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

learn more… | top users | synonyms (1)

1
vote
0answers
11 views

Comparing the orders of complexity.

I do have to evaluate a property defined as $$w = \sqrt{\det\left(\textbf{J}\,\textbf{J}^\text{T}\right)} = \prod_{i = 1}^n \sigma_i,$$ where $\textbf{J} \in R^{3 x n}$ and $n>3$. Using the ...
1
vote
0answers
33 views

Complexity analysis of alpha beta pruning of a full tree

I am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...
1
vote
0answers
13 views

How to obtain the minimizer parameter $\lambda$ for this computational complexity?

I'm trying to read a certain text, where they reach a computational complexity depending on scalars $a,b,c$ and a parameter $\lambda >0$ $$ O\left(\left\lceil\sqrt{\lambda a + \lambda^2 b^2} ...
1
vote
0answers
31 views

Kolmogorov complexity, no description mechanism can improve on additively optimal/universal one infinitely often

In An Introduction to Kolmogorov Complexity and Its Applications explaining the notion of additively optimal or universal it is written: The key point is not that the universal description method ...
1
vote
0answers
61 views

Why aren't all NP-complete problems strongly NP-complete, if any NP problem can be reduced to an NP-complete problem

So we know that : (1). A problem is NP-complete if every other problem in NP can be reduced to it in polynomial time (2). A problem is said to be strongly NP-complete if a strongly NP-complete problem ...
1
vote
0answers
35 views

How to properly detect rows to be swapped in a Gaussian elimination?

I'm trying to describe an algorithm for solving solvable linear systems. The Gaussian elimination is pretty straightforward in terms of adding multiples of rows. However, consider the following ...
1
vote
0answers
31 views

Prove that $S_2$ is closed under union and complement

I'm having trouble proving that $S_2$ is closed under union and complement, even though in this Wikipedia article it says that: It is immediate from the definition that $S_2$ is closed under union ...
1
vote
0answers
41 views

Is there a polynomial time algorithm for Poly-trees (oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
1
vote
0answers
47 views

Is there a “C vs NC”-problem, where C stands for “constant time”?

Appologies if this question is utterly naive. I know very little about complexity classes, but like to learn more. Consider the following problem. Given input $n$ (a natural number) we want to find ...
1
vote
0answers
40 views

What do log-equivalent and log-complete mean?

I'm reading the paper The Complexity of Satisfiability Problems by Thomas Schaefer(1978). In the paper, he mentions the phrases "log-equivalent" and "log-complete." Searching through the Google ...
1
vote
0answers
23 views

Simplicial maps between simplicial 2-manifolds

Suppose I have two simplicial two-manifolds ("triangle meshes") $M_1$ and $M_2$. I want to compute a surjective simplicial map between $M_1$ and $M_2$, i.e. a surjective function $\phi$ between the ...
1
vote
0answers
141 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
1
vote
0answers
21 views

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 ...
1
vote
0answers
47 views

Is there an optimal algorithm to calculate $2^n - 1$ in $\theta(n^n)$

The sequence $(f_{n})_{n \in \mathbb{N}}$ is defined by $f_{0} := 0, f_{1} := 1$ and $f_{n} := 3f_{n-1}-2f_{n-2}$ for $n \in \mathbb{N}_{0} \setminus \{0,1\}$. Is there an algorithm that takes an $n ...
1
vote
0answers
70 views

the sum of the reciprocals of the primes

The sum of the reciprocals of the primes is $\sum \limits_{p}\frac{1}{p} \approx N \ln\ln(N)$ what about this sum where $p_{3}=3,p_{5}=5,p_{n}=\sum \limits^{N}_{j=5}\frac{1}{p_{j}} \sum ...
1
vote
0answers
174 views

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 ...
1
vote
0answers
70 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 ...
1
vote
0answers
33 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 ...
1
vote
0answers
30 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 $?
1
vote
0answers
30 views

When does $A,A\cap B, A\cup B\in S$ imply $B\in S$?

Let $S\subset 2^{\Sigma^*}$ be some family of formal languages over some alphabet $\Sigma$. Consider the the following statement: $A,A\cap B, A\cup B\in S$ implies $B\in S$ For which ...
1
vote
0answers
38 views

Is it better to compute $A^tA$ once and then $Ax$ several times or compute $y=Ax$ and then $A^ty$ every time?

So I have this algorithm which given a matrix $A$ it assigns $A=A^tA$ outside the loop and then on the algorithm loop it solves multiple instances of $Ax$ for different $x$s, (meaning that it's ...
1
vote
0answers
329 views

Complexity of combination method

I have a question about complexity of combination two methods. Assume that I have method A with its complexity is $O(n)$ and second method that has complexity is $O(n^2)$. In which, n is number of ...
1
vote
0answers
102 views

smallest circuit

Let $SMALLESTCIRCUIT$ be the language consisting of all Boolean Circuits $C$ with the property that there is no smaller circuit $C^{'}$ that has the same truth table as $C$. (smaller means having ...
1
vote
0answers
28 views

Computation Complexity POLYLOGSPACE

POLYLOGSPACE is the complexity class $ \bigcup ^\infty _k_=_1 SPACE((logn)^k) $ (a) Show that, for any k, is $ A \in SPACE((logn)^k) $ and $ B \le _L A $, then $ B \in SPACE((logn)^k) $. (b) Show ...
1
vote
0answers
58 views

an instance of NP-complete

The cafeteria serves $m$ different kinds of food, $F = \{ f_i \}_{i = 1}^{m}$. The fruit are grouped into $n$ different types of bags $B_1, \cdots, B_n \subseteq F$. (The same kind of fruit might be ...
1
vote
0answers
54 views

Kolmogorov complexity of a computer?

Warning: Vague, unclear question ahead. Proceed at your own risk. The Shannon entropy and Kolmogorov complexity give you in broad informal terms how unpredictable a string is and to what degree the ...
1
vote
0answers
70 views

Oracles for TQBF

I've seen this question somewhere and I've been thinking about it a lot but couldnt think of an answer. Say you have oracles A and B for the TQFB (True Quantified Boolean Formula) decision problem, ...
1
vote
0answers
51 views

Combinatorial search by testing sets with fixed number of elements

I am struggling to see the complexity of the following combinatorial search problem. Could anyone help me? Consider a set $I$ of $n$ items known to contain $d$ defectives or less. Assume $d < ...
1
vote
0answers
40 views

Order of magnitudes comparisons

I need your help with the following. I need to determine how to order (functions) the following : \begin{align} &f(x)=(x/2)^{(x/2)} \\ &g(x)=x! \end{align} Note: I got both of them are ...
1
vote
0answers
127 views

Big-O estimate (smallest order)

I'm trying to give a big-O estimate for each of these functions, where I want to use a simple function $g$ of smallest order. I have them all done I just wanted to someone to run through and check ...
1
vote
0answers
79 views

Constructing a “one-way function” of two variables (a.k.a “stop my friend from hacking my game”)

This might be more of a computer science question than a mathematics one; I thought I'd start here but perhaps people might want to point me to a better forum, if this isn't the right one. ...
1
vote
0answers
62 views

Is finding a matrix out of some set with a given determinant a hard problem?

Given $n\ge 2\ \ ,\ u,v,k\ $ integers. Decision problem : Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist? In which complexity class is this problem ? Is it ...
1
vote
0answers
66 views

Fast checking Matrix multiplication in mod 10

I recently faced this problem in a programming contest: Given 3 square matrices N x N of size N up to 1000. All elements in 3 matrices are from 0 to 9. Check if matrix A x B equals to C, mod 10. In ...
1
vote
0answers
83 views

GCD of high order polynomials(modulo large prime)

I want to solve the following question: Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...
1
vote
0answers
178 views

Find a simple Theta bound for a geometric series

I'm working on a question like: find a simple g(n) (big theta) for $ f(n) = \sum _{i=1}^n 5^i $ My working starts with this $\frac{5-5^n}{1-5}$ which is not equivalent to the correct answer from ...
1
vote
0answers
784 views

Calculating time complexity of algorithms written in pseudocode.

Nowadays we are interested to find some algorithms with a prescribed running time. For example if for certain decisional problem $X$ there is an algorithm with running time $O(n^3)$ we try to break ...
1
vote
0answers
40 views

Is there a fast algorithm for computing the $(2^n)+1$ th last digit of $3^{2^n}$ in base $2$?

Is there an algorithm such that for some polynomial p, it always computes the $(2^n)+1$ th last digit of $3^{2^n}$ in base $2$ in at most p(n) steps for all nonnegative integers n? I'm only asking if ...
1
vote
0answers
37 views

Directed Hamiltonian Reduction

The reduction function given by Richard Karp in 'Reducibility among combinatorial problems' for Directed Hamiltonian Cycle $\leq_{p}$ Undirected Hamiltonian Cycle goes as follows : for input $G = ...
1
vote
0answers
148 views

Proving context free language membership is $P$ complete with respect to log-space reductions

This is exercise from Introduction to Automata theory, Languages and Computation, by Hopcrof, Ullman (first edition). I found example of polynomial reduction to some problems in logic, or graph ...
1
vote
0answers
29 views

Basic questions about descriptive complexity

I'm trying to learn descriptive complexity, and I'm having trouble on a basic level wrapping my head around what it means for a logical formula to define a computational language. I've tried and ...
1
vote
0answers
60 views

My notes on $\Bbb{Z}/p\Bbb{Z}$-theoretic computational complexity

(Question at the very bottom) Def 1. Let $F = \Bbb{Z}_p$ be a finite field. Then an $F^k$-machine is a machine with $k$ input / output memory slots. All computations are done in the field $F$ and ...
1
vote
0answers
35 views

Can cuts of size 2 be detected in linear time in an undirected, unweighted graph?

I'm having trouble finding any literature on the specific subject of 2-edge cut detection. It's not hard to come up with an algorithm that finds all 2-edge cuts in quadratic time, but it's not clear ...
1
vote
0answers
61 views

Is $\log^* (n+1)^{n+2} \in O(\log^* n)$?

I would like to know if $\log^* (n+1)^{n+2} \in O(\log^* n)$, where $\log^*$ is the iterated logarithm. I tried doing: $ \log^* (n+1)^{n+2} =\\ \log^{*}(\log(n+1)^{n+2})-1 =\\ \log^{*}((n+2) \cdot ...
1
vote
0answers
34 views

any approximation for $\sum_{i_3=3}^{n-k+3}\sum_{i_4=i_3+1}^{n-k+4}\sum_{i_5=i_4+1}^{n-k+5}\cdots\sum_{i_k=i_{k-1}+1}^{n} 1, (n \gg k)$?

Is there any approximation for $$\sum\limits_{i_3=3}^{n-k+3}\sum\limits_{i_4=i_3+1}^{n-k+4}\sum\limits_{i_5=i_4+1}^{n-k+5}\cdots\sum\limits_{i_k=i_{k-1}+1}^{n} 1, \quad (n \gg k)$$ ? We know that ...
1
vote
0answers
83 views

number of multiplication steps required to solve Ax = b

If we can factorize $A$ in $LU$, We can solve $Ax = b$ in 2 steps: Solve $Lc = b$ for c Solve $Ux = c$ for x As per the Linear algebra book by Gilbert Strang, each step takes $n^2/2$ number of ...
1
vote
0answers
129 views

Is Quadratically Constrained Quadratic Program (QCQP) in NP?

The general version of QCQP is NP-hard, but is it also NP-complete? That means, is there a non-deterministic algorithm, which solves QCQP in polynomial time complexity? If the general version of QCQP ...
1
vote
0answers
39 views

Why every regular language is in $\text{TIME}(n)$?

How can I prove that every regular language $R$ has linear time complexity, i.e. every regular language satisfies $$R \in \text{TIME}(n)$$
1
vote
0answers
315 views

A question on the computational complexity of Boruvka's algorithm

One algorithm that finds a minimum spanning tree in a graph in which all weights are distinct is Boruvka's Algorithm (also known as Sollin's Algorithm). On the page you would see once you clicked ...
1
vote
0answers
96 views

How to prove the NP-hardness of this scheduling problem

Suppose there are a set of $m$ jobs $J= \{J_1, J_2, \ldots, J_m\}$ and $n$ machines $M=\{M_1, M_2, \ldots, M_n\}$. Each job $J_i$ consists of $k_i$ unit operations, and there are totally K operations ...
1
vote
0answers
135 views

How to compute this recursion in linear time?

Can the following iterative update on a $n$-element vector $\mathbf{x}_t$ be computed in $O(n)$ computations? \begin{align*} \mathbf{x}_{t+1} & = a_t\mathbf{y}_t + \mathbf{A}_t \mathbf{x}_t \,,\\ ...