# Tagged Questions

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

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### Given inputs as positive integers $a$,$b$, and $c(i,j)$ where $i,j\leq a$, decide if there is a permutation $\tau$ such that

Given inputs as positive integers $a$,$b$, and $c(i,j)$ where $i,j\leq a$, decide if there is a permutation $\tau$ such that $$c(\tau(a),\tau(1))+\sum_{i=1}^{a-1} c(\tau(i),\tau(i+1))\leq b$$ Prove ...
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### Is $L = \{0^{i}1^{i}0^{j}1^{i} | i, j > 0\}$ a context free language?

Is the following argument correct? $L = (A \circ B) \cap C$ where, $A = \{0^{i}1^{i}$ $|$ $i > 0\}$ $B = \{0^{j}1^{i}$ $|$ $i, j > 0\}$ $C = \{0^{i}1^{j}0^{k}1^{i}$ $|$ $i, j, k > 0\}$ We ...
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### If $f(n) = \sum_{i = 1}^{n} (n / i) \log(n / i)$ and $g(n) = n ~ {\log^{2}}(n)$, then is $O(f) = O(g)$?

I was trying to prove that if $$f(n) = \sum_{i=1}^{n}\frac{n}{i} \log\frac{n}{i}$$ $$g(n) = n \log^2n$$ then $O(f(n)) = O(g(n))$ I am not sure that it is the case, but based on my simulation ...
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### Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
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### P vs NP and Countable vs Uncountable Decision Space

I have noticed that whenever the scope of a problem is pushed to infinity, problems in NP have an uncountably infinite decision space whereas problems in P seem to have a countably infinite decision ...
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### Find all groups that meets the condition

I have $n$ elements, each of them have two unsigned int attributes $x$ and $y$. Now I'd like to find out all the groups that fit the following condition: $A.x \geq B.y$ and $B.x \geq A.y$. The ...
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### Formula to calculate password cracking time in years, taking into account Moore's law and known adversary guessing power [closed]

We know that the biggest human rights violators in human history are capable of one trillion password guesses per second as of approximately January 2013. Assume that the 1 trillion guesses per ...
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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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### decidability of a given language

The language EGAL is $\{(A,B): A \text{ and } B \text{ are DFAs with } L(A) = L(B)\}$ How do I prove that such language is decidable by testing every word of $A$ and $B$ until a defined length ? i ...
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### Why isn't NP=coNP? [duplicate]

My understanding is that if a problem is in NP, there is a nondeterministic polynomial-time Turing machine that decides it. That is to say, if an NP problem has a solution, the NP machine has a ...
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### The meaning of 'worst case'

When giving bound on convergence rate, complexity and so on, people sometimes will specify it by 'worst case'. What is the meaning of 'worst case'?
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### Exponential vs Polynomial running time

As per this article: http://stackoverflow.com/questions/4317414/polynomial-time-and-exponential-time we know that exponential is worse than polynomial in terms of running time. Is it safe to say that ...
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### Confusion on Big $O$

I am so confused on the intuitive idea behind Big $O$ notation. $f(x)=O(g(x))$ iff there is a constant $C>0$ such that for large $x, |f(x)|\leq C|g(x)|$ and I have seen that in many places that ...
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### What's the conjectured optimal running time for an exponential function algorithm restricted to [0, 1]?

If such an algorithm were used, for each positive integer ''n'', what's the upper bound on the computation time for the ''n''th digit after the decimal place. The Schönhage–Strassen algorithm runs on ...
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### Required bits to communicate a partial order?

Suppose that you have a ranking (i.e. a strict complete partial order) over $n$ different objects, so that the objects can be ordered as $a>b>\cdots>n$. You want to communicate the exact ...
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### Number of operation to transform $(0,0,0)$ to $(a,b,c)$ with $2^h\leq a,b,c \leq 2^h-1$

Given a positive integer $h$, define: $$A_h=[2^h,2^{h}-1]\big \{2^h-1+\sum_{i\in A}2^i \Big/ A\subset[0,h-1]\big \}$$ (this is in terms of binary expressions: the set of all numbers having exactly $h$ ...
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### Average case complexity for checking if list is sorted

Consider the obvious algorithm for checking whether a list of integers is sorted: start at the beginning of the list, and scan along until we first find a successive pair of elements that is ...
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### A question on GCT

In http://ramakrishnadas.cs.uchicago.edu/gctriemann.ps it is stated that there is an unknown non-standard riemann hypothesis. AFAIK riemann hypothesis in AG was shown using Etale cohomology by Artin, ...
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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 ...