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

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On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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14 views

Complexity of recurrence containing geometic series.

What is the complexity of the recurrence $T(n) = 3T(\frac n2) + O(n)$? So far I have: $ O(n) \le cn$ for some constant $c$ Hence: $$T(n) \le 3T(\frac{n}{2}) + cn$$ After a recursion: $$T(n) \le ...
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2answers
34 views

Complexity of recursive algorithm.

An algorithm solves problems of size $n$ by recursively solving two subproblems of size $n - 1$ and then combining the solutions in constant time. What is the algorithms running time? Assume $ ...
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1answer
34 views

What is the number of words of length $h$ in a sequence of subsets of words?

Let $L=\{0,1\}^*$ (the set of binary words on $0$ and $1$), Given an integer $k$, and $S$ a finite subset of $L$ define recursively the following sequence of subsets of $L$: $$\begin{align} A_1 ...
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1answer
43 views

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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27 views

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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2answers
38 views

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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23 views

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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1answer
26 views

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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68 views

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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8 views

How to find a strictly increasing sequence of words $(t_i)_{0\leq i\leq n}$ of maximum length ?

Let $L=\{0,1\}^*$ be the set of all words consisting of $0$ or $1$, we define an order in $L$ by: $$\begin{align}\forall (x,y)\in L^2 && \big( x\leq y&&\Leftrightarrow y=x\text{ or ...
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183 views

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
26 views

Prove that $ (lg\; lg\; n)^k=o(lg^\epsilon n)$ for all $0<k,\epsilon$

I am stuck at this problem for a long time: Prove that $ (lg\; lg\; n)^k=o(lg^\epsilon n)$ for all $0<k,\epsilon$ I tried to show that $\lim_{x\to\infty}\frac{ (lg\; lg\; x)^k}{lg^\epsilon ...
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2answers
117 views

What is the algorithm to add binary numbers with boolean operations? [closed]

What is the algorithm to add up two binary numbers using only boolean operations (negation, conjunction, disjunction) in linear time? Also the program flow needs to be "linear" as well, meaning there ...
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2answers
59 views

what is the complexity of this type of algorithms (loop bounded)?

I have an algorithm which contains only the instructions of type: $X_i=X_j$ $X_i=X_i+1$ $\text{while }(X_i\le N)\text{ do }\{C\}$ where $C$ is another instruction $N$ is a global constant and ...
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1answer
80 views

Minimize the squared dot product of two specific vectors

Do you think there exists a efficient algorithm(non brute-force) for the following problem. I search the optimal solution for the following problem: Given a vector $u=(u_1, u_2,..., u_k)^T$ with ...
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34 views

Traversing multi-way tree, computational complexity

This is a computational challenge. I am looking for a clever simplification or heuristic. Imagine a multi-way tree. Each node has three child branches. Consider them to be decisions; do A, do B, do ...
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1answer
43 views

improved segmented sieve of erastothenes complexity

I improved the segmented sieve of erastothenes , my algorithm doesnt repeat the multiples of primes using the equation $p^{2}_{n}p_{j}+2p_{n}p_{j} \times c =N$ wich shows when at least two multiples ...
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44 views

Recurrence relationship of Hamiltonian backtracking

I'm struggling to understand how to express the recurrence relation in terms of N of a backtracking algorithm to find out if a Hamiltonian path exists. Where N is the number of vectors. After finding ...
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1answer
59 views

an strange set $ \Xi_A =$ {$ n \in N | \exists k^2 \in A $ s.t $ k^2 \leq n$} is decidable ?, an Interview questions?

We are some student that had an Interview for M.sc Entrance Exam. This interview has two part and one multiple choice question. We see 1 strange question that some definition is so strange for us, we ...
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52 views

Can you decouple the Traveling Salesman Problem from the number of cities?

I am studying the euclidian version of the Travelling Sales Man problem: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits ...
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46 views

Best time complexity for calculating the next, unique graph.

Whats the best time complexity, for a known algorithm, that when called generates the next, unique, graph, in order of node count? For example, the first result being the only single node graph, I ...
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23 views

Trying to determine the relationship of m and n in a Casting Out m under base n

While exploring $\mathbb{Z/n}$ I stumbled upon this It explains that Casting Out Nines works because our common counting system is decimal and thus there exist a congruence relation as follows ...
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1answer
16 views

Reducing a problem X to an np-complete problem Y.

Say I have a problem X that I can reduce to an NP-complete problem Y. Can I assume that problem X is in NP? Can it not be in NP?
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29 views

How would you prove this Big Omega complexity?

We're given $f(n)=\frac{1}{5}n^2-30n-5$ and $g(n)=n^2$, and are asked to prove $f \in \Omega(g)$. The exercise was posted, but no solution is given (this is not an assignment question). So by ...
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2answers
94 views

P vs NP - examples of P and NP

I'm currently studying 'p versus np'. Can someone help me in showing an example of a mathematical p problem and np problem? A clear worked example would be much appreciated. Many thanks in advance.
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1answer
52 views

On average, as a function of n, how many print statements are executed by the following algorithm?

On average, as a function of n, how many print statements are executed by the following algorithm? ...
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1answer
67 views

Better time complexity.

I am new to complexity theory and want to know, Which one is better time complexity(faster) for an algorithm ? \begin{equation} \frac{n^{k+\log_2(n)}}{\log_2(n)2^{n}} \end{equation} or ...
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1answer
102 views

Computing the order of a group element

This is a partially computer theoretic question, but is probably closer to math. I remember finding a paper from 1980's or so that had a proof of the fact that finding the order of a group element is ...
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1answer
43 views

Clarification over what NP means

I'm reading an informal definition of the decision class NP with a specific example being the standard knapsack problem and a decision variant of this problem. The example they are using is a ...
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14 views

Implications of deterministic Polynomial Identity Testing

It is well-known that polynomial identity testing (PIT) has a polynomial time Monte-Carlo algorithm. At the same time, no efficient deterministic algorithm is known. I came across a paper of Kabanets ...
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1answer
43 views

Natural Numbers and $A_x=\{y \in A \mid y \leq x\}$ [closed]

Suppose A is a arbitrary subset of Natural Numbers and $A_x=\{y \in A \mid y \leq x\}$ with respect to $ n \in A \Longleftrightarrow n \in A_n $ and $A_n$ is finte, which of them is True? a) A and ...
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What is the number of full binary trees of height less than $h$

Given a integer $h$ What is $N(h)$ the number of full binary trees of height less than $h$? For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by TravisJ in his partial answer) I can't ...
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39 views

Kirchoff's First Law (Algorithm Complexity)

I want a help in understanding the figure and the complexity of Kirchoff's First Law Kirchoff's First Law states that the number of incoming flows into a node must be equal to the number of outgoing ...
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23 views

Non-deterministic multiplication algorithms

Are there any algorithms for non-deterministic Turing machines that can compute the decision problem $mn=x$ (where $m=O(n),x=O(n^2)$) faster than the equivalent deterministic algorithm? Equivalently, ...
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32 views

Proof the Restricted Case of CVP is P-complete

Show that the following Restricted Case of CVP is P-complete: Like CVP, except the input circuit satisfying the following conditions: All gates are placed int layers; the inputs of a gate come from ...
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1answer
57 views

Finding the complement of a set by negating logical statements

As part of an exercise I've been given the assignment to find the complement of the following statement: L = {P⊆{0,1}*: P is a legal encoding of a C program, and P terminates on all but a finite set ...
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37 views

Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
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21 views

Notation in a fast perfect-power classification algorithm

I'm trying to implement a lengthy perfect-power classification algorithm with an interesting complexity of $O(\log_2(n)^{O(\sqrt{\log\log n\log\log\log n})}).$ To do so, I'm referring to Daniel J. ...
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109 views

the algorithm and computation cost for truncated SVD in rank k

It seems that the time cost of truncated SVD in rank k for matrix $A\in R^{m\times m}$ is $O(m^2 k)$. Could anyone show me some algorithms to calculate truncated SVD with the above time complexity?
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1answer
46 views

Formal definition of a set of graphs according to diameter restriction

I'm trying to find a formal way of defining the set of all directed graphs which their diameter is at most X, by formal definition I mean something of the following form: {G | G=(V,E) , G is a ...
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74 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
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25 views

Computational complexity of Gauß-Seidel method

I'm currently studying iterative methods to solve equations of the form $Ax=b$ One method that is presented in my script is the Gauß-Seidel method where one step is defined as: $$x^{(k+1)} = ...
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84 views

Why are there no continued fraction representation for $\pi$ obeying mathematical rules?

There are several irrational numbers that can be represented with continued fraction such that a mathematical rule arises in this continued fraction. For example, the Euler number $e$ can be ...
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1answer
75 views

How to solve this logarithmic inequality?

I've started a data structure course and I need some help with solving these logarithmic inequalities. It would also be helpful because later on these kind of calculation won't pose a problem later ...
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50 views

Finding the biggest decrease in a list of integers (Python)

Given an array of n integers, h0, . . . , hn−1, I want to find the largest decrease in values such that the largest decrease is max(hi − hj) such that i ≤ j. For example, given as an input the array ...
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121 views

Computation complexity with simple algebra expression reduction

I'm watching this computer science video on computational time complexity of a function where they introduce some maths and it doesn't make sense to me. I'm not even sure what the name for this maths ...
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27 views

why some people use the notation $a \leq O(n)$?

When describing the algorithm complexity denoted by $c$, some people use $c \leq O(n)$ instead of $c =O(n)$ to show complexity. I cannot understand why they should use $\leq $?
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154 views

How to prove worst case time complexity for binary search

I have the following problem: Show that the worst case time complexity for Binary Search is given by: $W(n) = \lfloor lg(n) \rfloor + 1$ when n is not restricted to being a power of 2. ...
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28 views

How to find asymptotic cost of matrix filling algorithm . Big O Notation

So I have a list X of N strings each of length M that will be called x_i for the ith index in X Example ...