Questions tagged [time-complexity]

Time complexity of decision problems or relations among time-bounded complexity classes. (Use the [analysis-of-algorithms] tag for the time taken by particular algorithms.)

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Number of outputs produced by levin search variant (SIMPLE)

Let $f$ be an inverting problem. If there is an algorithm A that invert $f$ in time $t(n)$, then the SIMPLE algorithm below invert $f$ in time $c.t(n)$ where $c$ is a constant depending only on $A$ ...
0 votes
3 answers
1k views

What is the problem in "closest pair problem" if all points share the same x-coordinate

The closest pair of points problem deals with the task to find a pair of points with the global minimum distance. There is a problem, when all points share the same x-coordinate, or at least a large ...
2 votes
2 answers
180 views

Are there any problems whose best known algorithms have running time $n^{\log \log n}$?

It’s well known that problems such as integer factorization have running times contained in $e^{\text{Poly} \log }$ which is the same $n^{ \text{Poly} \log }$ (actually the log term is itself in a ...
0 votes
1 answer
39 views

Selecting unique records from a large dataframe with many duplicate records

Suppose we have a dataframe with ~10M rows with ~9M duplicate records. What is the most time efficient way of selecting the unique records from this dataframe? Some sort of sampling algorithm?
-2 votes
0 answers
99 views

How do I convert a pseudo-polynomial algorithm to its equivalent exponential form?

I have a pseudo-polynomial algorithm that solves a decision problem $D$. How do I convert it in its "explicit" exponential form?
6 votes
1 answer
181 views

Solving All-Pairs Shortest Paths using a distance matrix in sub-cubic time

I'm working on a project centered around the All-Pairs Shortest Paths (APSP) problem. Common algorithms to APSP (Floyd-Warshall, Bellman-Ford, Johnson's) work with the standard definition of the ...
6 votes
1 answer
279 views

Lower-bounds under SETH

After reading a bit about SETH (the strong exponential time hypothesis), I see that a lot of lower bounds for problems in P can be proven if we assume SETH. But I notice that most of the ones that are ...
0 votes
0 answers
67 views

Simplify or bound using Big-O notation

I was following a research paper which have the following equation: $\left(1-\frac{1}{K}\right)^{K-i}\left[1-\left(1-\frac{1-p}{K}\right)^{i}\right]=\frac{i(1-p)}{K}+O\left(\left(\frac{i}{K}\right)^{2}...
2 votes
0 answers
75 views

Nondeterministic polynomial time languages with linearly bounded certificates

Define the class $X$ of languages by the condition that a language $L$ over alphabet $\Sigma$ is in $X$ iff there are a constant $c > 0$ and a polynomial-time checking relation $R$ such that for ...
5 votes
1 answer
207 views

What's the constant coefficient of the Coppersmith-Winograd algorithm?

Every source I can find just says "too big to be practical."
0 votes
0 answers
38 views

Does the linear algorithm of Farach for sorting suffixes imply a linear algorithm for sorting a special type of a sequence of integers?

I was looking at the lecture notes on a linear time suffix trie construction algorithm by Farach and I was wondering if his suffix sorting procedure implies anything about sorting integers.
4 votes
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Simulating a $k$ tape Turing machine with a 2 tape Turing machine

Let $k$ be an (fixed, $3$ for instance) integer, what is the fastest simulation of a $k$ tape Turing machine by a two tape Turing machine? That is we're looking for the best 2 tape TM $U$, such that ...
2 votes
0 answers
61 views

Can this relaxed subset-sum problem be solved with a smaller dynamic program? [closed]

Cross-post from CS.SE In the subset sum problem, the input is a list of positive integers $x_1,\ldots,x_n$ and an integer $T$, and the goal is to decide whether there is a subset of sum exactly $T$. ...
2 votes
0 answers
41 views

Is the following special case of multiway number partitioning NP-hard?

The following problem is a decision problem of multiway number partitioning (wikipedia) (Note that $k$ is also a part of an input in the following problem, while $k$ is a fixed number in wikipedia ...
6 votes
1 answer
335 views

Are all problems in the same time hierarchy related to each other?

In this problem, "runtimes" refer to worst-case complexity compared up to constant factor. Say you have two problems, A and B, in the same time hierarchy, and it is clear that algorithm P ...
1 vote
0 answers
29 views

How to deal with the time to minimize a function in a given interval?

I'm writing a paper in which I designed an algorithm running in $O(n^2m)\cdot T(f)$ to solve my problem, where $n,m$ is the size of input and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a function, and $T(...
1 vote
1 answer
110 views

3-SAT runtime if an optimal order to eliminate possible solutions is known

As a mental exercise I have been playing around with the 3-SAT problem, but I am having difficulty proving or disproving the usefulness of a current idea that I am playing around with. My current ...
17 votes
2 answers
423 views

similar matrices

Given two $n \times n$ matrices $A$ and $B$, the problem of deciding if there exist a permutation matrix $P$ such that $B = P^{-1}AP$ is equivalent to GI(Graph ...
0 votes
0 answers
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Subset Sum Problem exact Algorithm (hypothetical)

Let assume there is way to apply a divide & conquer approach to the classical subset sum solver of Horowitz and Sahni. And for this, we design a decomposition function, when applied to the ...
6 votes
1 answer
209 views

Lower bound for the OR problem

Let us have booleans $x_1, \cdots, x_n$. Any algorithm that determines $\bigvee_1^n x_i$ with probability at least $2/3$ requires $\Omega(n)$ time. It is not too difficult to prove this, but the proof ...
0 votes
0 answers
51 views

Finding the best $k-$subset which maximizes a matrix sum

Let $M\in \mathbb{R}^{N\times N}$ be a given matrix and $k\ge 2$ be a given integer. Then my question is the following optimization problem: Is there a polynomial-time solution to the following ...
2 votes
0 answers
111 views

Asymptotic complexity lower bounds of proof checking

This paper on universal search mentions (on pp. 6-7) that proof checking can be done in $O(n^2)$ where $n$ is the length of the proof. Is this optimal? I don't want to specify the problem too ...
0 votes
1 answer
115 views

Complexity for universal Counter Machine with {0,1}-valued registers

Consider a universal $\{0,1\}$-$k$-counter machine where each of the $k$ registers has a value in $\{0,1\}$ (as opposed to any non-negative integer in the usual formulation), and there are states $q_1,...
5 votes
3 answers
332 views

Computational Complexity of the Divisor summatory function

The divisor function $d(n)$, is the number of $(a,b)\in\mathbb {N^+}^2$ such that $a\times b =n$. For example, $d(2)=2$ because $2=1\times 2=2\times 1$ and d(6)=4 because $6=1\times 6=2\times 3=3\...
1 vote
1 answer
50 views

Graph classes where giving a q-clique edge cover makes testing for q-colouring easy

A $q$-clique of a graph is a complete subgraph on $q$ vertices. A $q$-clique edge cover of $G$ is a set of subgraphs of $G$ such that each subgraph is a $q$-clique and each edge of G is contained in ...
5 votes
2 answers
347 views

Are there problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not hard for $DTIME(n^{k-1})$ under nearly linear time reductions?

Background It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$. Although some natural problems are known to exist, many of them ...
3 votes
1 answer
187 views

Fastest approximate triangle counting algorithms in dense graphs

One may compute the number of triangles in a graph by matrix multiplication in time $O(n^\omega)$. There is also a very simple algorithm that runs in time $O(n^3/(\epsilon^2 T))$ (where $T$ is the ...
1 vote
0 answers
72 views

Additive error approximations of GapP functions

Consider a GapP function $g(x)$ for $x \in \{0, 1\}^{*}$. Consider an approximation $\tilde g(x)$ such that \begin{equation} \left|g(x) - \tilde g(x)\right| \leq \epsilon. \end{equation} Consider a ...
11 votes
2 answers
3k views

Complexity of Finding the Eigendecomposition of a *Symmetric* Matrix

This is a specialized version of a previous question: Complexity of Finding the Eigendecomposition of a Matrix . For NxN symmetric matrices, it is known that O(N^3) time suffices to compute the ...
12 votes
2 answers
203 views

Is the exponent in the rectangular matrix multiplication convex?

My question is regarding the paper "Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor". In the paper, the authors show an algorithm for multiplying a ...
1 vote
1 answer
56 views

Amortized time and worst case (non-amortized) separation

Assume a reasonable computation model (thinking about pointer machine or RAM model), is there a problem where there is a clear separation between amortized and worst case complexity? Say, if ...
9 votes
1 answer
214 views

Is the Triangle Finding decision problem in $coNTIME(\tilde{O}(n^2))$?

The Triangle Finding decision problem asks whether there exists a triangle in a graph $G$ containing $n$ vertices. A triangle is a triple of vertices $(a, b, c)$ such that $a$ is adjacent to $b$, $b$ ...
0 votes
0 answers
24 views

Looking for information about a problem of a least subset of vectors modulo 2 summing to another vector [duplicate]

I'm quite interested in the following algorithmic problem, on which I can't find any information. Phrased as a decision problem: Given a set of vectors $V$ in $\text{GF}(2)^n$, a vector $\mathbf u$ ...
13 votes
1 answer
496 views

NEXPTIME-completeness with more time for reductions

One thing that surprised me when learning about complexity theory is that for a complexity class C, we tend to define C-complete using polynomial time reductions, even when C is a very large ...
1 vote
1 answer
62 views

What is the run-time of the bin packing approximation algorithm?

The best approximation algorithm that I found for the bin packing problem is by Hoberg and Rothvoss (SODA, 2017). In their Theorem 1.2, they mention that their algorithm finds a solution with at most $...
2 votes
0 answers
82 views

Does the awards budget cut problem support a sub $O(n\log n)$ time solution?

There's a famous problem these days in the interview prep community (particularly in PRAMP) called the awards budget cut problem. The problem gives you an input of $n$ integers called grants $g_1 ... ...
-1 votes
1 answer
253 views

What is the time complexity of computing intersection and union of Nondeterministic Finite Automata (NFAs)?

Assume that $\mathcal{A} = (Q_A, \Sigma, \Delta_A, q_{i_A}, F_A)$ and $\mathcal{B} = (Q_B, \Sigma, \Delta_B, q_{i_B}, F_B)$ are two NFAs. What is the worst-case time complexity of computing $\mathcal{...
1 vote
0 answers
215 views

Is there a known lower-bound on what the exponent could be, even if it turned out that P=NP?

Underlying motivation for the question: if someone showed that $\text{P}=\text{NP}$ but the algorithm thus produced for, e.g., $3\text{-SAT}$, runs in time $\Omega(n^G)$ where $G$ is Graham's number, ...
5 votes
2 answers
253 views

Can we recover integers $a_i$ from the sum $a_0 + a_1e+a_2e^2+\cdots+a_ne^n$?

Since $e$ is transcendental, the function $f:\mathbb Z_{\geq 0}^{n+1}\to \mathbb R$ is injective, $$ f(\underset{\text{Integers}\ \geq\ 0}{\underbrace{a_0,a_1,\ldots, a_n}}) = a_0 + a_1e+a_2e^2+\cdots ...
6 votes
0 answers
181 views

Computing the $n$-th bit of the binary representation of $\pi$

I (only) learned today about the following fact: The $n$-th binary digit of $\pi$ is computable without calculating all the previous digits. This apparently has been discovered in 1995, and follows ...
2 votes
0 answers
147 views

Schönhage-Strassen algorithm: why don't we work over $\mathbb{C}$

I am trying to understand how Schönhage-Strassen works for integers by studying von zur Gathen and Gerhard's Modern computer algebra. However they only talk about multiplication of polynomials. In the ...
14 votes
1 answer
850 views

Are there languages decidable in linear time by RAM machines that have superlinear time complexity lower bounds for Multitape Turing machines?

Question: Are there languages decidable in linear time by RAM machines that have superlinear time complexity lower bounds for Multitape Turing machines? Background: I recently stumbled upon the ...
43 votes
7 answers
18k views

Complexity of Finding the Eigendecomposition of a Matrix

My question is simple: What is the worst-case running time of the best known algorithm for computing an eigendecomposition of an $n \times n$ matrix? Does eigendecomposition reduce to matrix ...
1 vote
1 answer
88 views

Can a NEXP machine simulate invalid queries to a promise problem oracle?

Let $A=(A_{YES},A_{NO})$ be some promise problem (such as xSAT, the Local Hamiltonian problem, etc). Suppose we want to show that a P machine with access to a the oracle A can always have its output ...
19 votes
1 answer
1k views

Has parameterized complexity led to better algorithms?

I know that for the vertex cover problem, if we know that the parameter $k$ (which is the number of vertices in the solution) is small, then we can expect to solve it feasibly in practice. So far, ...
2 votes
1 answer
208 views

Is it possible to reduce an NP language to a NEXP language with exponentially smaller input length?

Suppose we have an NP-complete language $L_1$ and a NEXP-complete language $L_2$. For any deterministic exptime machine $M_1$ with oracle access $M_1^{L_1}$, is it possible to find a deterministic ...
-4 votes
1 answer
33 views

Big O question concerning time complexity

Why is O(f(n)) − O(f(n)) not equal to 0? Full disclosure this is a question from practice problems for my theory class.
1 vote
0 answers
45 views

Find all paths between specially paired nodes in a DAG in linear time

If I have a DAG with 2n nodes partitioned into n pairs of nodes with e edges, is there a ...
2 votes
0 answers
182 views

Is $SUBLOG\subset DTIME(n)$?

In the course of trying to give a more natural answer to a previous question of mine involving the complexity classes $$SUBLOG=\{L\mid L \text{ is recognizable by a sublogarithmic-space TM} \}$$ and $...
1 vote
0 answers
116 views

Bounds on the construction of regular expressions' intersection operator

There are references on the exponential worst-case of the intersection operator for regular expressions (see [1]). However, I was wondering if there are similar results for the construction process ...

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