"麻省理工学院算法导论：MIT 顶级算法导论书，详尽介绍 20～40字"

MIT算法导论

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《麻省理工学院算法导论》是一本非常牛B的算法教材，由Thomas H. Cormen、Charles E. Leiserson、Ronald L. Rivest和Clifford Stein共同编写。该书由麻省理工学院电气工程与计算机科学系的教授所编写，由The MIT出版。该书第二版在剑桥、马萨诸塞州和伦敦的地方出版，并由McGraw-Hill Book Company、Boston、Burr Ridge、IL、Dubuque、IA、Madison、WI、New York、San Francisco、St. Louis、Montréal和Toronto等地的机构提供支持。《麻省理工学院算法导论》作为一本专注于算法学习的教材，在学术界具有很高的声誉和影响力。它全面而详尽地介绍了各种常见算法的原理、思想和应用。通过深度剖析算法的思维过程和设计技巧，读者可以全面掌握算法设计和分析的方法。该书分为多个章节，每一章都涵盖了特定主题的算法。其中包括排序、搜索、图算法、动态规划、贪心算法等。每个主题都详细讲解了相关算法的基本思想、实现方式和应用场景。此外，书中还包括大量的示例、习题和练习题，帮助读者加深对算法的理解和掌握。《麻省理工学院算法导论》第二版相比第一版加入了一些新的内容和案例，使其更加适应当前的技术和应用需求。此外，该书在讲解算法的同时，也对算法的复杂度进行了详细的分析和讨论。这对于读者理解和评估算法的效率和可行性非常有帮助。这本书的编写者都是麻省理工学院的顶级教授，他们在算法领域拥有极高的知名度和专业水平。他们的教学和研究经验使得这本书在内容的准确性和深度上都具有很高的可信度。同时，书中还加入了一些他们自己的研究成果和创新理念，使得该书更加丰富和有趣。《麻省理工学院算法导论》在全球范围内广泛应用于大学本科和研究生的计算机科学相关专业的教学中。无论是作为教材还是作为参考书，它都为学生提供了全面而系统的算法学习资源。除了学术界，该书也深受工程师和研究人员的欢迎，他们可以通过学习书中的算法和思想，提高解决实际问题的能力。综上所述，《麻省理工学院算法导论》是一本非常牛B的算法教材，由麻省理工学院的顶级教授编写而成。它全面而详尽地介绍了算法的基本原理、设计思想和应用场景。该书在学术界和工程实践中都具有很高的价值和影响力，是一本不可或缺的算法学习资源。

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By using an algorithm whose running time grows more slowly, even with a poor compiler,

computer B runs 20 times faster than computer A! The advantage of merge sort is even more

pronounced when we sort ten million numbers: where insertion sort takes approximately 2.3

days, merge sort takes under 20 minutes. In general, as the problem size increases, so does the

relative advantage of merge sort.

Algorithms and other technologies

The example above shows that algorithms, like computer hardware, are 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

• hardware with high clock rates, pipelining, and superscalar architectures,

• easy-to-use, intuitive graphical user interfaces (GUIs),

• object-oriented systems, and

• local-area and wide-area networking.

The answer is yes. Although there are some applications that do not explicitly require

algorithmic content at the application level (e.g., some simple web-based applications), most

also require a degree of algorithmic content on their own. For example, consider a web-based

service that determines how to travel from one location to another. (Several such services

existed at the time of this writing.) 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 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 efficiencies 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

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 [26], Brassard and Bratley [46, 47],

Goodrich and Tamassia [128], Horowitz, Sahni, and Rajasekaran [158], Kingston [179],

Knuth [182, 183, 185], Kozen [193], Manber [210], Mehlhorn [217, 218, 219], Purdom and

Brown [252], Reingold, Nievergelt, and Deo [257], Sedgewick [269], Skiena [280], and Wilf

[315]. Some of the more practical aspects of algorithm design are discussed by Bentley [39,

40] and Gonnet [126]. Surveys of the field of algorithms can also be found in the Handbook

of Theoretical Computer Science, Volume A [302] and the CRC Handbook on Algorithms

and Theory of Computation [24]. Overviews of the algorithms used in computational biology

can be found in textbooks by Gusfield [136], Pevzner [240], Setubal and Medinas [272], and

Waterman [309]

Chapter 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 will be introduced 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 readers who have done

computer programming and use it to show how we shall specify our algorithms. Having

specified the 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 a

, a

, . . .,a

.

• Output: A permutation (reordering) of the input sequence such that

The numbers that we wish to sort are also known as the keys.

In this book, we shall typically describe algorithms as programs written in a pseudocode that

is similar in many respects to C, Pascal, or Java. If you have been introduced to any of these

languages, you should have little trouble 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.

Figure 2.1: Sorting a hand of cards using insertion sort.

Our pseudocode for insertion sort is presented as a procedure called INSERTION-SORT,

which takes as a parameter an array A[1  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 length[A].) The input

numbers are sorted in place: the numbers are rearranged 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 INSERTION-SORT is finished.

INSERTION-SORT(A)

1 for j ← 2 to length[A]

2 do key ← A[j]

3 ▹ Insert A[j] into the sorted sequence A[1  j - 1].

4 i ← j - 1

5 while i > 0 and A[i] > key

6 do A[i + 1] ← A[i]

7 i ← i - 1

8 A[i + 1] ← key

Loop invariants and the correctness of insertion sort

Figure 2.2 shows how this algorithm works for A = 5, 2, 4, 6, 1, 3. The index j indicates

the "current card" being inserted into the hand. At the beginning of each iteration of the

"outer" for loop, which is indexed by j, the subarray consisting of elements A[1  j - 1]

constitute the currently sorted hand, and elements A[j + 1  n] correspond to the pile of cards

still on the table. In fact, elements A[1  j - 1] are the elements originally in positions 1

through j - 1, but now in sorted order. We state these properties of A[1  j -1] formally as a

loop invariant:

• At the start of each iteration of the for loop of lines 1-8, the subarray A[1  j - 1]

consists of the elements originally in A[1  j - 1] but in sorted order.

Figure 2.2: The operation of INSERTION-SORT on the array A = 5, 2, 4, 6, 1, 3. 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 A[j], 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 is moved to in line 8. (f) The

final sorted array.

We use loop invariants to help us understand why an algorithm is correct. We must show

three things about a loop invariant:

• 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. 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 is like the base case, and showing that the invariant holds from iteration to iteration is

like the inductive step.

The third property is perhaps the most important one, since we are using the loop invariant to

show correctness. It also differs from the usual use of mathematical induction, in which the

inductive step is used 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 = 2.

[1]

The subarray A[1  j - 1], therefore, consists of just the single

element A[1], which is in fact the original element in A[1]. 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 outer for loop works by

moving A[ j - 1], A[ j - 2], A[ j - 3], and so on by one position to the right until the

proper position for A[ j] is found (lines 4-7), at which point the value of A[j] is inserted

(line 8). A more formal treatment of the second property would require us to state and

show a loop invariant for the "inner" while loop. At this point, however, 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.

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