Computational complexity, a part of theoretical computer science that deals with understanding how efficiently a problem can be solved.

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Is factoring polynomials as hard as factoring integers?

There seems to be a consensus that factorization of integers is hard (in some precise computational sense.) Is it known whether polynomial factorization is computationally easy or hard? One thing I ...
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311 views

Finding the asymptotic behavior of the recurrence $T(n)=4T(\frac{n}{2})+n^2$ by using substitution method

I am trying to solve a recurrence by using substitution method. The recurrence relation is: $$T(n)=4T\left(\frac{n}{2}\right)+n^2$$ My guess is $T(n)$ is $\Theta (n\log n)$ (and I am sure about it ...
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What is the significance of the graph isomorphism problem?

It seems that graph isomorphism is an overwhelmingly interesting problem, particularly computationally. Why is that? What are the (theoretical and practical) implication of the existence of an ...
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30 views

Any problem computable in $k$ memory slots can be computed with polynomials.

Let our memory slots be represented by elements of $\Bbb{Z}_p$ for a prime $p$. $k$ memory slots would be $k$ copies of the ring: $R = (\Bbb{Z}_p)^k$. Suppose that for a problem $f : X \to Y$, ...
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Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
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$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
31
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763 views

What is the *middle* digit of $3^{100000}$?

The decimal representation of $3^{100000}$ has $47713$ digits. What is the $23857^{th}$ digit - i.e. the one in the $10^{23856}$'s place? There are lots of questions on this site asking for the ...
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11k views

What is the time complexity of Euclid's Algorithm (Upper bound,Lower Bound and Average)?

I looked it up online in many sites but none give a clear answer. They all give a lot of complicated mathematical stuff which is not only hard for me to grasp but also irrelevant as I simply want to ...
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300 views

Why isn't NP = coNP?

Suppose a language L is in NP. I think that means a nondeterministic Turing machine M can decide it in polynomial time. But then shouldn't it be in co-NP, because can't we define a new Turing machine ...
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Divisor summatory function for squares

The Divisor summatory function is a function that is a sum over the divisor function. $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor ...
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number of strictly increasing sequences of length $K$ with elements from $\{1, 2,\cdots,N\}$?

What is the number of strictly incremental sequences of length $K$ with elements from $\{1, 2,\cdots,N\}$ ? Is there any exact value? How about approximations?
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147 views

Finding the smallest set on which a group acts faithfully

Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the ...
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Minimum distance of a binary linear code

I need to find parameters $n$, $k$ and $d$ of a binary linear code from its Generator Matrix. How can I find parameter $d$ efficiently? I know the method that compute all the codewords and take ...
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515 views

List of calculation rules for asymptotic notation?

Background: I am working my way through CLR/CLRS's proof of the master theorem (section 4.4 in the 1st and 2nd editions of Introduction to Algorithms), and I'm doing my own write-up of this proof1 ...
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118 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
8
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684 views

Why does Strassen's algorithm work for $2\times 2$ matrices only when the number of multiplications is $7$?

I have been reading Introduction to Algorithms by Cormen. Before explaining Strassen algorithm the book says this: Strassen’s algorithm is not at all obvious. (This might be the biggest ...
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1answer
195 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
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134 views

How can the following language be determined in polynomial time

I'd love your help with understanding why the following is decidable and can be determinate in polynomial time ($L \in P$). $L=\{(\langle M \rangle,w)|M$ is a Turing machine with Q states and one ...
6
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1answer
941 views

Why is Dantzig's solution to the knapsack problem only approximate

For a bunch of items with values $v_i$ and weights $w_i$, and with a total weight $W$ that our bag can carry, how do we achieve maximum total value without breaking the bag? Dantzig proposed that we ...
4
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1answer
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Upper bound for $T(n) = T(n - 1) + T(n/2) + n$ with recursion-tree [duplicate]

I'm reading through Introduction to Algorithms, 3rd ed. and I got stuck on the following recurrence (exercise 4.4-5): $$T(n) = T(n - 1) + T(n/2) + n$$ The exercise asks you to find the upper bound ...
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Big O/little o true/false

These are all from Sipser's book, second edition. I was just hoping someone could verify/explain those that are more difficult for me. $2n = O(n)$: true $n^2 = O(n)$: false $n^2 = O(n\log^2 n)$: I ...
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How to find a function that is the upper bound of this sum?

The Problem Consider the recurrence $ T(n) = \begin{cases} c & \text{if $n$ is 1} \\ T(\lfloor(n/2)\rfloor) + T(\lfloor(n/4)\rfloor) + 4n, & \text{if $n$ is > 1} \end{cases}$ A. Express ...
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1answer
80 views

set of Kolmogorov-random strings is co-re

given RC = {x : C(x) ≥ |x|} is a set of Kolmogorov-random strings. How can I show that RC is co-re I have been reading this paper What Can be Efficiently Reduced to the Kolmogorov-Random ...
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63 views

Complexity classes and number of problems

Will every complexity class contain infinite number of problems? If they do not, do common complexity classes (e.g. P,NP,PSPACE,EXPTIME,EXPSPACE etc.) contain infinite number of problems?
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132 views

Determining computational complexity of stochastic processes

I have an program which implements a Markov chain Monte Carlo process on a system of N bits, stopping when the process converges. Let's use T to denote the average number of steps made by the Markov ...
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Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
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Computational complexity of least square regression operation

In a least square regression algorithm, I have to do the following operations to compute regression coefficients: Matrix multiplication, complexity: $O(C^2N)$ Matrix inversion, complexity: ...
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316 views

Is there any infinite set of primes for which membership can be decided quickly?

The AKS algorithm decides whether or not $n$ is prime in time $\tilde{O}((\log{n})^6)$. I am wondering if there is any faster algorithm to determine membership in some infinite set of primes. What I ...
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What is the 3SAT problem? [closed]

I don't get the 3SAT problem. Can someone explain the 3SAT problem as if I were 5 years old, ideally with examples? Thanks!
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Is there a theory that combines category theory/abstract algebra and computational complexity?

Category theory and abstract algebra deal with the way functions can be combined with other functions. Complexity theory deals with how hard a function is to compute. It's weird to me that I haven't ...
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How complicated is the set of tautologies?

Consider the set $\mathcal T$ of all tautologies in the propositional calculus in which the only operators allowed are $\to$ and $\neg$, and involving only the two variables $x$ and $y$. How ...
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249 views

Applications of computation on very large groups

I have been studying computational group theory and I am reading and trying to implement these algorithms. But what that is actually bothering me is, what is the practical advantage of computing all ...
8
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1answer
429 views

Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$

I was reading about Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$ by ultrafinitists. I am wondering if they were to deny the existence of $\lfloor e^{e^{e^{79}}} ...
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Does DTIME(O(n)) = REGULAR?

(I don't think that this is a good fit on cstheory, since I figure that this question already has a known answer. However, if you think that this would be a better fit there, please feel free to ...
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How many possible phone words exist for a phone number of length N when also counting words less than length N within that phone number?

The phone words problem find all possible words that can be derived from a phone keypad, "words" do not have to be English dictionary words, for this question, words can be any combination of letters ...
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1answer
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Definition of time-constructible function

What would be an intuitive notion of time-constructible functions ? Is there a function which is not time-constructible? In my own words I would say a function is time-constructible when 1. it is ...
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Using $O(n)$ to determine limits of form $1^{\infty},\frac{0}{0},0\times\infty,{\infty}^0,0^0$?

Is it sufficient to use $O(n)$ repeatedly on $1^{\infty},\frac{0}{0},0\times\infty,{\infty}^0,0^0$ to get determinate forms? For example if we look at $\frac{0}{0}$ then $$\frac{O(f(n))}{O(g(n))}$$ ...
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Complexity of counting the number of triangles of a graph

The trivial approach of counting the number of triangles in a simple graph $G$ of order $n$ is to check for every triple $(x,y,z) \in {V(G)\choose 3}$ if $x,y,z$ forms a triangle. This procedure ...
6
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1answer
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What is the simplest collatz like problem that is undecidable?

I have read that problems resemblings collatz have been shown to be undecidable. Conway proved that apparantly but Im not sure if the proof was constructive. So I wonder : What is the simplest ...
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How to maximize the number of operations in process

In my research project I have encountered the following problem, concerning a tuple of words in the formal language $L=\{0,1\}^*$, with $\epsilon$ denoting the empty word. If we are given an ordered ...
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1answer
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Why are Hornsat, 3sat and 2sat not equivalent?

I have been reading a little bit about complexity theory recently, and I'm having a bit of a stumbling block. The horn satisfiability problem is solvable in linear time, but the boolean satisfiability ...
6
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1answer
604 views

Is there an efficient algorithm to compute a minimal polynomial for the root of a polynomial with algebraic coefficients?

An algebraic number is defined as a root of a polynomial with rational coefficients. It is known that every algebraic number $\alpha$ has a unique minimal polynomial, the monic polynomial with ...
6
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304 views

Can a Pratt certificate for a prime be found in polynomial time?

Can a Pratt certificate for a prime be found in polynomial time? I guess this is the same as asking whether the AKS primality test provides extra information that allows $p-1$ to be factored quickly. ...
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Is there an equivalent concept of a “variety” for SAT?

Couldn't find anything via google - I was wondering what work is out there looking at SAT problems from the perspective akin to an algebraic variety, e.g. a set of variables $X_1=$true, $X_2=$false, ...
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Choosing primes uniformly at random

I'm interested in efficient methods of generating prime numbers in a given range [a, b] (or with a given number of bits/digits, etc.). By "efficient" I mean minimizing time, randomness, and perhaps ...
5
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Is there a log-space algorithm for divisibility?

Is there an algorithm to test divisibility in space $O(\log n)$, or even in space $O(\log(n)^k)$ for some $k$? Given a pair of integers $(a, b)$, the algorithm should return TRUE if $b$ is divisible ...
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Confusion related to the definition of NP problems

I have this confusion related to the definition of NP problems. According to wikipedia Intuitively, NP is the set of all decision problems for which the instances where the answer is "yes" have ...
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1answer
128 views

Giving an asymptotically tight bound on sum $\sum_{k=1}^n (\log_2 k)^2$

I am trying to give an asymptotically tight bound for the sum $\displaystyle\sum_{k=1}^{n} (\text{lg}k)^{2}$, where lg denotes the base 2 logarithm. I really have no idea where to begin, and I've ...
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1answer
67 views

Simple question on complexity

I just started to learn the complexity theory, and I have a simple question: Let's say that there's a language B in NP, such that there's no polynomial reduction from B to a given language A (which ...
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556 views

Polynomial complexity algorithm of partition problem with sets of equal size

Partition problem is well known ( http://en.wikipedia.org/wiki/Partition_problem ). Let's add an additional condition: sizes of both sets should be equal. Is there a pseudo-polynomial solution to ...