Questions tagged [time-complexity]

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

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Complexity of finite difference method

if I have a function $f:\mathbb R^N\to \mathbb R$ and i would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input, which is the computational complexity using the (ones sided) ...
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Time complexity for multiplying two lower triangular matrices

I was wondering, if multiplication of two $n \times n$ lower (or upper) triangular matrices has a more efficient algorithm than multiplication of two general $n \times n$ matrices? $$ \begin{bmatrix} ...
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Time complexity of binary search tree construction

"Every comparison-based algorithm to sort n elements must take Ω(nlogn) comparisons in the worst case. With this fact, what would be the complexity of constructing a n-node binary search tree and why?"...
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What is the evidence that the worst-case runtime of CDCL is exponential in the length of CNF?

Current SAT solvers have worst-case runtime exponential in the number of variables, leading many computer scientists to speculate that P does not equal to NP. However, the P vs NP problem cares only ...
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Program size versus program running time

Short "naive" question: Is it true that faster algorithms require longer programs ? Given a decision problem $A$ and a reasonable model of computation, there can be many ways (algorithms) to ...
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Running an algorithm for fixed amount of time on RAM model machine

Suppose there is a deterministic algorithm of size $O(1)$ that operates on an input of size $N$ on a RAM model machine. I want to run the algorithm for $O(\sqrt{N})$ time, pause the algorithm, print "...
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Is PP invariant under changing its cut-off from 1/2 to another number?

Suppose I have a fixed family of quantum circuits $\{C_i\}$ for which determining whether the maximum output acceptance probabilities are $p\geq 1/2$ or $p< 1/2$ is PP-hard. Now suppose I have the ...
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Are there complexity theory consequences of the collapse NEXP=EXP^NP?

It is clear that $NEXP\subseteq EXP^{NP}$, as a TM with exponential run time can simply query the NP oracle with an exponentially long query. However, it's not clear that the reverse $EXP^{NP}\...
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Complexity of solving systems of linear equations with hash preimages

Introduction: I'm researching a decision problem that I thought was in NP because there are certificates for its instances that have a polynomial number of elements. However, I realized that there are ...
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Sorting using comparisons that are not simple mappings of simple comparisons

The Python language has a sort(x) function that sorts a list based on the intrinsic comparison operator associated with the type of the elements of its input list x. One can also provide a cmp ...
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1answer
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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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Problem in deterministic time $n^p$ and not lower

I'm looking for any language $L$ candiate to be in $DTIME(n^p) -DTIME(n^{p-1})$ (it takes at least $n^{p-1}$ steps to determine if an input is in L with a 2-tape $TM$, but L is polynomially solvable). ...
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Determining if a word of specific length exists that is not accepted by a NFA

It is known that the problem of determining if an NFA accepts every word is PSPACE-COMPLETE, meaning it is also NP-Hard, but is this weaker version of the problem still NP-hard? Given an NFA and a ...
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Fast algorithms for evaluating functions with high Kolmogorov complexity

Motivation: I am motivated by a concrete example that occurs in neuroscience, dendritic computation, which may be approximated by functions computable on binary trees [1]. To be more precise, I ...
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Complexity of multi-objective optimization problems

How can we define and prove the worst-case complexity of multi-objective optimization problems (MOOP)? It is easy to see that, if one of the objectives is an NP-Hard optimization problem, then the ...
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Complexity of unbalanced bipartite isomorphism

For $i=1,2$, let $G_i=(A_i\cup B_i,E_i)$ be an undirected bipartite graph with bipartition $A_i$ and $B_i$, where $|A_1|=|A_2|=a$ and $|B_1|=|B_2|=b$ with $a\le b$. Question. Is the problem of ...
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Induction on all polynomial runtimes?

Has there ever been a proof technique to show that a language isn't in $\mathrm{P}$, by showing inductively there isn't any $k$ for which the language is in $\mathrm{TIME}(n^k)$? e.g.: $L\notin \...
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What is the run-time of LP?

Are there any further generalizations known to the result about run-time of a LP than what is stated in Theorem 1 of these lecture notes, https://nisheethvishnoi.files.wordpress.com/2018/05/lecture71....
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Are there common names for the subtiers of PTIME?

We all know P, or PTIME, I think, as a common name for the class of polynomial-time problems. Are there common names for the first few levels inside P; that is, for constant-time, linear-time, ...
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A sandwich Algorithm / Data Structure [closed]

$O(n^c)$ is asymptotically greater than $O(\log^d n) $ for all possible pair of values of $c$ and $d$. Can you give an example of a problem (or data structure) which has running time (or query/...
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Evidence integer multiplication is in linear time?

After millenia of quest we have identified two $n$ bit integers can be multiplied in $O(n\log n)$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-...
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When can convex optimization be considered to be exactly solvable?

If one is trying to find the global minima of a convex function using gradient descent then one will get a run-time which is a function of $\epsilon >0$ where $\epsilon$ measures the accuracy of ...
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Complexity of extracting a coefficient of a polynomial in multiple variables

I'm looking for efficient algorithms for problems of the following type: Let's say we have the variables $x_1,...,x_n$. Over these variables, we are given a function $p_1\cdot ... \cdot p_m$, ...
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What is a natural problem in theory of computation?

In Stephen Cook's paper on the P vs NP problem,[1] he states the following [2]: Feasibility Thesis: A natural problem has a feasible algorithm iff it has a polynomial-time algorithm. My question ...
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How to prove a general convex set is nonempty or empty in polynomial time?

The general convex set should be represented by a set of (generalized) inequalities $f_{i}(x)\leq 0 $ with $ f_{i}(x) $ being convex in $ x $. I know ellipsoid method and interior method, but I do ...
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Consequences/existence of problems without any “optimal” algorithm

Let $P$ be some kind of "problem" such as addition or graph coloring, that has an input size $n$. Let $S_P$ denote the set of algorithms $A_1, A_2, \dots$ which deterministically solve $P$. Based off ...
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Nondeterminstic Linear Time vs Other Complexity Classes

Is it known whether or not nondeterministic linear time contains $P$ and/or smaller classes such as Uniform-$NC^1$?
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what does NP ⊆ DTIME(…) mean?

Recently I've seen inside theory of a paper. This time complexity, DTIME, is completely new for me. Can somebody explain it? Also, the paper shows that the misinformation containment problem cannot ...
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Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?

Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
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Cost of in-place partitioning integer arrays

Suppose we are given an array $a\colon[n]\to[m]$ of length $n$ (and each entry is between 1 and m). We will denote the $i$th entry of the array as $a[i]$. Task: Permute the array $a$ in-place so that ...
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Given a subset of of the hypercube and an affine transform of it, find the affine map

This is a follow up to this resolved question. Suppose we are given a set of bitvectors $A\subseteq\mathbb{F}_2^d$ and an invertible affine transformed copy of it $$B=\{Mx + s\mid x\in A\}$$ for some ...
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Given a subset of the hypercube and a copy translated by s, find s

Problem: Suppose we are given an $n$ element subset $A\subseteq\{0,1\}^d$ of the $d$ dimensional hypercube and a translated copy $B= A+s$ by some secret $s\in\{0,1\}^d$. Find $s$ as fast as possible ...
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Hidden Constants in Complexity of Algorithms

For many problems, the algorithm with the best asymptotic complexity has a very large constant factor that is hidden by big O notation. This occurs in matrix multiplication, integer multiplication (...
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Faster than linear set intersection

Is there a probabilistic data structure, such as the bloom filter, for which we could perform intersection superlinearly (at the cost of false positives)? (I only know of a O(n) time complexity set ...
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Techniques to improve the efficiency of Dynamic Time Warping Algorithm

I am analyzing a set of time series that are shifted along the x-axis (see image below for clarification). I intend to average the time series and for that I would like to overlap all the start points ...
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Is counting simple cycles in $P$ for graphs of bounded tree width?

Motivation: Determining if a graph has a Hamiltonian cycle is $NP$-hard in general. However, determining if there is a Hamiltonian cycle is in polynomial time on graphs of bounded tree width, either ...
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Why is $BPP^{NP}$ in polynomial hierarchy? [closed]

Why is $BPP^{NP}$ in the polynomial hierarchy? I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. However, how does that imply $BPP^{NP}$ is inside the polynomial hierarchy?
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Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?

Is there a relation between BBH (black box hypothesis) and SETH (strong exponential time hypothesis)?
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Complexity of comparing extended integer power towers

Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...
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Finding 3SUM witness when promised a solution

Suppose we have a 3SUM instance given with the promise that there exists at least one solution. Is the trivial $O(n^2)$ (modulo logarithmic improvements) solution still the best algorithm or is there ...
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Arranging sets in a hierarchy

Suppose you have sets $S_1, \dots S_m$ such that $\sum_i |S_i| = n$. The goal is to arrange all the sets into a (possible unconnected) DAG such that $S_i$ is a parent (or ancestor) of $S_j$ iff $S_j \...
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how is time complexity defined in computational learning theory

In general, when we say an algorithm $A$ PAC learns $C$ in time $t$, we say $A$ takes time $t$ before outputting a hypothesis $h$, and the hypothesis can be evaluated (on every $x$) in time $t$. Now ...
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Time complexity of finding a point of infinite order on a rank 1 elliptic curve over Q

As an outsider, it sounds like a lot of progress has been made on understanding rank 1 elliptic curves over Q. Much of the BSD conjecture is known for rank 1, and Heegner points provide a way in ...
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Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)

In the wikipedia article on Time Complexity it is written that: The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with, ...
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Parallel building time of a k-d tree on n points with n processors

Given a point set with $n$ points to build a k-d tree on. We have $n$ processors available. What is the time-optimal building time for the k-d tree? A straight forward parallelization would be as ...
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Minimize The Number of Connected Components in Hit-map of A Boolean Matrix

Suppose there is a matrix with the value of 0 and 1. The hit-map of the matrix (0 is blue and 1 is red) create some connected component (see the following figure as an instance): Is there any ...
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Worst case polynomial in elimination theory under rank conditions?

Given $n$ polynomials $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})\in\mathbb Z[x_1,\dots,x_{2n}]$ where each of $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})$ is homogeneous of degree ...
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Evaluating addition chains

I hope this is a suitable place to ask this question. An addition chain of size $n$ is a sequence $x_1, \dots, x_n$, where $x_1$ is fixed to 1 and $x_i = x_j + x_k$ for some $j,k < i$. I am ...
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EXPSPACE proof and its implications

I'm dealing with the min-max regret 0-1 Integer Linear Programming problem (MMR-ILP, for short), which is formulated as below. \begin{equation} \label{eq:nip_obj} \min_{x \in \Phi} \sum_{i = 1}^n ...
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Time complexity of exponentiating s-sparse matrices

Could someone suggest me a reference which discusses the time complexity of algorithms meant for exponentiating (finding $e^A$ approximately given $A$) s-sparse matrices, along with their error rates? ...

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