At some point, you can’t get any further with linked lists, selection sort, and voodoo Big O, and you have to go get a real algorithms textbook and learn all that horrible math, at least a little. But which book? There are tons of them.

I haven’t read every algorithms book out there, but I have read four of them. Maybe my experience with these four can help guide your decision. The four books are *Algorithms*, by Dasgupta, Papadimitriou, and Vazirani (hereafter called Dasgupta); *Introduction to Algorithms*, by Cormen, Leiserson, Rivest, and Stein (hereafter called CLRS); *The Algorithm Design Manual*, by Steve Skiena (hereafter called Skiena); and *The Art of Computer Programming*, Volumes 1-3, by Donald Knuth. I’ll do a five-point comparison, going over the prose style, code use, mathematical heaviness, breadth and depth of topics, and position on the continuum between theoretical and practical of each book.

There’s one thing you should know before I start: Dasgupta is available for free online, while Knuth and CLRS aren’t (well, they probably are, but not legally). So that might make your decision for you. I haven’t been able to determine if Skiena is legally available online. The book is posted in either PDF or HTML on a few legit-ish looking sites, but Skiena’s own page for it doesn’t mention anything about it being freely available, so proceed with caution.

## Does it have to be one of these four?

Not at all. There are lots of good algorithms books that I’ve never read. You can even learn a lot by just reading CS.SE and Wikipedia. But these four are the ones I’ve personally used. Many people like one of these four, but they do reflect my taste and biases. But even if you decide to go with a different book, this overview might at least tell you what features to notice.

## Prose style

A good algorithms book will usually explain its topics in three ways: with a natural language prose explanation, with some kind of implementation code, and with mathematical notation. When you’re starting out, the prose is usually the most important part, because it takes you from “What the hell is a binary search?” to “Got it, how do I write actual code for a binary search?”

For me, the winner on prose style has to be Knuth. He’s just a masterful writer. You might have to read some of his sentences twice, but that’s only because he gets across twice as much information as a lesser writer. Knuth’s prose explanations got me to understanding on several topics I’d despaired of ever getting with other books, like B-trees and merge sort.

Skiena is also an excellent prose stylist. Where Knuth is elegant and flowing, like the John Milton of algorithms, Skiena is direct and sharp, like the Ernest Hemingway of algorithms. Knuth and Skiena are good at different things: Skiena is good at finding a simple, direct way to explain things which are traditionally made more complicated than they need to be. His overview of Big O is amazingly clear and simple, as long as you have at least some memory of calculus. On the other hand, some topics are inherently really complicated, and this is where Knuth shines: he’ll give you several different ways to view a complicated topic, and chances are at least one of them will work for you.

Knuth is much quicker to jump to math, while Skiena mostly eschews math and tries to keep everything intuitive. We’ll discuss that more in the section on mathematical heaviness.

Dasgupta has a pretty good prose style too, more patient and beginner-friendly than either Skiena or Knuth. Sometimes, I found Dasgupta too verbose and felt the authors belabored the obvious too much, but other times that was an asset rather than a weakness (e.g. in the material on Fast Fourier Transforms).

CLRS probably has the weakest prose style of these four books. It’s closer to the standard math text style, written with a formal and distant air, sometimes favoring precision over clarity. It’s strictly functional, but tolerable if you already have some understanding of the topic.

## Code use

Knuth’s big weakness is code. He uses two different notations, pseudocode and MIX assembly language, his imaginary assembly for his imaginary computer. I find both of them extremely difficult to follow.

The problems with MIX should be pretty obvious: it’s assembly language, so it’s just hard to read. Not only that, it’s pretty different from the MIPS, ARM, or x86 assembly that modern readers might have seen. It’s designed to be run on either a decimal or binary architecture and assumes a six-bit word. Knuth put a ton of effort into creating this imaginary machine and its assembly language. Since it’s made up, MIX assembly is still technically pseudocode; but MIX is to pseudocode as Quenya is to pseudo-languages. Newer editions of TAoCP use MMIX, which was redesigned to reflect modern assembly languages. I haven’t seen it yet, but I imagine it’s still hard to read since it’s assembly language.

Knuth also uses high-level pseudocode, but I find that hard to read too because it’s organized like an unstructured language with goto statements. If I were planning to implement the algorithms in an unstructured language, it would probably be fine, but I’ve always found that there’s a nontrivial translation process between Knuth’s unstructured pseudocode and structured pseudocode suitable for implementation in a modern language.

CLRS and Dasgupta both use high-level pseudocode that resembles Pascal, although not too slavishly. CLRS expresses some things as if they were methods or fields of an object, in a semi-object oriented way. Skiena also does some of this, but in addition he uses real C code.

A lot of modern books use C or Java throughout, which might be more to some people’s tastes. These books reflect my taste, and I like my algorithms language-independent. I don’t mind how Skiena uses C—he uses it mainly to show how to implement something when there’s a significant gap between theory and practice. But I’m glad he stuck to pseudocode for the most part.

## Mathematical Heaviness

In this category, going from heaviest to lightest, we have roughly Knuth > CLRS > Dasgupta > Skiena. Dasgupta and Skiena are pretty close, while there’s a significant gap between them and CLRS. There’s a minor gap between CLRS and Knuth, and it really depends on the topic, but CLRS just didn’t take on as many inherently mathematical topics as Knuth did (especially in Knuth’s Volume 2, Seminumerical Algorithms, where he deals with random number generation and efficient arithmetic).

In general, Dasgupta is geared towards people (like students) who just recently learned a little math and are going to learn a little more, but haven’t internalized it all yet. Skiena is geared towards people (like working programmers, or graduate students) who’ve learned some math and forgotten a lot of it. Skiena is also student friendly, but he stresses practicality more than anything, and sometimes, in practice, you gotta do some math.

CLRS is also for students, but it’s more at the graduate student level, especially the later “Selected Topics” chapters that can sometimes get pretty heavy. Knuth seems more geared towards graduate students near the end of their Ph.Ds and practicing researchers.

All four books require you to know some combinatorics and a little number theory, and to have at least a vague memory of limits and function growth from calculus. That’s about it for Skiena and Dasgupta, though Skiena seems to expect you to have internalized the math a little more than Dasgupta does. The basic chapters of CLRS are just a little beyond Skiena in math background, but some topics get quite heavily mathematical. Knuth gets pretty heavily mathematical in almost every topic.

I’ve been conflating two things in the discussion so far, since they’re sort of linked: actual knowledge of theorems and axioms in mathematics, and the ability to read mathematical writing and proofs and follow an author’s reasoning (what’s often called “mathematical maturity”). Knuth requires the most technical knowledge of mathematics of the four here, and he’ll also stretch your reading ability pretty far; but CLRS will also stretch your reading ability, and Dasgupta is no cakewalk either, although I do think Dasgupta qualifies as a picnic. Skiena doesn’t have any proofs, but he kind of sneaks in proof-like things with his “war stories”, which usually read quite a bit like proofs, extended with discussion of some of the approaches he tried that didn’t work out.

If you’re getting your first algorithms book and you’re a CS undergrad, Dasgupta or Skiena is easiest to follow. CLRS will stretch you a little, but it’s manageable. If you’re a math or physics undergrad, CLRS shouldn’t be too hard and Knuth might be doable.

## Breadth and Depth of Topics

Dasgupta is the loser here; not only does the book cover fewer topics than the others, the topics it chooses to cover are poorly organized and somewhat eccentric. I’m not sure why the authors threw in the awful chapter on quantum computing at the end; it’s totally incomprehensible unless you already understand quantum mechanics, which is no mean feat.

CLRS has the best balance of breadth and depth: it covers basic data structures; some more advanced ones like red-black trees, B-trees, and the union-find data structure; algorithmic paradigms like greediness and dynamic programming; graphs, shortest paths, and network flows; sorting; amortized analysis; and an assortment of other topics such as number theoretic algorithms, cryptography, linear programming, string matching, and NP completeness.

Skiena covers about the first third of CLRS, but he does a lot more with NP complete problems and how to design approximation schemes for them than CLRS does.

Knuth, of course, is the master of depth. Volume 1 covers math background and fundamental data structures; Volume 2 covers random number generation and arithmetic; Volume 3 covers searching and sorting, going through various sort routines and some more advanced data structures, such as B-trees, as well as developing the whole theory behind hash tables and how to choose hashing functions and table sizes; and Volume 4 covers combinatorial algorithms. Volume 4 is split into three subvolumes; right now, only Volume 4A has actually come out.

If you want to get just one book, I would get Skiena or CLRS. They include all the most important topics both in practice and for undergraduate classes. Skiena, as a bonus, even includes some computational geometry material, in case you’re doing video games or computer graphics.

## Theoretical vs Practical

Skiena is the most relentlessly practical of this bunch. Knuth was actually pretty practical at the time Volume 1 came out, but he became less and less practical as time went on because people stopped writing things in assembly or using tape drives. Both give you implementation tips to make your code perform better, although Knuth’s can be hard to capitalize on since you’re not writing in assembly.

CLRS and Dasgupta are both theoretical in the sense that they mostly ignore the computer. Everything in CLRS will work, but sometimes their approach is too “straight from the theory”, as I discovered when I implemented Karp-Rabin according to their discussion and put it on Code Review.SE after struggling with numerous overflow-related bugs, only to have someone suggest a different approach that rectified all the performance issues and handled overflow elegantly.

Dasgupta and CLRS are both still good books, and if you’re just starting out learning algorithms, don’t get too caught up on this issue. Write your code in Python or Ruby, some quick and easy language that also ignores the metal. You’ll get the idea of the implementation, and you can deal with the issues around implementing it in some other language later. (Python even looks like the pseudocode in CLRS.)

## Conclusion

I probably use CLRS the most, because of its breadth of coverage and because its pseudocode is so much easier to follow than Knuth’s. If I don’t understand the concept or the mathematical underpinnings, I’ll go to Knuth for the deep discussion, and the excellent prose style.

I just got Skiena recently, but in the future, I expect him to usurp CLRS somewhat. He covers most of the same material, but his prose style is more direct and his approach is more practical. Skiena is excellently organized and has a catalogue of problems and their algorithmic solutions in the second half of his book, good for browsing when you’re stumped.

I don’t use Dasgupta that much. Dasgupta was the text for my undergrad algorithms class, and while it was good in that capacity, it’s not really a book that continues to be useful once you’re past the first-semester course, mainly because of the lack of breadth in coverage and the eccentric organization and choice of topics.