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Questions tagged [reference-request]

Reference-request is used when the author needs to know about work related to the question.

557
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6answers
114k views

What's new in purely functional data structures since Okasaki?

Since Chris Okasaki's 1998 book "Purely functional data structures", I haven't seen too many new exciting purely functional data structures appear; I can name just a few: IntMap (also invented by ...
223
votes
37answers
107k views

What Books Should Everyone Read?

[Timeline] This question has the same spirit of what papers should everyone read and what videos should everybody watch. It asks for remarkable books in different areas of theoretical computer ...
110
votes
15answers
15k views

What Lecture Notes Should Everyone Read?

There has been several questions with the same scheme as this one: What papers should everyone read What books should everyone read What are the recent TCS books whose drafts are available online ...
108
votes
17answers
7k views

Examples of the price of abstraction?

Theoretical computer science has provided some examples of "the price of abstraction." The two most prominent are for Gaussian elimination and sorting. Namely: It is known that Gaussian elimination ...
97
votes
37answers
13k views

What are the recent TCS books whose drafts are available online?

Following the post What Books Should Everyone Read, I noticed that there are recent books whose drafts are available online. For instance, the Approximation Algorithms entry of the above post cites ...
78
votes
41answers
15k views

Funny TCS-related papers etc?

What is the funniest TCS-related published work you know? Please include only those that are intended to be funny. Works which are explicitly crafted to be intelligently humorous (rather than, say, ...
73
votes
20answers
8k views

Examples of “Unrelated” Mathematics Playing a Fundamental Role in TCS?

Please list examples where a theorem from mathematics which was not normally considered to apply in computer science was first used to prove a result in computer science. The best examples are those ...
71
votes
2answers
19k views

What is the actual time complexity of Gaussian elimination?

In an answer to an earlier question, I mentioned the common but false belief that “Gaussian” elimination runs in $O(n^3)$ time. While it is obvious that the algorithm uses $O(n^3)$ arithmetic ...
67
votes
13answers
18k views

Uses of algebraic structures in theoretical computer science

I'm a software practitioner and I'm writing a survey on algebraic structures for personal research and am trying to produce examples of how these structures are used in theoretical computer science (...
64
votes
11answers
4k views

What are good references to understanding the proof of the PCP theorem?

I'm familiar with a lot of results that use the PCP theorem (mainly in approximating algorithms), but I've never come across a clear explanation of the PCP theorem (ie, that $\mathsf{NP} = \mathsf{PCP}...
61
votes
14answers
11k views

Applications of topology to computer science

I'd like to write a survey on the applications of Topology in Computer Science. I plan to cover the history of topological ideas in Computer Science and also highlight a few current developments. It ...
60
votes
17answers
3k views

Applications of TCS to classical mathematics?

We in TCS often use powerful results and ideas from classical mathematics (algebra, topology, analysis, geometry, etc.). What are some examples of when it has gone the other way around? Here ...
54
votes
2answers
3k views

Can one amplify P=NP beyond P=PH?

In Descriptive Complexity, Immerman has Corollary 7.23. The following conditions are equivalent: 1. P = NP. 2. Over finite, ordered structures, FO(LFP) = SO. This can be thought of as "...
52
votes
7answers
3k views

For which problems in P is it easier to verify the result than to find it?

For (search versions) of NP-complete problems, verifying a solution is clearly easier than finding it, since the verification can be done in polynomial time, while finding a witness takes (probably) ...
44
votes
4answers
3k views

Generalized Ladner's Theorem

Ladner's Theorem states that if P ≠ NP, then there is an infinite hierarchy of complexity classes strictly containing P and strictly contained in NP. The proof uses the completeness of SAT under many-...
43
votes
5answers
2k views

Historical reasons for adoption of Turing Machine as primary model of computation.

It's my understanding that Turing's model has come to be the "standard" when describing computation. I'm interested to know why this is the case -- that is, why has the TM model become more widely-...
40
votes
12answers
2k views

Gröbner bases in TCS?

Does anyone know of interesting applications of Gröbner bases to theoretical computer science? Gröbner bases are used to solve multi-variate polynomial equations, an NP-hard problem in general. I was ...
40
votes
6answers
2k views

Which model of computation is “the best”?

In 1937 Turing described a Turing machine. Since then many models of computation have been decribed in attempt to find a model which is like a real computer but still simple enough to design and ...
39
votes
18answers
4k views

What hierarchies and/or hierarchy theorems do you know?

I am currently writing a survey on hierarchy theorems on TCS. Searching for related papers I noticed that hierarchy is a fundamendal concept not only in TCS and mathematics, but in numerous sciences, ...
39
votes
7answers
4k views

Truly random number generator: Turing computable?

I am seeking a definitive answer to whether or not generation of "truly random" numbers is Turing computable. I don't know how to phrase this precisely. This StackExchange question on "efficient ...
39
votes
3answers
3k views

Is the integer factorization problem harder than RSA factorization: $n = pq$?

This is a cross-post from math.stackexchange. Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers $e_i \in \mathbb{N},$ such that $...
38
votes
1answer
2k views

Prerequisite for learning GCT

It seems that Geometric Complexity Theory requires much knowledge of pure maths such as algebraic geometry, representation theory. While I am a CS student and do NOT have classes of very abstract ...
37
votes
9answers
4k views

References for TCS proof techniques

Are there any references (online or in book form) that organize and discuss TCS theorems by proof technique? Garey and Johnson do this for the various kinds of widget constructions needed for NP-...
37
votes
6answers
2k views

Geometric problems that are NP-complete in $R^3$ but tractable in $R^2$?

A number of geometric problems are easy when considered in $R^1$, but are NP-complete in $R^d$ for $d\geq2$ (including one of my favourite problems, unit disk cover). Does anyone know of a problem ...
37
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7answers
6k views

What do we know about provably correct programs?

The ever increasing complexity of computer programs and the increasingly crucial position computers have in our society leaves me wondering why we still don't collectively use programming languages in ...
37
votes
5answers
1k views

Results in Theoretical CS independent of ZFC

I'm going to ask a quite vague question, since the borderline between theoretical computer science and math is not always easy to distinguish. QUESTION: Are you aware of any interesting result in CS ...
36
votes
3answers
1k views

Is there a backup/replacement for the Complexity Zoo?

This is a non-technical question, but certainly relevant for the TCS community. If considered inappropriate, feel free to close. The Complexity Zoo webpage (http://qwiki.stanford.edu/index.php/...
36
votes
5answers
1k views

Complexity of testing for a value versus computing a function

In general we know that the complexity of testing whether a function takes a particular value at a given input is easier than evaluating the function at that input. For example: Evaluating the ...
35
votes
1answer
2k views

Multiplying n polynomials of degree 1

The problem is to compute the polynomial $(a_1 x + b_1) \times \cdots \times (a_n x + b_n)$. Assume that all coefficients fit in a machine word, i.e. can be manipulated in unit time. You can do $O(n \...
33
votes
3answers
2k views

complexity of greatest common divisor (gcd)

Consider the following counting problem (or the associated decision problem): Given two positive integers encoded in binary, compute their greatest common divisor (gcd). What is the smallest ...
32
votes
14answers
5k views

Book on Probability

While I have passed some courses on probability theory, both in the high school and the university, I have a hard time reading TCS papers when it comes to probability. It seems that the authors of ...
32
votes
11answers
22k views

Books on automata theory for self-study

I need a finite automata theory book with lots of examples that I can use for self-study and to prepare for exams.
32
votes
7answers
2k views

Algorithmic lens in the social sciences

Looking at questions through the algorithmic lens (i.e. from an algorithmic or complexity point of view) has become useful in disciplines outside of the 'standard domain' of computer science. In ...
32
votes
1answer
1k views

Does LOGLOG = NLOGLOG?

Define LOGLOG as the class of languages which can be computed in space O(loglog n) by a deterministic Turing machine (with two-way access to the input). Similarly define NLOGLOG as the class of ...
31
votes
7answers
7k views

Books on programming language semantics

I've been reading Nielson & Nielson's "Semantics with Applications", and I really like the subject. I'd like to have one more book on programming language semantics -- but I really can get only ...
30
votes
2answers
3k views

Quantum matrix multiplication?

It doesn't seem like this is known - but are there any interesting lower bounds on the complexity of matrix multiplication in the quantum computing model? Do we have any intuition that we can beat ...
29
votes
2answers
820 views

Polynomial method for complexity results

Polynomial methods, say Combinatorial Nullstellensatz and Chevalley–Warning theorem are powerful tools in additive combinatorics. By representing a problem with proper polynomials, they can guarantee ...
29
votes
0answers
881 views

Does $EXP\neq ZPP$ imply sub-exponential simulation of BPP or NP?

By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary. I think this is a proof: if $EXP\...
28
votes
6answers
2k views

Alternative proofs of Schwartz–Zippel lemma

I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz. Are there any other ...
28
votes
2answers
957 views

Kolmogorov's conjecture that $P$ has linear-size circuits

In his book, Boolean Function Complexity, Stasys Jukna mentions (page 564) that Kolmogorov believed that every language in P has circuits of linear size. No reference is mentioned and I couldn't find ...
27
votes
10answers
2k views

Probabilistic (randomized) algorithms before “modern” computer science appeared

Edit: I choice the answer with highest score by December 06, 2012. This is a soft question. The concept of (deterministic) algorithms dates back to BC. What about the probabilistic algorithms? In ...
27
votes
6answers
932 views

Well known classes of boolean formulas that require exponentially long resolution proofs

You might often find cutting plane methods, variable propagation, branch and bound, clause learning, intelligent backtracking or even handwoven human heuristics in SAT solvers. Yet for decades the ...
27
votes
5answers
2k views

Ecology and evolution through the algorithmic lens

The study of ecology and evolution is becoming increasingly more mathematical, but most of the theoretical tools seem to be coming from physics. However, in many cases the problems have a very ...
27
votes
1answer
801 views

Other applications of Karger-Stein branching amplification?

I just taught the Karger-Stein randomized mincut algorithm in my graduate algorithms class. This is a real algorithmic gem, so I can't not teach it, but it always leaves me frustrated, because I don'...
26
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3answers
1k views

Succinct Problems in $\mathsf{P}$

The study of Succinct representation of graphs was initiated by Galperin and Wigderson in a paper from 1983, where they prove that for many simple problems like finding a triangle in a graph, the ...
26
votes
3answers
1k views

Translating SAT to HornSAT

Is it possible to translate a boolean formula B into an equivalent conjunction of Horn clauses? The Wikipedia article about HornSAT seems to imply that it is, but I have not been able to chase down ...
26
votes
4answers
1k views

Compendium of the Best Approximation and Hardness Results for NP optimization problems

Do you know any up-to-date wiki dedicated to NP optimization problems with their best approximation and hardness result? Based on the feedback, it seems that it is safe to assume there is not such a ...
26
votes
1answer
851 views

Coloring complexity of graphs

Suppose $G$ is a graph with coloring number $d = \chi(G)$. Consider the following game between Alice and Bob. At each round, Alice picks a vertex, and Bob answers with a color in $\{1,\ldots,d-1\}$ ...
26
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1answer
1k views

Who first proposed using $x^2+y^2 < 1$ Monte Carlo algorithm to calculate Pi?

I'm sure everybody knows of Buffon's needle experiment in the 18th century, that is one of the first probabilistic algorithms to calculate $\pi$. The implementation of the algorithm in computers ...
25
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5answers
2k views

Subexponentially solvable hard graph problems

In light of the recent result of Arora, Barak, and Steurer, Subexponential Algorithms for Unique Games and Related Problems, I'm interested in graph problems that have subexponential time algorithms ...