fundamental
ideas used in algorithm analysis. Later parts of this book will build upon this base.
Chapter 1 provides an overview of algorithms and their place in modern computing systems. This chapter defines what an algorithm is and lists some examples.
It also makes a case that we should consider algorithms as a technology, alongside technologies such as fast hardware, graphical user interfaces, object-oriented
systems, and networks.
In Chapter 2, we see our first algorithms, which solve the problem of sorting
a sequence of n numbers. They are written in a pseudocode which, although not
directly translatable to any conventional programming language, conveys the structure of the algorithm clearly enough that you should be able to implement it in the
language of your choice. The sorting algorithms we examine are insertion sort,
which uses an incremental approach, and merge sort, which uses a recursive technique known as “divide-and-conquer.” Although the time each requires increases
with the value of n, the rate of increase differs between the two algorithms. We
determine these running times in Chapter 2, and we develop a useful notation to
express them.
Chapter 3 precisely defines this notation, which we call asymptotic notation. It
starts by defining several asymptotic notations, which we use for bounding algorithm running times from above and/or below. The rest of Chapter 3 is primarily
a presentation of mathematical notation, more to ensure that your use of notation
matches that in this book than to teach you new mathematical concepts.
4
Part I Foundations
Chapter 4 delves further into the divide-and-conquer method introduced in
Chapter 2. It provides additional examples of divide-and-conquer algorithms, including Strassen’s surprising method for multiplying two square matrices. Chapter 4 contains methods for solving recurrences, which are useful for describing
the running times of recursive algorithms. One powerful technique is the “master method,” which we often use to solve recurrences that arise from divide-andconquer algorithms. Although much of Chapter 4 is devoted to proving the correctness of the master method, you may skip this proof yet still employ the master
method.
Chapter 5 introduces probabilistic analysis and randomized algorithms. We typically use probabilistic analysis to determine the running time of an algorithm in
cases in which, due to the presence of an inherent probability distribution, the
running time may differ on different inputs of the same size. In some cases, we
assume that the inputs conform to a known probability distribution, so that we are
averaging the running time over all possible inputs. In other cases, the probability
distribution comes not from the inputs but from random choices made during the
course of the algorithm. An algorithm whose behavior is determined not only by its
input but by the values produced by a random-number generator is a randomized
algorithm. We can use randomized algorithms to enforce a probability distribution
on the inputs—thereby ensuring that no particular input always causes poor performance—or even to bound the error rate of algorithms that are allowed to produce
incorrect results on a limited basis.
Appendices A–D contain other mathematical material that you will find helpful
as you read this book. You are likely to have seen much of the material in the
appendix chapters before having read this book (although the specific definitions
and notational conventions we use may differ in some cases from what you have
seen in the past), and so you should think of the Appendices as reference material.
On the other hand, you probably have not already seen most of the material in
Part I. All the chapters in Part I and the Appendices are written with a tutorial
flavor.
1
The Role of Algorithms in Computing
What are algorithms? Why is the study of algorithms worthwhile? What is the role
of algorithms relative to other technologies used in computers? In this chapter, we
will answer these questions.
1.1 Algorithms
Informally, an algorithm is any well-defined computational procedure that takes
some value, or set of values, as input and produces some value, or set of values, as
output. An algorithm is thus a sequence of computational steps that transform the
input into the output.
We can also view an algorithm as a tool for solving a well-specified computational problem. The statement of the problem specifies in general terms the desired
input/output relationship. The algorithm describes a specific computational procedure for achieving that input/output relationship.
For example, we might need to sort a sequence of numbers into nondecreasing
order. This problem arises frequently in practice and provides fertile ground for
introducing many standard design techniques and analysis tools. Here is how we
formally define the sorting problem:
Input: A sequence of n numbers ha1 ; a2 ; : : : ; an i.
Output: A permutation (reordering) ha10 ; a20 ; : : : ; an0 i of the input sequence such
that a10 a20 an0 .
For example, given the input sequence h31; 41; 59; 26; 41; 58i, a sorting algorithm
returns as output the sequence h26; 31; 41; 41; 58; 59i. Such an input sequence is
called an instance of the sorting problem. In general, an instance of a problem
consists of the input (satisfying whatever constraints are imposed in the problem
statement) needed to compute a solution to the problem.
6
Chapter 1 The Role of Algorithms in Computing
Because many programs use it as an intermediate step, sorting is a fundamental
operation in computer science. As a result, we have a large number of good sorting
algorithms at our disposal. Which algorithm is best for a given application depends
on—among other factors—the number of items to be sorted, the extent to which
the items are already somewhat sorted, possible restrictions on the item values,
the architecture of the computer, and the kind of storage devices to be used: main
memory, disks, or even tapes.
An algorithm is said to be correct if, for every input instance, it halts with the
correct output. We say that a correct algorithm solves the given computational
problem. An incorrect algorithm might not halt at all on some input instances, or it
might halt with an incorrect answer. Contrary to what you might expect, incorrect
algorithms can sometimes be useful, if we can control their error rate. We shall see
an example of an algorithm with a controllable error rate in Chapter 31 when we
study algorithms for finding large prime numbers. Ordinarily, however, we shall
be concerned only with correct algorithms.
An algorithm can be specified in English, as a computer program, or even as
a hardware design. The only requirement is that the specification must provide a
precise description of the computational procedure to be followed.
What kinds of problems are solved by algorithms?
Sorting is by no means the only computational problem for which algorithms have
been developed. (You probably suspected as much when you saw the size of this
book.) Practical applications of algorithms are ubiquitous and include the following examples:
The Human Genome Project has made great progress toward the goals of identifying all the 100,000 genes in human DNA, determining the sequences of the
3 billion chemical base pairs that make up human DNA, storing this information in databases, and developing tools for data analysis. Each of these steps
requires sophisticated algorithms. Although the solutions to the various problems involved are beyond the scope of this book, many methods to solve these
biological problems use ideas from several of the chapters in this book, thereby
enabling scientists to accomplish tasks while using resources efficiently. The
savings are in time, both human and machine, and in money, as more information can be extracted from laboratory techniques.
The Internet enables people all around the world to quickly access and retrieve
large amounts of information. With the aid of clever algorithms, sites on the
Internet are able to manage and manipulate this large volume of data. Examples
of problems that make essential use of algorithms include finding good routes
on which the data will travel (techniques for solving such problems appear in
1.1 Algorithms
7
Chapter 24), and using a search engine to quickly find pages on which particular
information resides (related techniques are in Chapters 11 and 32).
Electronic commerce enables goods and services to be negotiated and exchanged electronically, and it depends on the privacy of personal information such as credit card numbers, passwords, and bank statements. The core
technologies used in electronic commerce include public-key cryptography and
digital signatures (covered in Chapter 31), which are based on numerical algorithms and number theory.
Manufacturing and other commercial enterprises often need to allocate scarce
resources in the most beneficial way. An oil company may wish to know where
to place its wells in order to maximize its expected profit. A political candidate
may want to determine where to spend money buying campaign advertising in
order to maximize the chances of winning an election. An airline may wish
to assign crews to flights in the least expensive way possible, making sure that
each flight is covered and that government regulations regarding crew scheduling are met. An Internet service provider may wish to determine where to place
additional resources in order to serve its customers more effectively. All of
these are examples of problems that can be solved using linear programming,
which we shall study in Chapter 29.
Although some of the details of these examples are beyond the scope of this
book, we do give underlying techniques that apply to these problems and problem
areas. We also show how to solve many specific problems, including the following:
We are given a road map on which the distance between each pair of adjacent
intersections is marked, and we wish to determine the shortest route from one
intersection to another. The number of possible routes can be huge, even if we
disallow routes that cross over themselves. How do we choose which of all
possible routes is the shortest? Here, we model the road map (which is itself
a model of the actual roads) as a graph (which we will meet in Part VI and
Appendix B), and we wish to find the shortest path from one vertex to another
in the graph. We shall see how to solve this problem efficiently in Chapter 24.
We are given two ordered sequences of symbols, X D hx1 ; x2 ; : : : ; xm i and
Y D hy1 ; y2 ; : : : ; yn i, and we wish to find a longest common subsequence of
X and Y . A subsequence of X is just X with some (or possibly all or none) of
its elements removed. For example, one subsequence of hA; B; C; D; E; F; Gi
would be hB; C; E; Gi. The length of a longest common subsequence of X
and Y gives one measure of how similar these two sequences are. For example,
if the two sequences are base pairs in DNA strands, then we might consider
them similar if they have a long common subsequence. If X has m symbols
and Y has n symbols, then X and Y have 2m and 2n possible subsequences,
8
Chapter 1 The Role of Algorithms in Computing
respectively. Selecting all possible subsequences of X and Y and matching
them up could take a prohibitively long time unless m and n are very small.
We shall see in Chapter 15 how to use a general technique known as dynamic
programming to solve this problem much more efficiently.
We are given a mechanical design in terms of a library of parts, where each part
may include instances of other parts, and we need to list the parts in order so
that each part appears before any part that uses it. If the design comprises n
parts, then there are nŠ possible orders, where nŠ denotes the factorial function.
Because the factorial function grows faster than even an exponential function,
we cannot feasibly generate each possible order and then verify that, within
that order, each part appears before the parts using it (unless we have only a
few parts). This problem is an instance of topological sorting, and we shall see
in Chapter 22 how to solve this problem efficiently.
We are given n points in the plane, and we wish to find the convex hull of
these points. The convex hull is the smallest convex polygon containing the
points. Intuitively, we can think of each point as being represented by a nail
sticking out from a board. The convex hull would be represented by a tight
rubber band that surrounds all the nails. Each nail around which the rubber
band makes a turn is a vertex of the convex hull. (See Figure 33.6 on page 1029
for an example.) Any of the 2n subsets of the points might be the vertices
of the convex hull. Knowing which points are vertices of the convex hull is
not quite enough, either, since we also need to know the order in which they
appear. There are many choices, therefore, for the vertices of the convex hull.
Chapter 33 gives two good methods for finding the convex hull.
These lists are far from exhaustive (as you again have probably surmised from
this book’s heft), but exhibit two characteristics that are common to many interesting algorithmic problems:
1. They have many candidate solutions, the overwhelming majority of which do
not solve the problem at hand. Finding one that does, or one that is “best,” can
present quite a challenge.
2. They have practical applications. Of the problems in the above list, finding the
shortest path provides the easiest examples. A transportation firm, such as a
trucking or railroad company, has a financial interest in finding shortest paths
through a road or rail network because taking shorter paths results in lower
labor and fuel costs. Or a routing node on the Internet may need to find the
shortest path through the network in order to route a message quickly. Or a
person wishing to drive from New York to Boston may want to find driving
directions from an appropriate Web site, or she may use her GPS while driving.
1.1 Algorithms
9
Not every problem solved by algorithms has an easily identified set of candidate
solutions. For example, suppose we are given a set of numerical values representing samples of a signal, and we want to compute the discrete Fourier transform of
these samples. The discrete Fourier transform converts the time domain to the frequency domain, producing a set of numerical coefficients, so that we can determine
the strength of various frequencies in the sampled signal. In addition to lying at
the heart of signal processing, discrete Fourier transforms have applications in data
compression and multiplying large polynomials and integers. Chapter 30 gives
an efficient algorithm, the fast Fourier transform (commonly called the FFT), for
this problem, and the chapter also sketches out the design of a hardware circuit to
compute the FFT.
Data structures
This book also contains several data structures. A data structure is a way to store
and organize data in order to facilitate access and modifications. No single data
structure works well for all purposes, and so it is important to know the strengths
and limitations of several of them.
Technique
Although you can use this book as a “cookbook” for algorithms, you may someday
encounter a problem for which you cannot readily find a published algorithm (many
of the exercises and problems in this book, for example). This book will teach you
techniques of algorithm design and analysis so that you can develop algorithms on
your own, show that they give the correct answer, and understand their efficiency.
Different chapters address different aspects of algorithmic problem solving. Some
chapters address specific problems, such as finding medians and order statistics in
Chapter 9, computing minimum spanning trees in Chapter 23, and determining a
maximum flow in a network in Chapter 26. Other chapters address techniques,
such as divide-and-conquer in Chapter 4, dynamic programming in Chapter 15,
and amortized analysis in Chapter 17.
Hard problems
Most of this book is about efficient algorithms. Our usual measure of efficiency
is speed, i.e., how long an algorithm takes to produce its result. There are some
problems, however, for which no efficient solution is known. Chapter 34 studies
an interesting subset of these problems, which are known as NP-complete.
Why are NP-complete problems interesting? First, although no efficient algorithm for an NP-complete problem has ever been found, nobody has ever proven
10
Chapter 1 The Role of Algorithms in Computing
that an efficient algorithm for one cannot exist. In other words, no one knows
whether or not efficient algorithms exist for NP-complete problems. Second, the
set of NP-complete problems has the remarkable property that if an efficient algorithm exists for any one of them, then efficient algorithms exist for all of them. This
relationship among the NP-complete problems makes the lack of efficient solutions
all the more tantalizing. Third, several NP-complete problems are similar, but not
identical, to problems for which we do know of efficient algorithms. Computer
scientists are intrigued by how a small change to the problem statement can cause
a big change to the efficiency of the best known algorithm.
You should know about NP-complete problems because some of them arise surprisingly often in real applications. If you are called upon to produce an efficient
algorithm for an NP-complete problem, you are likely to spend a lot of time in a
fruitless search. If you can show that the problem is NP-complete, you can instead
spend your time developing an efficient algorithm that gives a good, but not the
best possible, solution.
As a concrete example, consider a delivery company with a central depot. Each
day, it loads up each delivery truck at the depot and sends it around to deliver goods
to several addresses. At the end of the day, each truck must end up back at the depot
so that it is ready to be loaded for the next day. To reduce costs, the company wants
to select an order of delivery stops that yields the lowest overall distance traveled
by each truck. This problem is the well-known “traveling-salesman problem,” and
it is NP-complete. It has no known efficient algorithm. Under certain assumptions,
however, we know of efficient algorithms that give an overall distance which is
not too far above the smallest possible. Chapter 35 discusses such “approximation
algorithms.”
Parallelism
For many years, we could count on processor clock speeds increasing at a steady
rate. Physical limitations present a fundamental roadblock to ever-increasing clock
speeds, however: because power density increases superlinearly with clock speed,
chips run the risk of melting once their clock speeds become high enough. In order
to perform more computations per second, therefore, chips are being designed to
contain not just one but several processing “cores.” We can liken these multicore
computers to several sequential computers on a single chip; in other words, they are
a type of “parallel computer.” In order to elicit the best performance from multicore
computers, we need to design algorithms with parallelism in mind. Chapter 27
presents a model for “multithreaded” algorithms, which take advantage of multiple
cores. This model has advantages from a theoretical standpoint, and it forms the
basis of several successful computer programs, including a championship chess
program.
1.2 Algorithms as a technology
11
Exercises
1.1-1
Give a real-world example that requires sorting or a real-world example that requires computing a convex hull.
1.1-2
Other than speed, what other measures of efficiency might one use in a real-world
setting?
1.1-3
Select a data structure that you have seen previously, and discuss its strengths and
limitations.
1.1-4
How are the shortest-path and traveling-salesman problems given above similar?
How are they different?
1.1-5
Come up with a real-world problem in which only the best solution will do. Then
come up with one in which a solution that is “approximately” the best is good
enough.
1.2 Algorithms as a technology
Suppose computers were infinitely fast and computer memory was free. Would
you have any reason to study algorithms? The answer is yes, if for no other reason
than that you would still like to demonstrate that your solution method terminates
and does so with the correct answer.
If computers were infinitely fast, any correct method for solving a problem
would do. You would probably want your implementation to be within the bounds
of good software engineering practice (for example, your implementation should
be well designed and documented), but you would most often use whichever
method was the easiest to implement.
Of course, computers may be fast, but they are not infinitely fast. And memory
may be inexpensive, but it is not free. Computing time is therefore a bounded
resource, and so is space in memory. You should use these resources wisely, and
algorithms that are efficient in terms of time or space will help you do so.
12
Chapter 1 The Role of Algorithms in Computing
Efficiency
Different algorithms devised to solve the same problem often differ dramatically in
their efficiency. These differences can be much more significant than differences
due to hardware and software.
As an example, in Chapter 2, we will see two algorithms for sorting. The first,
known as insertion sort, takes time roughly equal to c1 n2 to sort n items, where c1
is a constant that does not depend on n. That is, it takes time roughly proportional
to n2 . The second, merge sort, takes time roughly equal to c2 n lg n, where lg n
stands for log2 n and c2 is another constant that also does not depend on n. Insertion sort typically has a smaller constant factor than merge sort, so that c1 < c2 .
We shall see that the constant factors can have far less of an impact on the running
time than the dependence on the input size n. Let’s write insertion sort’s running
time as c1 n n and merge sort’s running time as c2 n lg n. Then we see that where
insertion sort has a factor of n in its running time, merge sort has a factor of lg n,
which is much smaller. (For example, when n D 1000, lg n is approximately 10,
and when n equals one million, lg n is approximately only 20.) Although insertion
sort usually runs faster than merge sort for small input sizes, once the input size n
becomes large enough, merge sort’s advantage of lg n vs. n will more than compensate for the difference in constant factors. No matter how much smaller c1 is
than c2 , there will always be a crossover point beyond which merge sort is faster.
For a concrete example, let us pit a faster computer (computer A) running insertion sort against a slower computer (computer B) running merge sort. They each
must sort an array of 10 million numbers. (Although 10 million numbers might
seem like a lot, if the numbers are eight-byte integers, then the input occupies
about 80 megabytes, which fits in the memory of even an inexpensive laptop computer many times over.) Suppose that computer A executes 10 billion instructions
per second (faster than any single sequential computer at the time of this writing)
and computer B executes only 10 million instructions per second, so that computer A is 1000 times faster than computer B in raw computing power. To make
the difference even more dramatic, suppose that the world’s craftiest programmer
codes insertion sort in machine language for computer A, and the resulting code
requires 2n2 instructions to sort n numbers. Suppose further that just an average
programmer implements merge sort, using a high-level language with an inefficient
compiler, with the resulting code taking 50n lg n instructions. To sort 10 million
numbers, computer A takes
2 .107 /2 instructions
D 20,000 seconds (more than 5.5 hours) ;
1010 instructions/second
while computer B takes
1.2 Algorithms as a technology
13
50 107 lg 107 instructions
1163 seconds (less than 20 minutes) :
107 instructions/second
By using an algorithm whose running time grows more slowly, even with a poor
compiler, computer B runs more than 17 times faster than computer A! The advantage of merge sort is even more pronounced when we sort 100 million numbers:
where insertion sort takes more than 23 days, merge sort takes under four hours.
In general, as the problem size increases, so does the relative advantage of merge
sort.
Algorithms and other technologies
The example above shows that we should consider algorithms, like computer hardware, as a technology. Total system performance depends on choosing efficient
algorithms as much as on choosing fast hardware. Just as rapid advances are being
made in other computer technologies, they are being made in algorithms as well.
You might wonder whether algorithms are truly that important on contemporary
computers in light of other advanced technologies, such as
advanced computer architectures and fabrication technologies,
easy-to-use, intuitive, graphical user interfaces (GUIs),
object-oriented systems,
integrated Web technologies, and
fast networking, both wired and wireless.
The answer is yes. Although some applications do not explicitly require algorithmic content at the application level (such as some simple, Web-based applications),
many do. For example, consider a Web-based service that determines how to travel
from one location to another. Its implementation would rely on fast hardware, a
graphical user interface, wide-area networking, and also possibly on object orientation. However, it would also require algorithms for certain operations, such
as finding routes (probably using a shortest-path algorithm), rendering maps, and
interpolating addresses.
Moreover, even an application that does not require algorithmic content at the
application level relies heavily upon algorithms. Does the application rely on fast
hardware? The hardware design used algorithms. Does the application rely on
graphical user interfaces? The design of any GUI relies on algorithms. Does the
application rely on networking? Routing in networks relies heavily on algorithms.
Was the application written in a language other than machine code? Then it was
processed by a compiler, interpreter, or assembler, all of which make extensive use
14
Chapter 1 The Role of Algorithms in Computing
of algorithms. Algorithms are at the core of most technologies used in contemporary computers.
Furthermore, with the ever-increasing capacities of computers, we use them to
solve larger problems than ever before. As we saw in the above comparison between insertion sort and merge sort, it is at larger problem sizes that the differences
in efficiency between algorithms become particularly prominent.
Having a solid base of algorithmic knowledge and technique is one characteristic
that separates the truly skilled programmers from the novices. With modern computing technology, you can accomplish some tasks without knowing much about
algorithms, but with a good background in algorithms, you can do much, much
more.
Exercises
1.2-1
Give an example of an application that requires algorithmic content at the application level, and discuss the function of the algorithms involved.
1.2-2
Suppose we are comparing implementations of insertion sort and merge sort on the
same machine. For inputs of size n, insertion sort runs in 8n2 steps, while merge
sort runs in 64n lg n steps. For which values of n does insertion sort beat merge
sort?
1.2-3
What is the smallest value of n such that an algorithm whose running time is 100n2
runs faster than an algorithm whose running time is 2n on the same machine?
Problems
1-1 Comparison of running times
For each function f .n/ and time t in the following table, determine the largest
size n of a problem that can be solved in time t, assuming that the algorithm to
solve the problem takes f .n/ microseconds.
Notes for Chapter 1
1
second
15
1
minute
1
hour
1
day
1
month
1
year
1
century
lg n
p
n
n
n lg n
n2
n3
2n
nŠ
Chapter notes
There are many excellent texts on the general topic of algorithms, including those
by Aho, Hopcroft, and Ullman [5, 6]; Baase and Van Gelder [28]; Brassard and
Bratley [54]; Dasgupta, Papadimitriou, and Vazirani [82]; Goodrich and Tamassia
[148]; Hofri [175]; Horowitz, Sahni, and Rajasekaran [181]; Johnsonbaugh and
Schaefer [193]; Kingston [205]; Kleinberg and Tardos [208]; Knuth [209, 210,
211]; Kozen [220]; Levitin [235]; Manber [242]; Mehlhorn [249, 250, 251]; Purdom and Brown [287]; Reingold, Nievergelt, and Deo [293]; Sedgewick [306];
Sedgewick and Flajolet [307]; Skiena [318]; and Wilf [356]. Some of the more
practical aspects of algorithm design are discussed by Bentley [42, 43] and Gonnet
[145]. Surveys of the field of algorithms can also be found in the Handbook of Theoretical Computer Science, Volume A [342] and the CRC Algorithms and Theory of
Computation Handbook [25]. Overviews of the algorithms used in computational
biology can be found in textbooks by Gusfield [156], Pevzner [275], Setubal and
Meidanis [310], and Waterman [350].
2
Getting Started
This chapter will familiarize you with the framework we shall use throughout the
book to think about the design and analysis of algorithms. It is self-contained, but
it does include several references to material that we introduce in Chapters 3 and 4.
(It also contains several summations, which Appendix A shows how to solve.)
We begin by examining the insertion sort algorithm to solve the sorting problem
introduced in Chapter 1. We define a “pseudocode” that should be familiar to you if
you have done computer programming, and we use it to show how we shall specify
our algorithms. Having specified the insertion sort algorithm, we then argue that it
correctly sorts, and we analyze its running time. The analysis introduces a notation
that focuses on how that time increases with the number of items to be sorted.
Following our discussion of insertion sort, we introduce the divide-and-conquer
approach to the design of algorithms and use it to develop an algorithm called
merge sort. We end with an analysis of merge sort’s running time.
2.1
Insertion sort
Our first algorithm, insertion sort, solves the sorting problem introduced in Chapter 1:
Input: A sequence of n numbers ha1 ; a2 ; : : : ; an i.
Output: A permutation (reordering) ha10 ; a20 ; : : : ; an0 i of the input sequence such
that a10 a20 an0 .
The numbers that we wish to sort are also known as the keys. Although conceptually we are sorting a sequence, the input comes to us in the form of an array with n
elements.
In this book, we shall typically describe algorithms as programs written in a
pseudocode that is similar in many respects to C, C++, Java, Python, or Pascal. If
you have been introduced to any of these languages, you should have little trouble
2.1 Insertion sort
17
♣♣
♣
♣♣
10
5♣ ♣
4 ♣♣
♣♣ ♣
♣
♣
♣
♣♣ ♣
7
♣
0
♣♣
♣
5♣
♣♣
♣
4 2♣
♣
♣
♣ ♣♣ ♣
♣♣
7
♣
2
♣
1
Figure 2.1 Sorting a hand of cards using insertion sort.
reading our algorithms. What separates pseudocode from “real” code is that in
pseudocode, we employ whatever expressive method is most clear and concise to
specify a given algorithm. Sometimes, the clearest method is English, so do not
be surprised if you come across an English phrase or sentence embedded within
a section of “real” code. Another difference between pseudocode and real code
is that pseudocode is not typically concerned with issues of software engineering.
Issues of data abstraction, modularity, and error handling are often ignored in order
to convey the essence of the algorithm more concisely.
We start with insertion sort, which is an efficient algorithm for sorting a small
number of elements. Insertion sort works the way many people sort a hand of
playing cards. We start with an empty left hand and the cards face down on the
table. We then remove one card at a time from the table and insert it into the
correct position in the left hand. To find the correct position for a card, we compare
it with each of the cards already in the hand, from right to left, as illustrated in
Figure 2.1. At all times, the cards held in the left hand are sorted, and these cards
were originally the top cards of the pile on the table.
We present our pseudocode for insertion sort as a procedure called I NSERTION S ORT, which takes as a parameter an array AŒ1 : : n containing a sequence of
length n that is to be sorted. (In the code, the number n of elements in A is denoted
by A:length.) The algorithm sorts the input numbers in place: it rearranges the
numbers within the array A, with at most a constant number of them stored outside
the array at any time. The input array A contains the sorted output sequence when
the I NSERTION -S ORT procedure is finished.
18
Chapter 2 Getting Started
1
2
3
4
5
6
(a)
5
2
4
6
1
3
1
2
3
4
5
6
(d)
2
4
5
6
1
3
1
2
3
4
5
6
(b)
2
5
4
6
1
3
1
2
3
4
5
6
(e)
1
2
4
5
6
3
1
2
3
4
5
6
(c)
2
4
5
6
1
3
1
2
3
4
5
6
(f)
1
2
3
4
5
6
Figure 2.2 The operation of I NSERTION -S ORT on the array A D h5; 2; 4; 6; 1; 3i. Array indices
appear above the rectangles, and values stored in the array positions appear within the rectangles.
(a)–(e) The iterations of the for loop of lines 1–8. In each iteration, the black rectangle holds the
key taken from AŒj , which is compared with the values in shaded rectangles to its left in the test of
line 5. Shaded arrows show array values moved one position to the right in line 6, and black arrows
indicate where the key moves to in line 8. (f) The final sorted array.
I NSERTION -S ORT .A/
1 for j D 2 to A:length
2
key D AŒj
3
// Insert AŒj into the sorted sequence AŒ1 : : j 1.
4
i D j 1
5
while i > 0 and AŒi > key
6
AŒi C 1 D AŒi
7
i D i 1
8
AŒi C 1 D key
Loop invariants and the correctness of insertion sort
Figure 2.2 shows how this algorithm works for A D h5; 2; 4; 6; 1; 3i. The index j indicates the “current card” being inserted into the hand. At the beginning
of each iteration of the for loop, which is indexed by j , the subarray consisting
of elements AŒ1 : : j 1 constitutes the currently sorted hand, and the remaining
subarray AŒj C 1 : : n corresponds to the pile of cards still on the table. In fact,
elements AŒ1 : : j 1 are the elements originally in positions 1 through j 1, but
now in sorted order. We state these properties of AŒ1 : : j 1 formally as a loop
invariant:
At the start of each iteration of the for loop of lines 1–8, the subarray
AŒ1 : : j 1 consists of the elements originally in AŒ1 : : j 1, but in sorted
order.
We use loop invariants to help us understand why an algorithm is correct. We
must show three things about a loop invariant:
2.1 Insertion sort
19
Initialization: It is true prior to the first iteration of the loop.
Maintenance: If it is true before an iteration of the loop, it remains true before the
next iteration.
Termination: When the loop terminates, the invariant gives us a useful property
that helps show that the algorithm is correct.
When the first two properties hold, the loop invariant is true prior to every iteration
of the loop. (Of course, we are free to use established facts other than the loop
invariant itself to prove that the loop invariant remains true before each iteration.)
Note the similarity to mathematical induction, where to prove that a property holds,
you prove a base case and an inductive step. Here, showing that the invariant holds
before the first iteration corresponds to the base case, and showing that the invariant
holds from iteration to iteration corresponds to the inductive step.
The third property is perhaps the most important one, since we are using the loop
invariant to show correctness. Typically, we use the loop invariant along with the
condition that caused the loop to terminate. The termination property differs from
how we usually use mathematical induction, in which we apply the inductive step
infinitely; here, we stop the “induction” when the loop terminates.
Let us see how these properties hold for insertion sort.
Initialization: We start by showing that the loop invariant holds before the first
loop iteration, when j D 2.1 The subarray AŒ1 : : j 1, therefore, consists
of just the single element AŒ1, which is in fact the original element in AŒ1.
Moreover, this subarray is sorted (trivially, of course), which shows that the
loop invariant holds prior to the first iteration of the loop.
Maintenance: Next, we tackle the second property: showing that each iteration
maintains the loop invariant. Informally, the body of the for loop works by
moving AŒj 1, AŒj 2, AŒj 3, and so on by one position to the right
until it finds the proper position for AŒj (lines 4–7), at which point it inserts
the value of AŒj (line 8). The subarray AŒ1 : : j then consists of the elements
originally in AŒ1 : : j , but in sorted order. Incrementing j for the next iteration
of the for loop then preserves the loop invariant.
A more formal treatment of the second property would require us to state and
show a loop invariant for the while loop of lines 5–7. At this point, however,
1 When
the loop is a for loop, the moment at which we check the loop invariant just prior to the first
iteration is immediately after the initial assignment to the loop-counter variable and just before the
first test in the loop header. In the case of I NSERTION -S ORT , this time is after assigning 2 to the
variable j but before the first test of whether j A: length.
20
Chapter 2 Getting Started
we prefer not to get bogged down in such formalism, and so we rely on our
informal analysis to show that the second property holds for the outer loop.
Termination: Finally, we examine what happens when the loop terminates. The
condition causing the for loop to terminate is that j > A:length D n. Because
each loop iteration increases j by 1, we must have j D n C 1 at that time.
Substituting n C 1 for j in the wording of loop invariant, we have that the
subarray AŒ1 : : n consists of the elements originally in AŒ1 : : n, but in sorted
order. Observing that the subarray AŒ1 : : n is the entire array, we conclude that
the entire array is sorted. Hence, the algorithm is correct.
We shall use this method of loop invariants to show correctness later in this
chapter and in other chapters as well.
Pseudocode conventions
We use the following conventions in our pseudocode.
Indentation indicates block structure. For example, the body of the for loop that
begins on line 1 consists of lines 2–8, and the body of the while loop that begins
on line 5 contains lines 6–7 but not line 8. Our indentation style applies to
if-else statements2 as well. Using indentation instead of conventional indicators
of block structure, such as begin and end statements, greatly reduces clutter
while preserving, or even enhancing, clarity.3
The looping constructs while, for, and repeat-until and the if-else conditional
construct have interpretations similar to those in C, C++, Java, Python, and
Pascal.4 In this book, the loop counter retains its value after exiting the loop,
unlike some situations that arise in C++, Java, and Pascal. Thus, immediately
after a for loop, the loop counter’s value is the value that first exceeded the for
loop bound. We used this property in our correctness argument for insertion
sort. The for loop header in line 1 is for j D 2 to A:length, and so when
this loop terminates, j D A:length C 1 (or, equivalently, j D n C 1, since
n D A:length). We use the keyword to when a for loop increments its loop
2 In
an if-else statement, we indent else at the same level as its matching if. Although we omit the
keyword then, we occasionally refer to the portion executed when the test following if is true as a
then clause. For multiway tests, we use elseif for tests after the first one.
3 Each
pseudocode procedure in this book appears on one page so that you will not have to discern
levels of indentation in code that is split across pages.
4 Most block-structured languages have equivalent constructs, though the exact syntax may differ.
Python lacks repeat-until loops, and its for loops operate a little differently from the for loops in
this book.
2.1 Insertion sort
21
counter in each iteration, and we use the keyword downto when a for loop
decrements its loop counter. When the loop counter changes by an amount
greater than 1, the amount of change follows the optional keyword by.
The symbol “//” indicates that the remainder of the line is a comment.
A multiple assignment of the form i D j D e assigns to both variables i and j
the value of expression e; it should be treated as equivalent to the assignment
j D e followed by the assignment i D j .
Variables (such as i, j , and key) are local to the given procedure. We shall not
use global variables without explicit indication.
We access array elements by specifying the array name followed by the index in square brackets. For example, AŒi indicates the ith element of the
array A. The notation “: :” is used to indicate a range of values within an array. Thus, AŒ1 : : j indicates the subarray of A consisting of the j elements
AŒ1; AŒ2; : : : ; AŒj .
We typically organize compound data into objects, which are composed of
attributes. We access a particular attribute using the syntax found in many
object-oriented programming languages: the object name, followed by a dot,
followed by the attribute name. For example, we treat an array as an object
with the attribute length indicating how many elements it contains. To specify
the number of elements in an array A, we write A:length.
We treat a variable representing an array or object as a pointer to the data representing the array or object. For all attributes f of an object x, setting y D x
causes y:f to equal x:f . Moreover, if we now set x:f D 3, then afterward not
only does x:f equal 3, but y:f equals 3 as well. In other words, x and y point
to the same object after the assignment y D x.
Our attribute notation can “cascade.” For example, suppose that the attribute f
is itself a pointer to some type of object that has an attribute g. Then the notation
x:f :g is implicitly parenthesized as .x:f /:g. In other words, if we had assigned
y D x:f , then x:f :g is the same as y:g.
Sometimes, a pointer will refer to no object at all. In this case, we give it the
special value NIL.
We pass parameters to a procedure by value: the called procedure receives its
own copy of the parameters, and if it assigns a value to a parameter, the change
is not seen by the calling procedure. When objects are passed, the pointer to
the data representing the object is copied, but the object’s attributes are not. For
example, if x is a parameter of a called procedure, the assignment x D y within
the called procedure is not visible to the calling procedure. The assignment
x:f D 3, however, is visible. Similarly, arrays are passed by pointer, so that
22
Chapter 2 Getting Started
a pointer to the array is passed, rather than the entire array, and changes to
individual array elements are visible to the calling procedure.
A return statement immediately transfers control back to the point of call in
the calling procedure. Most return statements also take a value to pass back to
the caller. Our pseudocode differs from many programming languages in that
we allow multiple values to be returned in a single return statement.
The boolean operators “and” and “or” are short circuiting. That is, when we
evaluate the expression “x and y” we first evaluate x. If x evaluates to FALSE,
then the entire expression cannot evaluate to TRUE, and so we do not evaluate y.
If, on the other hand, x evaluates to TRUE, we must evaluate y to determine the
value of the entire expression. Similarly, in the expression “x or y” we evaluate the expression y only if x evaluates to FALSE. Short-circuiting operators
allow us to write boolean expressions such as “x ¤ NIL and x:f D y” without
worrying about what happens when we try to evaluate x:f when x is NIL.
The keyword error indicates that an error occurred because conditions were
wrong for the procedure to have been called. The calling procedure is responsible for handling the error, and so we do not specify what action to take.
Exercises
2.1-1
Using Figure 2.2 as a model, illustrate the operation of I NSERTION -S ORT on the
array A D h31; 41; 59; 26; 41; 58i.
2.1-2
Rewrite the I NSERTION -S ORT procedure to sort into nonincreasing instead of nondecreasing order.
2.1-3
Consider the searching problem:
Input: A sequence of n numbers A D ha1 ; a2 ; : : : ; an i and a value .
Output: An index i such that D AŒi or the special value NIL if does not
appear in A.
Write pseudocode for linear search, which scans through the sequence, looking
for . Using a loop invariant, prove that your algorithm is correct. Make sure that
your loop invariant fulfills the three necessary properties.
2.1-4
Consider the problem of adding two n-bit binary integers, stored in two n-element
arrays A and B. The sum of the two integers should be stored in binary form in
2.2 Analyzing algorithms
23
an .n C 1/-element array C . State the problem formally and write pseudocode for
adding the two integers.
2.2 Analyzing algorithms
Analyzing an algorithm has come to mean predicting the resources that the algorithm requires. Occasionally, resources such as memory, communication bandwidth, or computer hardware are of primary concern, but most often it is computational time that we want to measure. Generally, by analyzing several candidate
algorithms for a problem, we can identify a most efficient one. Such analysis may
indicate more than one viable candidate, but we can often discard several inferior
algorithms in the process.
Before we can analyze an algorithm, we must have a model of the implementation technology that we will use, including a model for the resources of that
technology and their costs. For most of this book, we shall assume a generic oneprocessor, random-access machine (RAM) model of computation as our implementation technology and understand that our algorithms will be implemented as
computer programs. In the RAM model, instructions are executed one after another, with no concurrent operations.
Strictly speaking, we should precisely define the instructions of the RAM model
and their costs. To do so, however, would be tedious and would yield little insight
into algorithm design and analysis. Yet we must be careful not to abuse the RAM
model. For example, what if a RAM had an instruction that sorts? Then we could
sort in just one instruction. Such a RAM would be unrealistic, since real computers
do not have such instructions. Our guide, therefore, is how real computers are designed. The RAM model contains instructions commonly found in real computers:
arithmetic (such as add, subtract, multiply, divide, remainder, floor, ceiling), data
movement (load, store, copy), and control (conditional and unconditional branch,
subroutine call and return). Each such instruction takes a constant amount of time.
The data types in the RAM model are integer and floating point (for storing real
numbers). Although we typically do not concern ourselves with precision in this
book, in some applications precision is crucial. We also assume a limit on the size
of each word of data. For example, when working with inputs of size n, we typically assume that integers are represented by c lg n bits for some constant c 1.
We require c 1 so that each word can hold the value of n, enabling us to index the
individual input elements, and we restrict c to be a constant so that the word size
does not grow arbitrarily. (If the word size could grow arbitrarily, we could store
huge amounts of data in one word and operate on it all in constant time—clearly
an unrealistic scenario.)
24
Chapter 2 Getting Started
Real computers contain instructions not listed above, and such instructions represent a gray area in the RAM model. For example, is exponentiation a constanttime instruction? In the general case, no; it takes several instructions to compute x y
when x and y are real numbers. In restricted situations, however, exponentiation is
a constant-time operation. Many computers have a “shift left” instruction, which
in constant time shifts the bits of an integer by k positions to the left. In most
computers, shifting the bits of an integer by one position to the left is equivalent
to multiplication by 2, so that shifting the bits by k positions to the left is equivalent to multiplication by 2k . Therefore, such computers can compute 2k in one
constant-time instruction by shifting the integer 1 by k positions to the left, as long
as k is no more than the number of bits in a computer word. We will endeavor to
avoid such gray areas in the RAM model, but we will treat computation of 2k as a
constant-time operation when k is a small enough positive integer.
In the RAM model, we do not attempt to model the memory hierarchy that is
common in contemporary computers. That is, we do not model caches or virtual
memory. Several computational models attempt to account for memory-hierarchy
effects, which are sometimes significant in real programs on real machines. A
handful of problems in this book examine memory-hierarchy effects, but for the
most part, the analyses in this book will not consider them. Models that include
the memory hierarchy are quite a bit more complex than the RAM model, and so
they can be difficult to work with. Moreover, RAM-model analyses are usually
excellent predictors of performance on actual machines.
Analyzing even a simple algorithm in the RAM model can be a challenge. The
mathematical tools required may include combinatorics, probability theory, algebraic dexterity, and the ability to identify the most significant terms in a formula.
Because the behavior of an algorithm may be different for each possible input, we
need a means for summarizing that behavior in simple, easily understood formulas.
Even though we typically select only one machine model to analyze a given algorithm, we still face many choices in deciding how to express our analysis. We
would like a way that is simple to write and manipulate, shows the important characteristics of an algorithm’s resource requirements, and suppresses tedious details.
Analysis of insertion sort
The time taken by the I NSERTION -S ORT procedure depends on the input: sorting a
thousand numbers takes longer than sorting three numbers. Moreover, I NSERTION S ORT can take different amounts of time to sort two input sequences of the same
size depending on how nearly sorted they already are. In general, the time taken
by an algorithm grows with the size of the input, so it is traditional to describe the
running time of a program as a function of the size of its input. To do so, we need
to define the terms “running time” and “size of input” more carefully.
2.2 Analyzing algorithms
25
The best notion for input size depends on the problem being studied. For many
problems, such as sorting or computing discrete Fourier transforms, the most natural measure is the number of items in the input—for example, the array size n
for sorting. For many other problems, such as multiplying two integers, the best
measure of input size is the total number of bits needed to represent the input in
ordinary binary notation. Sometimes, it is more appropriate to describe the size of
the input with two numbers rather than one. For instance, if the input to an algorithm is a graph, the input size can be described by the numbers of vertices and
edges in the graph. We shall indicate which input size measure is being used with
each problem we study.
The running time of an algorithm on a particular input is the number of primitive
operations or “steps” executed. It is convenient to define the notion of step so
that it is as machine-independent as possible. For the moment, let us adopt the
following view. A constant amount of time is required to execute each line of our
pseudocode. One line may take a different amount of time than another line, but
we shall assume that each execution of the ith line takes time ci , where ci is a
constant. This viewpoint is in keeping with the RAM model, and it also reflects
how the pseudocode would be implemented on most actual computers.5
In the following discussion, our expression for the running time of I NSERTION S ORT will evolve from a messy formula that uses all the statement costs ci to a
much simpler notation that is more concise and more easily manipulated. This
simpler notation will also make it easy to determine whether one algorithm is more
efficient than another.
We start by presenting the I NSERTION -S ORT procedure with the time “cost”
of each statement and the number of times each statement is executed. For each
j D 2; 3; : : : ; n, where n D A:length, we let tj denote the number of times the
while loop test in line 5 is executed for that value of j . When a for or while loop
exits in the usual way (i.e., due to the test in the loop header), the test is executed
one time more than the loop body. We assume that comments are not executable
statements, and so they take no time.
5 There are some subtleties here. Computational steps that we specify in English are often variants
of a procedure that requires more than just a constant amount of time. For example, later in this
book we might say “sort the points by x-coordinate,” which, as we shall see, takes more than a
constant amount of time. Also, note that a statement that calls a subroutine takes constant time,
though the subroutine, once invoked, may take more. That is, we separate the process of calling the
subroutine—passing parameters to it, etc.—from the process of executing the subroutine.
26
Chapter 2 Getting Started
I NSERTION -S ORT .A/
1 for j D 2 to A:length
2
key D AŒj
3
// Insert AŒj into the sorted
sequence AŒ1 : : j 1.
4
i D j 1
5
while i > 0 and AŒi > key
6
AŒi C 1 D AŒi
7
i D i 1
8
AŒi C 1 D key
cost
c1
c2
times
n
n1
0
c4
c5
c6
c7
c8
n1
n
P 1
n
t
PjnD2 j
.t 1/
PjnD2 j
.t
j D2 j 1/
n1
The running time of the algorithm is the sum of running times for each statement executed; a statement that takes ci steps to execute and executes n times will
contribute ci n to the total running time.6 To compute T .n/, the running time of
I NSERTION -S ORT on an input of n values, we sum the products of the cost and
times columns, obtaining
T .n/ D c1 n C c2 .n 1/ C c4 .n 1/ C c5
n
X
j D2
C c7
n
X
tj C c6
n
X
.tj 1/
j D2
.tj 1/ C c8 .n 1/ :
j D2
Even for inputs of a given size, an algorithm’s running time may depend on
which input of that size is given. For example, in I NSERTION -S ORT, the best
case occurs if the array is already sorted. For each j D 2; 3; : : : ; n, we then find
that AŒi key in line 5 when i has its initial value of j 1. Thus tj D 1 for
j D 2; 3; : : : ; n, and the best-case running time is
T .n/ D c1 n C c2 .n 1/ C c4 .n 1/ C c5 .n 1/ C c8 .n 1/
D .c1 C c2 C c4 C c5 C c8 /n .c2 C c4 C c5 C c8 / :
We can express this running time as an C b for constants a and b that depend on
the statement costs ci ; it is thus a linear function of n.
If the array is in reverse sorted order—that is, in decreasing order—the worst
case results. We must compare each element AŒj with each element in the entire
sorted subarray AŒ1 : : j 1, and so tj D j for j D 2; 3; : : : ; n. Noting that
6 This characteristic does not necessarily hold for a resource such as memory. A statement that
references m words of memory and is executed n times does not necessarily reference mn distinct
words of memory.
2.2 Analyzing algorithms
n
X
j D2
j D
27
n.n C 1/
1
2
and
n
X
n.n 1/
.j 1/ D
2
j D2
(see Appendix A for a review of how to solve these summations), we find that in
the worst case, the running time of I NSERTION -S ORT is
n.n C 1/
1
T .n/ D c1 n C c2 .n 1/ C c4 .n 1/ C c5
2
n.n 1/
n.n 1/
C c7
C c8 .n 1/
C c6
2
2
c
c6
c7 2
c5 c6 c7
5
C
C
n C c1 C c2 C c4 C
C c8 n
D
2
2
2
2
2
2
.c2 C c4 C c5 C c8 / :
We can express this worst-case running time as an2 C bn C c for constants a, b,
and c that again depend on the statement costs ci ; it is thus a quadratic function
of n.
Typically, as in insertion sort, the running time of an algorithm is fixed for a
given input, although in later chapters we shall see some interesting “randomized”
algorithms whose behavior can vary even for a fixed input.
Worst-case and average-case analysis
In our analysis of insertion sort, we looked at both the best case, in which the input
array was already sorted, and the worst case, in which the input array was reverse
sorted. For the remainder of this book, though, we shall usually concentrate on
finding only the worst-case running time, that is, the longest running time for any
input of size n. We give three reasons for this orientation.
The worst-case running time of an algorithm gives us an upper bound on the
running time for any input. Knowing it provides a guarantee that the algorithm
will never take any longer. We need not make some educated guess about the
running time and hope that it never gets much worse.
For some algorithms, the worst case occurs fairly often. For example, in searching a database for a particular piece of information, the searching algorithm’s
worst case will often occur when the information is not present in the database.
In some applications, searches for absent information may be frequent.
28
Chapter 2 Getting Started
The “average case” is often roughly as bad as the worst case. Suppose that we
randomly choose n numbers and apply insertion sort. How long does it take to
determine where in subarray AŒ1 : : j 1 to insert element AŒj ? On average,
half the elements in AŒ1 : : j 1 are less than AŒj , and half the elements are
greater. On average, therefore, we check half of the subarray AŒ1 : : j 1, and
so tj is about j=2. The resulting average-case running time turns out to be a
quadratic function of the input size, just like the worst-case running time.
In some particular cases, we shall be interested in the average-case running time
of an algorithm; we shall see the technique of probabilistic analysis applied to
various algorithms throughout this book. The scope of average-case analysis is
limited, because it may not be apparent what constitutes an “average” input for
a particular problem. Often, we shall assume that all inputs of a given size are
equally likely. In practice, this assumption may be violated, but we can sometimes
use a randomized algorithm, which makes random choices, to allow a probabilistic
analysis and yield an expected running time. We explore randomized algorithms
more in Chapter 5 and in several other subsequent chapters.
Order of growth
We used some simplifying abstractions to ease our analysis of the I NSERTION S ORT procedure. First, we ignored the actual cost of each statement, using the
constants ci to represent these costs. Then, we observed that even these constants
give us more detail than we really need: we expressed the worst-case running time
as an2 C bn C c for some constants a, b, and c that depend on the statement
costs ci . We thus ignored not only the actual statement costs, but also the abstract
costs ci .
We shall now make one more simplifying abstraction: it is the rate of growth,
or order of growth, of the running time that really interests us. We therefore consider only the leading term of a formula (e.g., an2 ), since the lower-order terms are
relatively insignificant for large values of n. We also ignore the leading term’s constant coefficient, since constant factors are less significant than the rate of growth
in determining computational efficiency for large inputs. For insertion sort, when
we ignore the lower-order terms and the leading term’s constant coefficient, we are
left with the factor of n2 from the leading term. We write that insertion sort has a
worst-case running time of ‚.n2 / (pronounced “theta of n-squared”). We shall use
‚-notation informally in this chapter, and we will define it precisely in Chapter 3.
We usually consider one algorithm to be more efficient than another if its worstcase running time has a lower order of growth. Due to constant factors and lowerorder terms, an algorithm whose running time has a higher order of growth might
take less time for small inputs than an algorithm whose running time has a lower
2.3 Designing algorithms
29
order of growth. But for large enough inputs, a ‚.n2 / algorithm, for example, will
run more quickly in the worst case than a ‚.n3 / algorithm.
Exercises
2.2-1
Express the function n3 =1000 100n2 100n C 3 in terms of ‚-notation.
2.2-2
Consider sorting n numbers stored in array A by first finding the smallest element
of A and exchanging it with the element in AŒ1. Then find the second smallest
element of A, and exchange it with AŒ2. Continue in this manner for the first n 1
elements of A. Write pseudocode for this algorithm, which is known as selection
sort. What loop invariant does this algorithm maintain? Why does it need to run
for only the first n 1 elements, rather than for all n elements? Give the best-case
and worst-case running times of selection sort in ‚-notation.
2.2-3
Consider linear search again (see Exercise 2.1-3). How many elements of the input sequence need to be checked on the average, assuming that the element being
searched for is equally likely to be any element in the array? How about in the
worst case? What are the average-case and worst-case running times of linear
search in ‚-notation? Justify your answers.
2.2-4
How can we modify almost any algorithm to have a good best-case running time?
2.3 Designing algorithms
We can choose from a wide range of algorithm design techniques. For insertion
sort, we used an incremental approach: having sorted the subarray AŒ1 : : j 1,
we inserted the single element AŒj into its proper place, yielding the sorted
subarray AŒ1 : : j .
In this section, we examine an alternative design approach, known as “divideand-conquer,” which we shall explore in more detail in Chapter 4. We’ll use divideand-conquer to design a sorting algorithm whose worst-case running time is much
less than that of insertion sort. One advantage of divide-and-conquer algorithms is
that their running times are often easily determined using techniques that we will
see in Chapter...
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