Questions tagged [time-complexity]

The amount of time resources (number of atomic operations or machine steps) required to solve a problem expressed in terms of input size. If your question concerns algorithm analysis, use the [runtime-analysis] tag instead. If your question concerns whether or not a computation will *ever* finish, use the [computability] tag instead. Time-complexity is perhaps the most important sub-topic of complexity theory.

Filter by
Sorted by
Tagged with
78
votes
6answers
18k views

How can we assume that basic operations on numbers take constant time?

Normally in algorithms we do not care about comparison, addition, or subtraction of numbers -- we assume they run in time $O(1)$. For example, we assume this when we say that comparison-based sorting ...
37
votes
3answers
4k views

Decision problems vs “real” problems that aren't yes-or-no

I read in many places that some problems are difficult to approximate (it is NP-hard to approximate them). But approximation is not a decision problem: the answer is a real number and not Yes or No. ...
26
votes
2answers
5k views

Data structure with search, insert and delete in amortised time $O(1)$?

Is there a data structure to maintain an ordered list that supports the following operations in $O(1)$ amortized time? GetElement(k): Return the $k$th element of the list. InsertAfter(x,y): Insert ...
28
votes
4answers
34k views

The time complexity of finding the diameter of a graph

What is the time complexity of finding the diameter of a graph $G=(V,E)$? ${O}(|V|^2)$ ${O}(|V|^2+|V| \cdot |E|)$ ${O}(|V|^2\cdot |E|)$ ${O}(|V|\cdot |E|^2)$ The diameter of a ...
13
votes
2answers
18k views

Time complexity of a triple-nested loop

Please consider the following triple-nested loop: ...
21
votes
2answers
4k views

Can one show NP-hardness by Turing reductions?

In the paper Complexity of the Frobenius Problem by Ramírez-Alfonsín, a problem was proved to be NP-complete using Turing reductions. Is that possible? How exactly? I thought this was only possible by ...
14
votes
2answers
15k views

algorithm time analysis “input size” vs “input elements”

I'm still a bit confused with the terms "input length" and "input size" when used to analyze and describe the asymptomatic upper bound for an algorithm Seems that input length for the algorithm ...
2
votes
4answers
2k views

minimum subset of dominating 2D points

From an initial set $S$ of 2D points, how to efficiently compute a minimum(-size) dominating subset $M$ ? $M$ is a dominating subset of $S$ if for any $(x,y)$ in $S$ there is at least one point (a,b) ...
58
votes
8answers
185k views

What is a the fastest sorting algorithm for an array of integers?

I have come across many sorting algorithms during my high school studies. However, I never know which is the fastest (for a random array of integers). So my questions are: Which is the fastest ...
25
votes
3answers
7k views

Is it really possible to prove lower bounds?

Given any computational problem, is the task of finding lower bounds for such computation really possible? I suppose it boils down to how a single computational step is defined and what model we use ...
14
votes
2answers
591 views

Is there an efficient algorithm for expression equivalence?

e.g. $xy+x+y=x+y(x+1)$ ? The expressions are from ordinary high-school algebra, but restricted to arithmetic addition and multiplication (e.g. $2+2=4; 2.3=6$), with no inverses, subtraction or ...
1
vote
1answer
154 views

If a language is X-complete, is its complement is X-complete as well?

I'm looking for an information about closure of complexity complete classes. Is it true that any language, if the language is X-complete, then its complement is X-complete? Why? I was thinking ...
60
votes
8answers
4k views

Algorithmic intuition for logarithmic complexity

I believe I have a reasonable grasp of complexities like $\mathcal{O}(1)$, $\Theta(n)$ and $\Theta(n^2)$. In terms of a list, $\mathcal{O}(1)$ is a constant lookup, so it's just getting the head of ...
12
votes
5answers
3k views

Efficient algorithm to compute the $n$th Fibonacci number

The $n$th Fibonacci number can be computed in linear time using the following recurrence: ...
7
votes
1answer
1k views

Time Complexity of Regular Languages

I wonder how I can go about proving that if a language L is decidable in o(nlog(n)) then L must be regular. I should probably mention that by "decidable" I mean "being decidable by single-tape ...
6
votes
2answers
533 views

Modeling the problem of finding all stable sets of an argumentation framework as SAT

As a continuation of my previous question i will try to explain my problem and how i am trying to convert my algorithm to a problem that can be expressed in a CNF form. Problem: Find all stable sets ...
5
votes
1answer
802 views

Are there strongly-polynomial algorithms that take more than polynomial time?

In [1] strongly-polynomial is defined as either: The algorithm runs in strongly polynomial time if the algorithm is a polynmomial space algorithm and performs a number of elementary arithmetic ...
5
votes
1answer
633 views

Confusion about the Time Hierarchy Theorem and relativization

I know that $\mathsf{P}^A = \mathsf{EXP}$ for any $\mathsf{EXPTIME}$-complete language $A$. Is it true that $\mathsf{DTIME}^A(n^k) = \mathsf{EXP}$ for any fixed $k$ and any $\mathsf{EXPTIME}$-...
12
votes
4answers
1k views

Does the complexity of strongly NP-hard or -complete problems change when their input is unary encoded?

Does the difficulty of a strongly NP-hard or NP-complete problem (as e.g. defined here) change when its input is unary instead of binary encoded? What difference does it make if the input of a ...
5
votes
7answers
4k views

Is there a meaningful difference between O(1) and O(log n)?

A computer can only process numbers smaller than say $2^{64}$ in a single operation, so even an $O(1)$ algorithm only takes constant time if $n<2^{64}$. If I somehow had an array of $2^{1000}$ ...
16
votes
3answers
4k views

Why not to take the unary representation of numbers in numeric algorithms?

A pseudo-polynomial time algorithm is an algorithm that has polynomial running time on input value (magnitude) but exponential running time on input size(number of bits). For example testing whether ...
12
votes
3answers
4k views

How fast can we find all Four-Square combinations that sum to N?

A question was asked at Stack Overflow (here): Given an integer $N$, print out all possible combinations of integer values of $A,B,C$ and $D$ which solve the equation $A^2+B^2+C^2+D^2 = N$. This ...
28
votes
1answer
1k views

Is there a 'string stack' data structure that supports these string operations?

I'm looking for a data structure that stores a set of strings over a character set $\Sigma$, capable of performing the following operations. We denote $\mathcal{D}(S)$ as the data structure storing ...
19
votes
2answers
456 views

Problems that provably require quadratic time

I'm looking for examples of problem which has a lower bound of $\Omega(|x|^2$) for input $x$. The problem needs to have the following properties: $\Omega(n^2)$ runtime proof for any algorithm - ...
9
votes
3answers
7k views

Time complexity of base conversion

Why can't arbitrary base conversion be as fast as converting from base $b$ to base $b^k$ ? Seems to be a big time complexity difference! I am also interested in reading material about it. Old. ...
8
votes
4answers
2k views

Why is not known whether integer factorization can be done in polynomial time knowing how to do primality tests efficiently?

First of all, I have just started studying computer science by myself and maybe I just need some clarification of what "polynomial time" means regarding the time complexity of an algorithm and ...
7
votes
2answers
1k views

Quantum computers, parallel computing and exponential time

I've read that quantum computers can solve 'certain problems' exponentially better than classical computers. As I think I understand it, it's NOT the same to say that quantum computers take any ...
5
votes
1answer
3k views

Multitape Turing machines against single tape Turing machines

Introduction: I recently learned that a multi-tape Turing Machine $\text{TM}_k$ is no more "powerful" than a single tape Turing machine $\text{TM}$. The proof that $\text{TM}_k \equiv \text{TM}$ is ...
7
votes
2answers
628 views

How many strings are close to a given set of strings?

This question has been prompted by Efficient data structures for building a fast spell checker. Given two strings $u,v$, we say they are $k$-close if their Damerau–Levenshtein distance¹ is small, i.e. ...
6
votes
3answers
427 views

Shouldn't complexity theory consider the time taken for different operations?

I have read the answer found here which considers the size of integers when doing comparisons and how that affects on the basic cost of comparison. I am trying to understand why each basic operation ...
5
votes
3answers
304 views

Complexity inversely propotional to $n$

Is it possible an algorithm complexity decreases by input size? Simply $O(1/n)$ possible?
5
votes
3answers
252 views

Why does NTIME consider the length of the longest computation?

In Sipser's textbook "Introduction to the Theory of Computation, Second Edition," he defines nondeterministic time complexity as follows: Let $N$ be a nondeterministic Turing machine that is a ...
4
votes
2answers
212 views

Why Djikstra's algorithm is said to have $\mathcal{O}(|V|^2)$ complexity?

Djikstra's algorithm assigns some number to non-removed vertex each time it finds a path from removed vertex to it. Number of assignments is $\mathcal{O}(|V|^2)$. However, complexity of assignment is ...
4
votes
1answer
419 views

Oracle Turing Machine EXP^EXP

I'm reading Arora Barak and in that it is written that when $O \in \mathrm{P}$, then $\mathrm{P}^O = \mathrm{P}$. Can this be generalized? Intuitively, I think that $\mathrm{NP}^\mathrm{NP} \neq \...
2
votes
1answer
367 views

Generalized Geography with repetitions

Consider the "Generalized Geography" game: on directed graph G with selected start node, players take turns moving along edges, without ever going back to previously visited nodes. Last player to ...
2
votes
1answer
2k views

How can I prove DP completeness?

We defined the class $\text{DP}$ like this: $$\text{DP} := \{ A \setminus B : A, B \in \text{NP} \}$$ We say a problem $P$ is $\text{DP}$ complete iff $P \in \text{DP}$ and $X \leq P \forall X \in \...
1
vote
1answer
2k views

Complexity of matrix inverse via Gaussian elimination

I'm trying to determine the exact complexity of finding an $n\times n$ matrix inverse of $A$. If it is known that the complexity of Gaussian elimination is $\frac{2}{3}n^3 + \frac{1}{2}n^2+O(n)$, then ...
1
vote
1answer
526 views

Euclid's Algorithm Time Complexity

I have a question about the Euclid's Algorithm for finding greatest common divisors. gcd(p,q) where p > q and q is a n-bit integer. I'm trying to follow a time complexity analysis on the algorithm (...
0
votes
1answer
956 views

Relationship between an integer N and the number of bits n required to represent the integer

I'm trying to understand the time complexity of the following code in terms of n. Pseudocode for trial division: I understand that the time complexity of the algorithm is O(sqrt(N)). However, can ...
0
votes
2answers
74 views

Can you always prove the asymptotic bound of a recurrence of the form aT(n/b) + f(n) using the substitution method?

To make my question more concrete, here is an example I am stuck on. I want to prove that $T(n) = 8T(\frac{n}{2}) + n^3$ is asymptotic bound by $n^3\log(n)$ using the substution method. That is $T(n)$...
54
votes
4answers
84k views

Find median of unsorted array in $O(n)$ time

To find the median of an unsorted array, we can make a min-heap in $O(n\log n)$ time for $n$ elements, and then we can extract one by one $n/2$ elements to get the median. But this approach would take ...
39
votes
3answers
65k views

What exactly is polynomial time? [duplicate]

I'm trying to understand algorithm complexity, and a lot of algorithms are classified as polynomial. I couldn't find an exact definition anywhere. I assume it is the complexity that is not exponential....
20
votes
3answers
10k views

How hard is finding the discrete logarithm?

The discrete logarithm is the same as finding $b$ in $a^b=c \bmod N$, given $a$, $c$, and $N$. I wonder what complexity groups (e.g. for classical and quantum computers) this is in, and what ...
8
votes
2answers
5k views

Shor's Algorithm speed

I'm a fledgling computer science scholar, and I'm being asked to write a paper which involves integer factorization. As a result, I'm having to look into Shor's algorithm on quantum computers. For ...
18
votes
2answers
6k views

Difference between time complexity and computational complexity

For measuring the complexity of an algorithm, is it time complexity, or computational complexity? What is the difference between them? I used to calculate the maximum (worst) count of basic (most ...
14
votes
1answer
9k views

Can a Big-Oh time complexity contain more than one variable?

Let us say for instance I am doing string processing that requires some analysis of two strings. I have no given information about what their lengths might end up being, so they come from two distinct ...
10
votes
3answers
24k views

A d-ary heap problem from CLRS

I got confused while solving the following problem (questions 1–3). Question A d-ary heap is like a binary heap, but(with one possible exception) non-leaf nodes have d children instead of 2 ...
11
votes
4answers
25k views

Evaluating the average time complexity of a given bubblesort algorithm.

Considering this pseudo-code of a bubblesort: ...
10
votes
1answer
968 views

Collection of APX-hard problems

Everyone knows "Garey & Johnson", which is my go-to reference whenever I need a problem to transform from for an NP-hardness proof. However I recently find myself in need of an APX-hardness proof, ...
3
votes
1answer
2k views

Prerequisites of computational complexity theory

what's the prerequisite topics needed for understanding computational complexity theory and analysis of algorithm ...including big-O and Big-theta notations and these staff. I want a mathematical ...