I've read both books, and I think that they're aimed towards different audiences. CLRS is aimed towards computer scientists. It offers a fairly rigorous treatment of algorithms, mathematically proving correctness and performance. The Algorithm Design Manual, on the other hand, is aimed at programmers who are willing to take claims of performance and correctness at face value if it leaves room to discuss further optimization and compositions of algorithms into solutions for problems.

EDIT: Though, that's not to say I wholly endorse CLRS. I think Knuth's The Art Of Computer Programming is more detailed and more mathematically rigorous than CLRS. Unfortunately TAoCP is multiple volumes, while CLRS manages to be a single (substantial) volume.

I can't agree more with your judgement. I've read all these three books and the fact is just like what you mentioned. CLRS and TAoCP use a lot of space to prove the algorithms correct(especially TAoCP, it even uses different ways to prove one algorithm) and that makes me feel dizzy. Perhaps I'm just a programmer not a computer scientist.

But I have to say that use some time to read CLRS and TAoCP is worthwhile. It will help u know more about algorithms and avoid the condition that just know how but not know why.

I agree with your analysis. From my review of Skiena (http://www.mccaughan.org.uk/g/books/alg-design.html): "Put differently, this is a book for algorithmic engineers rather than for algorithmic scientists." And: "You really wouldn't want this to be your only algorithms book. It doesn't have enough in it, it doesn't go deeply enough into anything, and a certain amount of the bracing mathematical rigour that the book so conspicuously lacks is ... well, good for the soul, I suppose. So go buy Cormen, Leiserson & Rivest. Then, unless you're already an expert algorist, buy this one too."