For the past several years mathematics majors in the computing track at the University of Pennsylvania have taken a course in continuous algorithms (numerical analysis) in the junior year, and in discrete algorithms
in the senior year. This book has grown out of the senior course as I have been teaching it recently. It has also been tried out on a large class of computer science and mathematics majors, including seniors and graduate students, with good results.

Selection by the instructor of topics of interest will be very important, because normally I’ve found that I can’t cover anywhere near all of this material in a semester. A reasonable choice for a first try might
be to begin with Chapter 2 (recursive algorithms) which contains lots of motivation. Then, as new ideas are needed in Chapter 2, one might delve into the appropriate sections of Chapter 1 to get the concepts
and techniques well in hand. After Chapter 2, Chapter 4, on number theory, discusses material that is extremely attractive, and surprisingly pure and applicable at the same time. Chapter 5 would be next, since
the foundations would then all be in place. Finally, material from Chapter 3, which is rather independent of the rest of the book, but is strongly connected to combinatorial algorithms in general, might be studied
as time permits.

Throughout the book there are opportunities to ask students to write programs and get them running. These are not mentioned explicitly, with a few exceptions, but will be obvious when encountered. Students
should all have the experience of writing, debugging, and using a program that is nontrivially recursive, for example. The concept of recursion is subtle and powerful, and is helped a lot by hands-on practice. Any of the algorithms of Chapter 2 would be suitable for this purpose. The recursive graph algorithms are particularly recommended since they are usually quite foreign to students’ previous experience and therefore have great learning value.

In addition to the exercises that appear in this book, then, student assignments might consist of writing occasional programs, as well as delivering reports in class on assigned readings. The latter might be found
among the references cited in the bibliographies in each chapter.

I am indebted first of all to the students on whom I worked out these ideas, and second to a number of colleagues for their helpful advice and friendly criticism. Among the latter I will mention Richard Brualdi, Daniel Kleitman, Albert Nijenhuis, Robert Tarjan and Alan Tucker. For the no-doubt-numerous
shortcomings that remain, I accept full responsibility.

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