如何设计算法

How to Design Algorithms

Designing the right algorithm for a given application is a difficult job. It requires a major creative act, taking a problem and pulling a solution out of the ether. This is much more difficult than taking someone else's idea and modifying it or tweaking it to make it a little better. The space of choices you can make in algorithm design is enormous, enough to leave you plenty of freedom to hang yourself.

设计一个正确的算法是一件困难的工作，因为它需要创新，从以太真空中发掘出一个解方案来解决问题。算法设计比对现有的方案进行改良要难得多，因为算法设计的可选择空间太，过多的自由反而成了一种约束。

This book is designed to make you a better algorithm designer. The techniques presented in Part I of this book provide the basic ideas underlying all combinatorial algorithms. The problem catalog of Part II will help you with modeling your application and point you in the right direction of an algorithm or implementation. However, being a successful algorithm designer requires more than book knowledge; it requires a certain attitude, the right problem-solving approach. It is difficult to teach this mindset in a book; yet getting it is essential to become a successful designer.

本书的设计目标是让你成为一个更好的算法设计者。本书第一部分展示有关组合算法的基本原理和基本思想；第二部分的问题清单帮助你为你的问题建模，并且为你指明实现正确算法的方向。尽管如此，要成为一个成功的算法设计者光有书本知识是不够的，面对问题的态度（attitude）和选择正确的方法更重要。书本容易传授知识，很难传授人的心态（mindset)和思考方式；而这种心态和思考却是成为成功的算法设计者的根本条件。

The key to algorithm design (or any other problem-solving task) is to proceed by asking yourself a sequence of questions to guide your thought process. What if we do this? What if we do that? Should you get stuck on the problem, the best thing to do is move onto the next question. In any group brainstorming
session, the most useful person in the room is the one who keeps asking, ``Why can't we do it this way?'' not the person who later tells them why. Because eventually she will stumble on an approach that can't be shot down.

算法设计（或其它问题解决任务）的关键是一系列持续的自我反问，这些反问引导我们思维的前进。“如果这样做会怎样？”，“如果那样做又会怎样？”……如果 你被一个问题掐住了，最好的办法就是先搁一下，换一个问题换一个前进的方向试试。在每组头脑风暴会议中，最有价值的人是不断提出为什么的人，不是尔后解说为什么的人。因为我们常常被一些习以为常的东西所拌倒，掉进自己设置的陷阱。

kemin：如果问题解决是一种思考过程，那么思考的形式（过程的严谨性、细致性和正确性）很重要，而思考的内容也不容忽视。因为引导我们思考前进的方式 除反问本身外，反问的内容也很重。就比如参加头脑风暴的材料一样。人大脑的思维功能是硬编码的，人与人之间没有思维规律——质的区别，只是思维的清晰度和 灵敏度——量的差别。人与人之间智力的差别更多体现在思维内容的量上，体现在对外部世界的事实掌握的广度和深度上。

Towards this end, we provide below a sequence of questions to guide your search for the right algorithm for your problem. To use it effectively, you must not only ask the questions, but answer them. The key is working through the answers carefully, by writing them down in a log. The correct answer to, ``Can I do
it this way?'' is never ``no,'' but ``no, because ....'' By clearly articulating（明确有力地表达） your reasoning as to why something doesn't work, you can check if it really holds up or whether you have just glossed（掩盖） over a possibility that you didn't want to think hard enough about. You will be surprised how often the reason you can't find a convincing（使人信服的） explanation for something is because your conclusion is wrong.

在末尾我们提供一个反问问题的列表，你不但要反问自己这些问题，更重要是仔细回答这些问题，最好把答案写下来。回答诸如问题“我可以使用这种方式吗？”的 不是一个“不能”就完了，而是“不能，因为……”。通过仔细明确的回答“为什么不能”时，你会发现到底是“真的不能“，还是只是你自己不愿意去深入思考掩 盖了”能“。如果你不曾训练出严谨的思考方式，当你这样做时你会惊讶的发现，为了说明某些东西但却找不到一个令人信服的解释的原因常常是因为你的结论本身 是错的。

An important distinction to keep aware of during any design process is the difference between strategy and tactics（战略）. Strategy represents the quest for the big picture, the framework around which we construct our path to the goal. Tactics are used to win the minor battles we must fight along the way. In problem
solving, it is important to check repeatedly whether you are thinking on the right level. If you do not have a global strategy of how you are going to attack your problem, it is pointless to worry about the tactics.

在设计过程中特别重要区分策略和战略的概念。策略是对全局的一个探索，一个构筑通向目标路径的指导框架。战略则是用来解决通向大目标过程的较小的问题。如果你对关于如何对付所面临的问题没有一个全局的策略，那关心战略是不得要领的，予事无补的。在解题领域，不断修正思维的层次（thinking on the right level）是很重要战略。
（--莱布尼兹曾经将人的解题思考过程比喻成晃筛子，把脑袋里面的东西都给抖落出来，然后正在搜索的注意力会抓住一切细微的、与问题有关的东西。事实上，要做到能够令注意力抓住这些有关的东西，就必须时刻将问题放在注意力层面，否则即使关键的东西抖落出来了也可能没注意到。）

An example of a strategic question is, ``How best can I model my application as a graph algorithm problem?'' A tactical question might be, ``Should I use an adjacency邻接 list or adjacency matrix data structure to represent my graph?'' Of course, such tactical decisions are critical to the ultimate quality of the solution, but they can be properly evaluated only in light of a successful strategy.

一个策略问题的例子是：“我如何才能更好地把我的问题建模成图问题？”。而一个战略问题可能是这样：“我是用邻接列表还是邻接矩阵来实现我的图结构？”。当然，这种战略选择是对解决方案的最终质量起着重要作用；不过战略价值的体现还是基于正确的策略的选择。

When faced with a design problem, too many people freeze up in their thinking. After reading or hearing the problem, they sit down and realize that they don't know what to do next. They stare（凝视） into space, then panic（惊惶）, and finally end up settling（沉淀; 决定） for the first thing that comes to mind. Avoid this fate（天数; 运气; 命运 ）. Follow the sequence of questions provided below and in most of the catalog problem sections. We'll tell you what to do next!

初学者在面对问题时常常表现出思维凝滞、手足无措和盲目解题。参考以下的反问问题列表和本书的问题清单，我们告诉你应该怎么做。

Obviously, the more experience you have with algorithm design techniques such as dynamic programming, graph algorithms, intractability, and data structures, the more successful you will be at working through the list of questions. Part I of this book has been designed to strengthen this technical background. However, it pays to work through these questions regardless of how strong your technical skills are. The earliest and most important questions on the list focus on obtaining a detailed understanding of the problem and do not require specific expertise.

当然本反问问题列表对读者有背景要求，要求读者对算法设计技术（动态规划、图算法、难解性和数据结构）的熟悉程度。本书第一部分的目标就是对这些技术背景进行强化。不过，不管你的技术背景怎样，通读这些问题对你解题还是很有裨益的。

This list of questions was inspired by a passage in that wonderful book about the space program The Right Stuff [Wol79]. It concerned the radio transmissions from test pilots just before their planes crashed.
One might have expected that they would panic, so that ground control would hear the pilot yelling Ahhhhhhhhhhh --, terminated only by the sound of smacking into a mountain. Instead, the pilots ran through a list of what their possible actions could be. I've tried the flaps. I've checked the engine. Still got
two wings. I've reset the --. They had ``the Right Stuff.'' Because of this, they sometimes managed to miss the mountain.

I hope this book and list will provide you with ``the Right Stuff'' to be an algorithm designer. And I hope it prevents you from smacking into any mountains along the way.

1. Do I really understand the problem?

你真的完全理解你的问题吗？

1. What exactly does the input consist of?
2. What exactly are the desired results or output?
3. Can I construct an example input small enough to solve by hand? What happens when I try to solve it?
4. How important is it to my application that I always find an exact, optimal answer? Can I settle for something that isusually pretty good?
5. How large will a typical instance of my problem be? Will I be working on 10 items? 1,000 items? 1,000,000 items?
6. How important is speed in my application? Must the problem be solved within one second? One minute? One hour? One day?
7. How much time and effort can I invest in implementing my algorithm? Will I be limited to simple algorithms that can be coded up in a day, or do I have the freedom to experiment with a couple of approaches and see which is best?
8. Am I trying to solve a numerical problem? A graph algorithm problem? A geometric problem? A string problem? A set problem? Might my problem be formulated in more than one way? Which formulation seems easiest?

2. Can I find a simple algorithm or heuristic for the problem?

我能够为问题找到一个简单的算法或试错式算法吗？

1. Can I find an algorithm to solve my problem correctly by searching through all subsets or arrangements and picking the best one?
   1. If so, why am I sure that this algorithm always gives the correct answer?
   2. How do I measure the quality of a solution once I construct it?
   3. Does this simple, slow solution run in polynomial or exponential time? Is my problem small enough that this brute-force solution will suffice?
   4. If I can't find a slow, guaranteed correct algorithm, why am I certain that my problem is sufficiently well-defined to have a correct solution?
2. Can I solve my problem by repeatedly trying some simple rule, like picking the biggest item first? The smallest item first? A random item first?
   1. If so, on what types of inputs does this heuristic work well? Do these correspond to the data that might arise in my application?
   2. On what types of inputs does this heuristic work badly? If no such examples can be found, can I show that it always works well?
   3. How fast does my heuristic come up with an answer? Does it have a simple implementation?

3. Is my problem in the catalog of algorithmic problems in the back of this book?

我的问题有在本书问题清单中吗？

•  
  1. If it is, what is known about the problem? Is there an implementation available that I can use?
  2. If I don't see my problem, did I look in the right place? Did I browse through all the pictures? Did I look in the index under all possible keywords?
  3. Are there relevant resources available on the World-Wide Web? Did I do a Lycos, Alta Vista, or Yahoo search? Did I go to the WWW page associated with this book, ?

4. Are there special cases of the problem that I know how to solve exactly?

问题的特殊情况我以前解决过吗？

•  
  1. Can I solve the problem efficiently when I ignore some of the input parameters?
  2. What happens when I set some of the input parameters to trivial values, such as 0 or 1? Does the problem become easier to solve?
  3. Can I simplify the problem to the point where I can solve it efficiently? Is the problem now trivial or still interesting?
  4. Once I know how to solve a certain special case, why can't this be generalized to a wider class of inputs?
  5. Is my problem a special case of a more general problem in the catalog?

5. Which of the standard algorithm design paradigms are most relevant to my problem?

哪一个标准算法设计模式可用于我的问题？

•  
  1. Is there a set of items that can be sorted by size or some key? Does this sorted order make it easier to find the answer?
  2. Is there a way to split the problem in two smaller problems, perhaps by doing a binary search? How about partitioning the elements into big and small, or left and right? Does this suggest a divide-and-conquer algorithm?
  3. Do the input objects or desired solution have a natural left-to-right order, such as characters in a string, elements of a permutation, or the leaves of a tree? If so, can I use dynamic programming to exploit this order?
  4. Are there certain operations being repeatedly done on the same data, such as searching it for some element, or finding the largest/smallest remaining element? If so, can I use a data structure to speed up these queries? What about a dictionary/hash table or a heap/priority queue?
  5. Can I use random sampling to select which object to pick next? What about constructing many random configurations and picking the best one? Can I use some kind of directed randomness like simulated annealing in order to zoom in on the best solution?
  6. Can I formulate my problem as a linear program? How about an integer program?
  7. Does my problem seem something like satisfiability, the traveling salesman problem, or some other NP-complete problem? If so, might the problem be NP-complete and thus not have an efficient algorithm? Is it in the problem list in the back of Garey and Johnson [GJ79]?

6. Am I still stumped?

•  
  1. Am I willing to spend money to hire an expert to tell me what to do? If so, check out the professional consulting services mentioned in Section .
  2. Why don't I go back to the beginning and work through these questions again? Did any of my answers change during my latest trip through the list?

Problem solving is not a science, but part art and part skill. It is one of the skills most worth developing. My favorite book on problem solving remains Pólya's How to Solve It [Pol57], which features a catalog of problem solving techniques that are fascinating to browse through, both before and after you have a
problem.