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Questions tagged [asymptotics]

Questions about asymptotic notations and analysis

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Does converting adjacency matrix representation of graph of size $n \times n$ to adjacency list always require $O(n^2)$ time?

Assume that I have the adjacency matrix representation of a graph in $0,1$ values. Does converting it to a corresponding adjacency list representation always have a time complexity of $O(n^2)$?
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Why integer division is of equal complexity as multiplication

I am trying to understand the fact that integer division is no more difficult than integer multiplication. I found some references - here and this lecture note. Wikipedia says if there is a way to ...
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Analyzing asymptotic notation $\sqrt n = O(\log^2 n)$

I am trying to determine whether $f(n) = \sqrt n$ is in $O(g(n))$, $\Omega(g(n))$, or $\Theta(g(n))$ where $g(n) = \log^2 n$. The answer says that only $f(n) = \Omega(g(n))$ is correct, but why isn't ...
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Algorithm to find sum of all values in a 2D matrix when i<j [closed]

Given an array A of n integers, output an n by n array B in which B[i,j] (for all i=j values of B[i,j] are left unspecified so ignore that case). My approach: The most natural way to do this is by ...
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What is the asymptotic complexity of the following code snippet?

for (i = 2; i < n; i = i * i) { for (j = 1; j < i / 2; j = j + 1) { sum = sum + 1; } } I know that the outer loop can run for a maximum of $n^2$ ...
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1answer
36 views

Asymptotic relation between n! and (n+1)!

I am having difficulty writing this formally. I know that by L'Hospital's rule we can reduce it to $\lim_{n \to \infty} \frac{n+1}{n}$ which is a constant and hence $n = \theta (n+1)!$. But I am not ...
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Master Method: $T(n) = 10T\Big(\frac{n}{2}\Big) + \frac{n^4}{\log(n)}$

I'm having a hard time trying to understand how to solve this recurrence relation using the Master Method: $$T(n) = 10T\Big(\frac{n}{2}\Big) + \frac{n^4}{\log(n)}$$ First, we have: $a = 10,\ b = 2$ ...
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Algorithm comparison

I am learning Big O and Big theta notation and confused the certain case. I have two functions, function 1(f1) $$ n * n^{1/2} $$ function 2(f2) $$ 1.001^n $$ in smaller cases (10,000) f1 is much ...
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The applicability of the Master Theorem and calculation of asymptotic limits

Given the following recursive equation $T(n)=3T(\dfrac{n}{8})+ Θ(n^{1/3})$ I want to know how to explain the applicability of the Master theorem in a rigorous way and what means asymtotic limits of ...
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Meaning of $ e^x = 1 + x + Θ(x^2)$?

In the CLRS chapter 3: When $x → 0$, the approximation of $e^x$ by $1+x$ is quite good: $$e^x = 1 + x + Θ(x^2).$$ How is it to be interpreted, what is the role of asymptotic notation here?
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Big Oh/Omega/Theta Notations [duplicate]

I am trying to learn Asymptotic notations from the famous Algorithms, Dasgupta,Vazirani book. I found some exercises at the end of the chapter, which I have attempted to solve, but I am not sure if I ...
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Running Time of Sorting Algorithm

Determine the asymptotic running time of the sorting algorithm maxSort. Algorithm maxSort(A) Input: An integer array A Output: Array A sorted in non-decreasing order ...
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What is the depth of recursion if we split an array into $\log_2(n)$ with each recursive call?

We have a function which takes an array as input. It breaks an array into $\log_2(n)$ parts with equal sizes where $n$ is the size of the subarray. It keeps breaking each of the subarrays until there ...
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1answer
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Are the following Big Oh Notations equivalent?

In the context of Upper bounds computaion and Big Oh Notation, I was wondering if the following could be proved... if they are equivalent. $\mathcal{O}((log(n))^{-1}) = (\mathcal{O}(log(n)))^{-1}$ $\...
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$f(n) = o(n^c) \rightarrow \exists \epsilon > 0 \ s.t. f(n) = O(n^{c-\epsilon})$

I'm trying to prove that for arbitrary $c > 0$, $f(n) = o(n^c) \rightarrow \exists \epsilon > 0 \ s.t. f(n) = O(n^{c-\epsilon})$ Intuitively, this seems to be true to me (little-o implies ...
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Running Time for Finding Maximum

Consider the algorithm findMax that finds the maximum entry in an integer array. Algorithm findMax($A$) Input: An integer array $A$ Output: The maximum entry of $A$ ...
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How to calculate time complexity of a randomized search algorithm?

Example: Finding an element from a sorted array Let's say we have an algorithm that accepts a sorted array of length N as its input. Then in each iteration it randomly selects an element from the ...
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Are Θ,Ο,Ω, etc. considered functions of function? [duplicate]

Are Θ,Ο,Ω, etc. considered functions of function? Or just sets of function?
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1answer
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Bounds on “well dispersed” sparse matrices

Suppose we have an $n\times n$ zero/one matrix $M$, with $k$ ones. Let us say that the extent of $M$ is the maximum of $i+j$ over all ones at positions $(i,j)$ of the matrix, and the quality $q(M)$ is ...
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Performance of Modified Dijkstra's algorithm with Binariy heap as Priority Queue

we know the performance of Dijkstra's algorithm with binary heap is O(log |V |) for delete_min, O(log |V |) for insert/ decrease_key, so the overall run time is O((|V|+|E|)log|V|). Now let's modify ...
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Terminology for worst-case N-complexity on $O(1)$ insert after amortisation

Normally, when discussing amortisation and worst-case complexity, amortisation negates the worst-case scenarios, and the BigO describes the average for the operation (the way it's used in interviews ...
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Are Big-Theta functions asymptotic monotonically non decreasing?

For example, suppose $f(n) = \Theta(n^2)$, then does that mean for any sufficiently large $n$, $f(n) \le f(n+1)$? Is it a general case for all Big-Theta?
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Is $f(cn)$ always $O(f(n))$ for constant $c$ and any function $f$?

This seems to be true for any function I can think of, but I'm not quite sure how to prove it. Is there a proof of this proposition for any such function or a counter-example?
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What is the Big-Ω of the following function?

For the following function: $$ \sum_{n=1}^{2n}x+x^2 $$ It is easy to see the (tightest) Big-Oh is $O(n^3)$, but I am not so sure about the Big-Omega. Here is my attempt: $$ \sum_{n=1}^{2n}x+x^2 $$ $...
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d-ary heap implementation vs Fibonacci heap implementation Dijkstra performance comparions

Let's say that Dijkstra’s algorithm with the priority queue using a d-ary heap. if adjusting d, we can try to achieve the best runtimes for the algorithm with d being $\sim |E|/|V|$. Then for a ...
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When is this even possible (even for a dense graphs) $|E| = \Theta (|V|^2)$

Wikipedia says that "a dense graph is a graph in which the number of edges is close to the maximal number of edges." and "The maximum number of edges for an undirected graph is $|V|(|V|-1)/2$". Then ...
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Run time analysis of inner loop [duplicate]

What is the run time of the following piece of code in Big-Oh notation? The first loop runs n times in the worst case. But I am having difficulty in finding run time of nested loop which runs V / deno[...
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forming recurrence equations from code [closed]

Please could someone help me with understanding how to form recurrence equations when reading code? I'm having some trouble in my class: ...
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Runtime explanation of this function [duplicate]

I am trying to understand the runtime complexity of the below code in terms of n. I know that it is $Θ(n^{4/3})$, but I don't get why. I thought the outer loop runs $log(n)$ times, the second one ...
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Is $O(T+\log T)= O(T\log T)$?

Is $O(T+\log T)= O(T\log T)$? I think this is true but I do not know how to show it mathematically? Please show it using the definition. Also, if it is true, is the following true? $O((T+\log T)^{...
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Missing part of the proof of Master Theorem's case 2 (with ceilings and floors) in CLRS?

I am trying to go through the proof of the Master Theorem in Introduction to Algorithms of Cormen, Leiserson, Rivest, Stein (CLRS). The theorem providers an asymptotic analysis for recurrence ...
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1answer
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Why $2R\sigma\sqrt{T+logT+1}=\tilde{{O}}(\sigma\sqrt{T})$?

On page 17 on the paper Online Learning with Predictable Sequences, we find a regret of an algorithm equal to $$ \text{Reg}_T=\frac{R^2}{\eta}+\frac{\eta}{2}\sigma^2(T+logT+1) $$ where $T$ is the ...
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Can I say the two cases of Recursion Tree are always either $\theta{(n)}$ or $\theta({n\log{n}})$

Given positive constants: $c_1, c_2, ..., c_k, c^\prime$, assume that $T(n) = T(c_1n) + T(c_2n) + ...+ T(c_kn) + c^\prime n$ There are two cases: $c_1 + c_2 + ...+ c_k < 1$ $c_1 + c_2 + ...+ c_k ...
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How to properly calculate dependent nested loops for big-O [duplicate]

I am revising for my algorithms exam and I have come across one topic in particular that I do not quite understand; which is how to analyse dependent nested loops. I know if we have a 2-nested loop, ...
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Show that for any real constants $a$ and $b$, where $b > 0$, $(n + a)^b = \Theta(n^b)$

I'm currently studying growth of function chapter in Introduction to Algorithm. In exercise 3.1-2 the question is: Show that for any real constants $a$ and $b$, where $b>0$, $(n + a)^b = \...
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Big O analysis trying to follow a logic

Can someone please help me understand why(the derivation) "and m = 2n+1 for each n."? I am trying to follow the logic of the solution provide while myself have a different approach. Here is my ...
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50 views

Let $f(n)=\Omega(n), g(n)=O(n)$ and $h(n)=\theta(n)$ then $f(n).g(n)+h(n)$ is?

Let $f(n)=\Omega(n), g(n)=O(n)$ and $h(n)=\theta(n)$ then $f(n).g(n)+h(n)$ is? My attempt: Lets $f(n)=g(n)=n$, then $f(n).g(n)+h(n)=\Omega(n^2)+\theta(n)=\Omega(n^2)$ But given answer is $O(n)$. ...
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O(·) is not a function, so how can a function be equal to it?

I totally understand what big $O$ notation means. My issue is when we say $T(n)=O(f(n))$ , where $T(n)$ is running time of an algorithm on input of size $n$. I understand semantics of it. But $T(n)$ ...
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How come O(n) + O(logn) = O(logn)

How come O(n) + O(logn) = O(logn)? When talking for example about an algorithm that has two operations. One of them takes O(n) and the other O(logn) and in the end we say that the total complexity is ...
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Is this a valid use of big-O notation?

Suppose that $m=O(n^{c+1/2})$ for some real $c>0$ and $x=O(\sqrt{\log m})$. Are the following two computations valid? I understand that I'm abusing notations a bit to get at the desired results. ...
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What is the solution of $T(n, m) = T(n, m-1) + T(n-1, m) + c$?

Consider the recurrence $$ T(n,m) = T(n,m-1) + T(n-1,m) + c, $$ with base cases $T(n,0) = T(0,m) = 1$. This is the complexity of a recursive algorithm for the longest common subsequence, I know that ...
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How to compute the complexity of $T(n) = T(n-2)+T(n-3)+2T(n/3)$?

$T(n) = T(n-2)+T(n-3)+2T(n/3)$ and $T(n)=1$ for $n<4$. I tried to compute the complexity of $T(n) = T(n-2)+T(n-3)+2T(n/3)$ using the recursion tree but it's not clear enough for me to make a guess ...
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Big O space complexity of this isAnagram method

I'm currently debating with some friends what is the Big O space complexity of this isAnagram method: ...
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Find an asymptotic bound for $T(n)=n^2+T(\frac{n}{2})+T(\frac{n}{4})+T(\frac{n}{8})+…+T(\frac{n}{2^k})$

Given is the following recurrence relation: $T(n)=n^2+T(\frac{n}{2})+T(\frac{n}{4})+T(\frac{n}{8})+...+T(\frac{n}{2^k})$ where $k$ is some constant and $n = 2^t$ for some $t\in \mathbb{Z}$. I'm ...
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Is “super-exponential” a precise definition of algorithmic complexity?

I cannot seem to find a precise definition of what "super-exponential" is supposed to refer to when one's talking about an algorithm's time complexity. For instance, if an algorithm runs for $nC(n-1)$...
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Showing $2^x$ is a lower bound

How do I show that $2^x - x^2 \in \Omega(2^x)$? Basically, I know that this means that $\exists a, x_0 \in \mathbb{R^+}, \forall x \in \mathbb{N}, a.2^x \leq 2^x - x^2$. I worked around a bit with ...
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Big O order of a function

I'm doing some practice questions on Big O notation and came across this question. What is the Big O order of function 𝑓(𝑛) = 𝑛^2 + 𝑛 log2(𝑛) + log2(𝑛). Show your working. My answer is O(n^2) ...
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1answer
44 views

How to find sets of polynomially bounded numbers whose subset sums are different?

Let $n$ be any positibe integer and set $N=\{1,\dots,n\}$. Now select two arbitrary but different subsets of $N$, say $S$ and $S'$. We are interested in finding a function $\pi(A)=\sum_{i\in A}a_i$ ...
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2answers
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Is the runtime of binary search big omega of logarithm of n?

My question is that can we say that runtime of the binary search is $\Omega(\log n)$? I know it is both $\Omega(1)$ and $O(1)$ for the best case, and $\Omega(\log n)$ and $O(\log n)$ for the worst ...
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1answer
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Big O and constants [duplicate]

I've already asked this question on stack overflow, but guys have suggested me to ask my question here. Let's consider classic big O notation definition (proof link): The $O(f(n))$ is the set of all ...