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.
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.
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.
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 your reasoning as to why something doesn't work, you can check if it really holds up or whether you have just glossed 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 explanation for something is because your conclusion is wrong.
An important distinction to keep aware of during any design process is the difference between strategy and tactics. 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. 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 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.
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 into space, then panic, and finally end up settling for the first thing that comes to mind. Avoid this fate. Follow the sequence of questions provided below and in most of the catalog problem sections. We'll tell you what to do next!
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.
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.
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.
