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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32 views

Polynomial time delay and Amortized analysis

In order to estimate the complexity of output sensitive algorithm, we usually use Polynomial time delay or Amortized analysis. Then how to select the analysis methods, why we don't only use Amortized ...
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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 ...
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Is it possible to find closest number to a given irrational one, which is obtained by dividing 2 numbers in a given range faster than O(n^2)?

Let's say I have a range from 11 to 99 I need to find: abs(a/b)-k = min, a nd b - integer, k-an irrational number I can just look at all pairs of numbers in ...
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328 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 ...
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Why query complexity is said to lower bound time complexity?

Generally, query complexity is thought of as a lower bound on time complexity. It might be true if data is read sequentially. However, in quantum computation, state preparation routines exist that ...
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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(...
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Algorithm to find all vertices on some path between two points of a given set

Suppose I have a simple undirected (assume connected) graph $G=(V, E)$ and a set $V’$ of vertices of $G$. My goal is to find all vertices that lie on some simple path between two distinct vertices of $...
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1answer
102 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 ...
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417 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 ...
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55 views

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 ...
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201 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 ...
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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 ...
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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 ...
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1answer
107 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,...
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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\...
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1answer
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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 ...
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2answers
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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 ...
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1answer
168 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 ...
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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 ...
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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 ...
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2answers
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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 ...
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1answer
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 ...
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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 $b$, $b$ ...
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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 $\mathbf u$ ...
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1answer
470 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 ...
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1answer
56 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 $...
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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 ... ...
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165 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{...
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211 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, ...
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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 ...
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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 ...
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132 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 ...
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1answer
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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 ...
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7answers
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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 ...
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1answer
86 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 ...
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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, ...
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1answer
195 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 ...
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1answer
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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.
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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 ...
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180 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 $...
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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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12answers
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What are some algorithms where space complexity tends to be the limiting factor in practice?

Time complexity can't be any lower than space complexity (at least one operation is required to use a unit of memory), so what are some algorithms where space actually tends to be the limiting factor? ...
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1answer
109 views

Complexity of acyclicity of a "nondeterministic" graph

By "nondeterministic" I mean the graph is a collection of sets of "candidate" edges sharing a single destination: $E \subseteq 2^V \times V$. The problem is whether it is possible ...
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887 views

Can we compute $n$ from the bits of $3^n$ in $O(n)$ time?

I'm seeking an efficient algorithm for the problem: Input: The positive integer $3^n$ (stored as bits) for some integer $n \geq 0$. Output: The number $n$. Question: Can we compute $n$ from the bits ...
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131 views

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

Every source I can find just says "too big to be practical."
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162 views

Time complexity of Succinct-CVP

I want to know what is the best known lower time complexity of Succinct-CVP? The succinct version of many P-complete problems are EXP-complete and Succinct-CVP is EXP-complete too (It is because of ...
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2answers
160 views

Complexity of Set Difference

Given $k$ sets $S_1$, $S_2$, $\dots$, $S_k$ in the universe $U = \{1, 2, \dots, n\}$, is there a way to preprocess the $k$ sets such that there is an output-sensitive query algorithm that computes $...
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2answers
339 views

3SUM Complexity—A special(?) Case

In the paper “Consequences of Faster Alignment of Sequences” by Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann which appeared in ICALP 2014 and is available here the following version of ...
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478 views

Implication of solving 3SUM problem of a certain size on the Exponential Time Hypothesis

In the recent question 3SUM Complexity—A special(?) Case I asked about why the set size $O(n^3)$ was an interesting value for the 3SUM Problem and got a nice answer. My reference was the paper “...
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Most general setting for fine-grained exponential-time complexity classes?

Consider the class of functions computable in time $(b+o(1))^n = 2^{\log_2{(b)} \times n + o(n)}$ on a $2$-tape Turing machine. By the Hennie-Stearns theorem, the same functions are computable in ...

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