Questions tagged [computational-complexity]
Use for questions about the efficiency of a specific algorithm (the amount of resources, such as running time or memory, that it requires) or for questions about the efficiency of any algorithm solving a given problem.
3,578 questions
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What is the complexity of computing the earth mover's distance?
Definition (earth mover's distance). The integral probability metric between two distributions $P$ and $Q$ is defined as
\begin{align}
d_{F}(P,Q)
=\sup_{f\in F}\Big|\mathbb{E}_{X\sim P}[f(X)]-...
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What is a "linear time transformation" for algorithms? By extension, what does "linear time transformable" mean?
I'm reading a slice of Preparata and Shamos's book on Computational Geometry. I'm enjoying it so far, and I feel it's pretty well-written. However, some of the language is unusual and can't be found ...
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Is the polynomial bound on the certificate in the verifier definition of the complexity class NP redundant?
All formal verifier-based definitions of NP that I can find online ask for a polynomial bound on the certificate. For example, the Wikipedia page for NP defines:
"A language $L$ is in NP if and ...
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Are there problems without asymptotically optimal algorithms?
Let $f: \mathbb N \to \mathbb N$ computable. Can it be the case that given any Turing machine that computes $f$ in $O(g(n))$ steps there is another Turing machine that computes $f$ in $O(h(n))$ steps ...
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Lower bound on a scheduling-like problem using a virtual tree
Problem
Let $v_1, v_2, \dots , v_n$ be processes such that $v_i$ must wake up at time $i$ and take a decision.
When $v_i$ wakes up to take a decision, it must take in account the decisions made by the ...
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construct the circuit which checks whether the $j$-th bit of the sum of given binary numbers $k$ and $m$ is 1
A boolean circuit C has n inputs and m outputs, and is constructed with AND, OR, and NOT gates. Each gate has fan-in 2 except the NOT gate which has fan-in 1. The out-degree can be any number. A ...
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Calculate polynomial's coefficient that is trace of matrix multiplication
Let $A_i,\ B_i$ be $M \times M$ real matrices.
Define polynomial function $f:\mathbb{R} \to \mathbb{R}$ such that
$$f(x)=\text{tr}[(A_1+B_1x)(A_2+B_2x)\cdots(A_n+B_nx)]$$
And I want to calculate ...
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Efficiently enumerating minimal clique covers of a graph and its subgraphs
I am studying a certain set of graphs using algorithms, and one of my goals is to solve the following problem efficiently. If $H$ is a (labelled) graph, let $\mathscr{S}:=\mathscr{S}(H)$ be a list of ...
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Algorithms in which the two given complexities are possible/relevant. [closed]
This post is about finding the algorithm, in which the given expression for time-complexity is applicable, as time-complexity serves to directly represent the code(block) at hand.
It is given in the ...
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Polynomial hierarchy with satisfied boolean formulas [closed]
Some research paper define $\Sigma_i SAT$ in this way,
$$\Sigma_i \text{QBF}=\{\exists x_1\forall x_2\exists x_3\cdots Q_ix_i\varphi(x_1\cdots x_i) \text{ where } \varphi(x_1\cdots x_i) \text{is an ...
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Computing the norm in a multiquadratic number field
Let $M = \mathbb{Q}(\sqrt{m_1},\sqrt{m_2},\dots,\sqrt{m_n})$ be a multiquadratic number field of degree $2^n$ over $\mathbb{Q}$.
Suppose $\alpha$ is a primitive element of $M$.
Is it possible to ...
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Is there an algorithm for fast computation of k subsequent values of a homogeneous linear recurrence of order k?
Problem
Consider a linear recurrence of order $k$ with known initial values $\{a_i\}_{i=1..k}$ and constant coefficients $\{b_i\}_{i=1..k}$ in a (perhaps not algebraically closed) field:
$$ a_n = \...
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Algorithm with worst-case time complexity as $\log_2(n)$
Was searching for an algorithm that has the worst-case time complexity as $O(\lg(n)).$
Have read that binary-search algorithm, has both the worst-case and best-case time-complexity as $\theta (\lg(n)),...
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Application of Savitch theorem [closed]
We consider a representation of directed graphs that assumes layers in the graph. That is, we denote by $G = \{V_1, V_2, \cdots , V_k; E\}$ the graph whose vertex set is $V =\cup_{i=1}^kV_i$ and $E$ ...
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Relation between permanent and row sums
I am studying about permanent of the matrix and I stumbled upon some equations that describe the permanent or bound its value using the row sums of the matrix.
For example, Ryser formula states that: ...
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Computing primes of the form $x^2+ny^2$ [closed]
The book Primes of the form $x^2+ny^2$ by Cox discusses in detail the problem of when a prime $p$ can be written as $x^2+ny^2$, and along the way introduces more and more sophisticated number theory ...
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Computing $\|\boldsymbol{X}\boldsymbol{X}^{\text{T}}\|_1$ in $\mathcal{O}(n)$ time for $\boldsymbol{X}\in\mathbb{R}^{n\times r}$ where $r\ge 2$ fixed
Let us consider some $\boldsymbol{X}\in\mathbb{R}^{n\times r}$, where $r\ll n$. When $r=1$, i.e., $\boldsymbol{X}$ is essentially a vector $\boldsymbol{x}\in\mathbb{R}^n$, then we have
$$
\|\...
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Reduction from np hard to np complete [closed]
I have couple of confusions regarding reduction:
(1) If A is NPC and B is NPC then reduction from A to B and B to A possible.
(2) And if A is NPC and B is not known to be NPC, then reduction from A ...
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need to prove that $DSPACE(O(2^n)) \neq EXP$
this question is from my computational complexity HW.
I'm not sure if my solution is correct:
If $DSPACE(O(2^n)) = EXP$, than we can take language $ L \in DTIME(2^{2^n})$ which not in $EXP$ (from the ...
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Strict bound for recurrence formula
Given the recurrence $$T(n) = (2k - 1)T(\frac{n}{k})+2^kn$$ I need to show (or disprove) that for every $\epsilon>0$ there exists a $k$ such that $T(n)=o(n^{1+\epsilon})$.
So far, I've been able to ...
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Inexact Approximation Oracles with Multiplicative Error?
I have a linear program (LP) with $n$ variables and $2^m$ constraints:
$$
\min \sum_{i \in [n]} x_i \quad \text{subject to} \quad x_i \geq y_j \quad \forall j \in 2^m, \, i \in [n].
$$
I can compute ...
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language $L,$ if $L\in \textbf{ZPP}$, then there exists a Turing machine that flips coins
I Followed from Barak and Arora book:
For a language $L$, we define $x\in L \Leftrightarrow L(x) =1$.
$\textbf{RP}$- is the collection of all languages $L$ for which there exists a polynomial coin-...
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How to evaluate the time complexity of this prime generating algorithm?
I’m working on a sieve algorithm to generate primes, and I’m having a hard time evaluating its time complexity.
The idea:
Fact 1: Any positive integer $>1$, except $2,3$ and their multiples, are of ...
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Rate of convergence in probability
I am reading the paper
In this paper, they proved theorem 7 which stated in the following way
Theorem 7: Let $p, q, X, Y$ be defined as in Problem 1, and assume $0 \leq k(x, y) \leq K$. Then:
$$
\Pr_{...
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Complexity of Finding Independent Sets in Undirected Graphs
Consider the following two problems on undirected graphs:
A: Given G(V,E), does G have an independent set of size |V|-4 ?
B: Given G(V,E), does have an independent set of size 5?
I think both ...
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