Computational Complexity: A Modern Approach 1st Edition by Sanjeev Arora (PDF)

Description

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Quality of paperback version is not good: has printing errors (all ≠ signs such as in P≠NP are printed as =, with some artifact lines above); two of the pages were stuck together along the right edge and had to be manually cut apart. Otherwise, in terms of content, good textbook. It’s good enough to study from.

⭐This is a very comprehensive text containing many results. However the amount of typos is astounding, and when appearing in the middle of a proof, it often takes a lot of extra effort on the part of the reader to figure out if there is some subtlety afoot or if it’s just a dumb copy-paste error that made it past the editor. For example, the proof of Savitch’s theorem begins as follows:”…Let L in NSPACE(S(n)) be a language decided by a TM M such that for every x in {0,1}^n, the configuration graph G = G_{M,x} has at most M = 2^O(S(n)) vertices…”So wait, is M a TM deciding L or is it a power of 2? As a newbie to the theory it took me a while to figure out that this was ridiculous (I mean, at face value it seems ridiculous but was it ACTUALLY ridiculous? As it turns out, yes. It was ridiculous).What’s more, the proofs in general are very difficult to follow, but it’s my understanding that this is in part just the style of the material, so this isn’t factored into my 3 star review. However, on more than one occasion I’ve seen much more elegant approaches to some of the proofs here (nicer reductions, etc) that make life a lot nicer. In future editions it would be nice to see some effort going into making the proofs more elegant and getting rid of the gross amount of typos. A math text should NOT leave the reader wondering if the proof that they just read is actually valid. I trust the statements of the theorems but after almost 100 pages of reading I have learned almost nothing about how to actually prove them.To end on a positive note, the motivation behind a lot of the text is very nice, and with some cleaning up this could easily be a 5 star review. Further, there are no other texts trying to do what Arora and Barak are doing (namely, compiling all modern results in Complexity Theory), making this a very valuable resource. I may start looking at original material and using this as a source to structure the theory at a high level, leaving lower level verification to other sources – from that point of view, this text is very well suited.Wish I could rate this higher, I really want to like it!

⭐For the most part, Arora and Barak provide a relatively solid introduction to complexity; however, as a previous reviewer noted, the number of typos is astounding. In some cases, it is difficult to discern whether a typo is just a typo or a mistake on the part of the authors. I expected a higher standard of editing. There is also a great deal of hand-waving in the proofs, which is incredibly frustrating. A textbook should explain proofs, not hint at their existence and leave the more challenging aspects of the proof to the reader.I might suggest purchasing a supplemental text if this is the only book for your course. If you cannot, then read carefully, and always check the arithmetic and individual steps presented in proofs.

⭐I’m more of an engineer than mathematician (and not much of an engineer) so enrolling in a 4th-year undergrad/graduate course in computational complexity was probably a mistake. I find this book readable until there are mathematical statements like this definition:For every function T : {maps natural numbers to natural numbers} and L {subset of} {0, 1}*, we say that L {is an element of} NTIME(T(n)) if there is aconstant c > 0 and a cT (n)-time NDTM M such that for every x {element of} {0, 1}*, x {is an element of} L {if and only if} M(x) = 1(where the stuff in {braces} is written in mathematical notation that won’t work in this review). I wish that such statements were accompanied by a conversational English translation. Otherwise the English is quite readable.The Kindle version has difficulty with the notation, which makes me recommend against. The Bookerly font displays most of the symbols correctly, but chokes on lower-case phi, among others. Worse, negation bars don’t appear at all, so the thing is just an exercise in frustration. The other fonts are utterly useless, with broken symbols all over some pages – but at least you won’t get fooled.

⭐The contents of the book are certainly up-to-date and the presentation is quite nice. The book assumes some prior knowledge in the field and presents its contents based on that, so it’s a good read for people with good proof/math background. I liked that the book spared a good space to present some non-traditional complexity topics in it like cryptography. Apart from that issue of some typos, I’d recommend it for its simplicity and elegant way of presentation. It’s a book to have in your library.

⭐Not recommended as a first introduction to computational complexity. But after a thorough reading of Sipser’s book “introduction to computational complexity” it prepares you for the next level. It provides an amazing amount of information.

⭐Text is [relatively] short and sweet, to the point. But definitely not a introductory textbook. If you’re just starting out on complexity and automata, get Sipser’s book instead!

⭐Really good intro to CC, updated and rigorous enough.

⭐This book would be quite good as a text book, but I did not really get on with the style or the presentations of the arguments. It is however a modern text book and has good coverage of the topics.

⭐I had used this book when I took the Computational Complexity Theory course at college and I loved this bookNow I’m working on my personal library and this title is a mustWell written, a bit physically damaged but nothing to regretImportant: I had to bought this book twice because the first time it came broken. Is incredibly how Amazon doesn’t guarantee and verify the quality of its products before shipping

⭐In the book “Computational Complexity”, the exposition of topics chosen is simple and clear. Most of the chapters contain important results that are only stated and left for the reader to prove. This is not easy because every result in this subject uses non-trivial techniques or that the proof borrows subtle ideas from other subjects like Probability or Combinatorics. Unfortunately some of the results mentioned in this book have been verified to be incorrect. A corrigendum is eagerly expected to be supplied by the authors.

⭐There are some mistakes and typos nevertheless a good text on the topic

⭐boring book

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