算法第四版：程序员必知必会的50个算法

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"《算法（第四版）》是由Robert Sedgewick和Kevin Wayne合著的一本经典算法教程，专注于介绍算法和数据结构的基础知识，特别关注排序、搜索、图处理和字符串处理。该书提供了50个程序员必备的算法实例，所有代码以Java语言实现，采用模块化编程风格，便于读者理解和修改。书中附带的配套网站提供了更多代码、测试数据、练习、教学材料等丰富的学习资源。" 在《算法（第四版）》中，作者深入浅出地讲解了算法设计与分析的关键概念。以下是该书涉及的一些主要知识点： 1. **排序算法**：包括经典的快速排序、归并排序、堆排序、冒泡排序、插入排序、选择排序以及更高效的计数排序、基数排序和桶排序等。这些算法不仅解释了其工作原理，还讨论了它们的时间复杂度和空间复杂度。 2. **搜索算法**：涵盖了线性搜索、二分搜索、哈希表搜索以及更高级的二叉查找树、AVL树、红黑树等自平衡搜索树。这些搜索算法在不同场景下有不同的效率和适用性。 3. **数据结构**：如数组、链表、栈、队列、堆、图、树等，这些都是实现各种算法的基础。书中详细阐述了它们的特性、操作方法以及如何在实际问题中选择合适的数据结构。 4. **图处理算法**：包括深度优先搜索、广度优先搜索、最小生成树算法（如Prim和Kruskal）、最短路径算法（如Dijkstra和Floyd-Warshall）等。这些算法在解决网络问题、社交网络分析等方面有着广泛应用。 5. **字符串处理**：讲解了字符串匹配算法，如朴素匹配、KMP算法、Boyer-Moore算法，以及Trie树、后缀数组等高效的数据结构用于字符串处理。 6. **动态规划**：介绍了如何使用动态规划解决背包问题、最长公共子序列、最短路径等问题，强调了状态转移方程的构建和优化。 7. **递归与分治**：通过递归函数和分治策略来解决复杂问题，如快速傅里叶变换、大整数乘法等。 8. **算法设计技巧**：包括贪心算法、回溯法、分支限界法等，帮助读者掌握解决问题的通用策略。 9. **复杂度分析**：讲解了如何分析和估算算法的时间复杂度和空间复杂度，以及大O记法在算法效率评估中的应用。 10. **算法实现**：提供的Java代码实现具有模块化特点，易于理解并可作为实际项目的基础。这本书不仅适合初学者入门，也对有经验的开发者提供了深入理解算法的宝贵资源。通过阅读和实践，读者可以提升算法设计能力，为解决实际问题提供强大的工具。

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he objective of this book is to study a broad variety of important and useful

algorithms—methods for solving problems that are suited for computer imple-

mentation. Algorithms go hand in hand with data structures—schemes for or-

ganizing data that leave them amenable to efﬁcient processing by an algorithm. This

chapter introduces the basic tools that we need to study algorithms and data structures.

First, we introduce our basic programming model. All of our programs are imple-

mented using a small subset of the Java programming language plus a few of our own

libraries for input/output and for statistical calculations. Section 1.1 is a summary of

language constructs, features, and libraries that we use in this book.

Next, we emphasize data abstraction, where we deﬁne abstract data types (ADTs) in

the service of modular programming. In Section 1.2 we introduce the process of im-

plementing an ADT in Java, by specifying an applications programming interface (API)

and then using the Java class mechanism to develop an implementation for use in client

code.

As important and useful examples, we next consider three fundamental ADTs: the

bag, the queue, and the stack. Section 1.3 describes APIs and implementations of bags,

queues, and stacks using arrays, resizing arrays, and linked lists that serve as models and

starting points for algorithm implementations throughout the book.

Performance is a central consideration in the study of algorithms. Section 1.4 de-

scribes our approach to analyzing algorithm performance. The basis of our approach is

the scientiﬁc method: we develop hypotheses about performance, create mathematical

models, and run experiments to test them, repeating the process as necessary.

We concl ude w it h a c ase s tudy wh ere we co nsid er solu tion s to a connectivity problem

that uses algorithms and data structures that implement the classic union-ﬁnd ADT.

From <www.wowebook.com>

ptg

A l g o r i t h m s When we write a computer program, we are generally implementing a

method that has been devised previously to solve some problem. This method is often

independent of the particular programming language being used—it is likely to be

equally appropriate for many computers and many programming languages. It is the

method, rather than the computer program itself, that speciﬁes the steps that we can

take to solve the problem. The term algorithm is used in computer science to describe

a ﬁnite, deterministic, and effective problem-solving method suitable for implementa-

tion as a computer program. Algorithms are the stuff of computer science: they are

central objects of study in the ﬁeld.

We ca n de ﬁne an al gor ithm by d esc rib ing a pro cedure fo r so lvi ng a prob lem in a

natural language, or by writing a computer program that implements the procedure,

as shown at right for Euclid’s algorithm for ﬁnding the greatest common divisor of

two numbers, a variant of which was devised

over 2,300 years ago. If you are not familiar

with Euclid’s algorithm, you are encour-

aged to work Exercise 1.1.24 and Exercise

1.1.25, perhaps after reading Section 1.1. In

this book, we use computer programs to de-

scribe algorithms. One important reason for

doing so is that it makes easier the task of

checking whether they are ﬁnite, determin-

istic, and effective, as required. But it is also

important to recognize that a program in a

particular language is just one way to express

an algorithm. The fact that many of the al-

gorithms in this book have been expressed

in multiple programming languages over the

past several decades reinforces the idea that each algorithm is a method suitable for

implementation on any computer in any programming language.

Most algorithms of interest involve organizing the data involved in the computa-

tion. Such organization leads to data structures, which also are central objects of study

in computer science. Algorithms and data structures go hand in hand. In this book we

take the view that data structures exist as the byproducts or end products of algorithms

and that we must therefore study them in order to understand the algorithms. Simple

algorithms can give rise to complicated data structures and, conversely, complicated

algorithms can use simple data structures. We shall study the properties of many data

structures in this book; indeed, we might well have titled the book Algorithms and Data

Structures.

Compute the greatest common divisor of

two nonnegative integers p and q as follows:

If q is 0, the answer is p. If not, divide p by q

and take the remainder r. The answer is the

greatest common divisor of q and r.

public static int gcd(int p, int q)

{

if (q == 0) return p;

int r = p % q;

return gcd(q, r);

}

Euclid’s algorithm

Java-language description

English-language description

4 CHAPTER 1

■

Fundamentals

From <www.wowebook.com>

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When we use a computer to help us solve a problem, we typically are faced with a

number of possible approaches. For small problems, it hardly matters which approach

we use, as long as we have one that correctly solves the problem. For huge problems (or

applications where we need to solve huge numbers of small problems), however, we

quickly become motivated to devise methods that use time and space efﬁciently.

The primary reason to learn about algorithms is that this discipline gives us the

potential to reap huge savings, even to the point of enabling us to do tasks that would

otherwise be impossible. In an application where we are processing millions of objects,

it is not unusual to be able to make a program millions of times faster by using a well-

designed algorithm. We shall see such examples on numerous occasions throughout

the book. By contrast, investing additional money or time to buy and install a new

computer holds the potential for speeding up a program by perhaps a factor of only 10

or 100. Careful algorithm design is an extremely effective part of the process of solving

a huge problem, whatever the applications area.

When developing a huge or complex computer program, a great deal of effort must

go into understanding and deﬁning the problem to be solved, managing its complex-

ity, and decomposing it into smaller subtasks that can be implemented easily. Often,

many of the algorithms required after the decomposition are trivial to implement. In

most cases, however, there are a few algorithms whose choice is critical because most

of the system resources will be spent running those algorithms. These are the types of

algorithms on which we concentrate in this book. We study fundamental algorithms

that are useful for solving challenging problems in a broad variety of applications areas.

The sharing of programs in computer systems is becoming more widespread, so

although we might expect to be using a large fraction of the algorithms in this book, we

also might expect to have to implement only a small fraction of them. For example, the

Java libraries contain implementations of a host of fundamental algorithms. However,

implementing simple versions of basic algorithms helps us to understand them bet-

ter and thus to more effectively use and tune advanced versions from a library. More

important, the opportunity to reimplement basic algorithms arises frequently. The pri-

mary reason to do so is that we are faced, all too often, with completely new computing

environments (hardware and software) with new features that old implementations

may not use to best advantage. In this book, we concentrate on the simplest reasonable

implementations of the best algorithms. We do pay careful attention to coding the criti-

cal parts of the algorithms, and take pains to note where low-level optimization effort

could be most beneﬁcial.

The choice of the best algorithm for a particular task can be a complicated process,

perhaps involving sophisticated mathematical analysis. The branch of computer sci-

ence that comprises the study of such questions is called analysis of algorithms. Many

5CHAPTER 1

■

Fundamentals

From <www.wowebook.com>

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of the algorithms that we study have been shown through analysis to have excellent

theoretical performance; others are simply known to work well through experience.

Our primary goal is to learn reasonable algorithms for important tasks, yet we shall also

pay careful attention to comparative performance of the methods. We should not use

an algorithm without having an idea of what resources it might consume, so we strive

to be aware of how our algorithms might be expected to perform.

Summary of topics As an overview, we describe the major parts of the book, giv-

ing speciﬁc topics covered and an indication of our general orientation toward the

material. This set of topics is intended to touch on as many fundamental algorithms as

possible. Some of the areas covered are core computer-science areas that we study in

depth to learn basic algorithms of wide applicability. Other algorithms that we discuss

are from advanced ﬁelds of study within computer science and related ﬁelds. The algo-

rithms that we consider are the products of decades of research and development and

continue to play an essential role in the ever-expanding applications of computation.

Fundamentals (Chapter 1) in the context of this book are the basic principles and

methodology that we use to implement, analyze, and compare algorithms. We consider

our Java programming model, data abstraction, basic data structures, abstract data

types for collections, methods of analyzing algorithm performance, and a case study.

Sorting algorithms (Chapter 2) for rearranging arrays in order are of fundamental

importance. We consider a variety of algorithms in considerable depth, including in-

sertion sort, selection sort, shellsort, quicksort, mergesort, and heapsort. We also en-

counter algorithms for several related problems, including priority queues, selection,

and merging. Many of these algorithms will ﬁnd application as the basis for other algo-

rithms later in the book.

Searching algorithms (Chapter 3) for ﬁnding speciﬁc items among large collections

of items are also of fundamental importance. We discuss basic and advanced methods

for searching, including binary search trees, balanced search trees, and hashing. We

note relationships among these methods and compare performance.

Graphs (Chapter 4) are sets of objects and connections, possibly with weights and

orientation. Graphs are useful models for a vast number of difﬁcult and important

problems, and the design of algorithms for processing graphs is a major ﬁeld of study.

We con side r d epth -ﬁrs t s earch , bre adt h-ﬁr st se arch, co nnec tiv it y p ro ble ms, a nd s ev -

eral algorithms and applications, including Kruskal’s and Prim’s algorithms for ﬁnding

minimum spanning tree and Dijkstra’s and the Bellman-Ford algorithms for solving

shortest-paths problems.

6 CHAPTER 1

■

Fundamentals

From <www.wowebook.com>

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