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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.

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 tag instead. If your question concerns whether or not a computation will ever finish, use the tag instead. Time-complexity is perhaps the most important sub-topic of .

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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 is $O(n\log n)$, but when numbers are too big to…
user742
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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 $O(n \log n)$ time. Can we do the same by some…
Luv
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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 currently known sorting algorithm? Theoretically, is it…
gen
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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 the list. $\Theta(n)$ is where I'd walk the entire…
Khanzor
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What exactly is polynomial time?

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. Do linear/constant/quadratic complexities…
Oleksiy
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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. Also for each desired approximation factor,…
Ran G.
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Why is push_back in C++ vectors constant amortized?

I am learning C++ and noticed that the running time for the push_back function for vectors is constant "amortized." The documentation further notes that "If a reallocation happens, the reallocation is itself up to linear in the entire…
David Faux
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Largest set of cocircular points

Given $n$ points with integer coordinates in the plane, determine the maximum number of points that lie on the same circle (on its circumference, not its interior). This can be done in $O(n^3)$ easily by trying $\binom{n}{3}$ combinations of…
chubakueno
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Problems that are polynomially "hard" to compute but "easy" to verify

In the (unlikely) event that $P=NP$ with a constructive proof of a polynomial time algorithm that solves 3SAT, obviously things will be very different. However, practically, it could happen that the degree of the polynomial run time is very large…
zyl1024
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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 graph $G$ is the maximum of the set of shortest path distances…
Gigili
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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 the set of strings $S$. Add-Prefix-Set on…
Alex ten Brink
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Is there a name for the class of algorithms that are the most efficient for a particular task?

This would be analogous to the Kolmogorov complexity of a string, except that in this case, I'm interested in the algorithm that solves a given problem using the least number of steps. We would therefore have to be able to show that any other…
Feynmanfan85
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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 for the proof, but given that, do we really prove…
hsalin
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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 the new element y into the list immediately after…
A T
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Understanding of big-O massively improved when I began thinking of orders as sets. How to apply the same approach to big-Theta?

Today I revisited the topic of runtime complexity orders – big-O and big-$\Theta$. I finally fully understood what the formal definition of big-O meant but more importantly I realised that big-O orders can be considered sets. For example, $n^3 + 3n…
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