# Questions tagged [asymptotics]

Questions about asymptotic notations and analysis

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### What does $|V|=O(|E|)$ mean?

I was reading about Dijkstra's algorithm from this Stanford University lecture presentation. On page 18 it says Dijkstra's algorithm is $O(|V|\log|V|+|E|\log|V|)$ and I understand why. But then it ...
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### How to find kth largest element in (max) priority queue in O(m) time?

Here is my exercise. FINDLARGEST(k): return the elements in the heap with key >=k" ... "expand the priority queue (max-heap) so that it supports FINDLARGEST(k) in O(m) time, where m is the number ...
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### Useful conditions for proving super polynomial lower bound for some kind of recurrences

Given a recurrence of the form $\forall n,m.\ \ T(n,m)=\begin{cases}1,&,m=1\\\sum_i{T(n_i,m_i)}&,\text{else}\end{cases}$ Note: both $n_i$ and $m_i$ are dependent on $n,m$ so they should have ...
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### Struggling to understand the symbolism around the big oh formal definition

I'm struggling to understand what exactly T(n), and f(n) is in the above text: When we compute the time complexity T(n) of an ...
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### Mapping every character to its next occurrence based on the number of unique characters between the occurrences

To optimize my LF mapping, I was asked to do the following. Given a string, say $abaxyxwxbx$ I need to encode it in a way where every index stores the value of the number of unique characters ...
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### Is there an O(1) solution to find the kth-smallest element in an implicit min-heap?

I know this would be an O(k log n) operation on a traditional heap, and I know there are ways to maintain Kth-smallest over a stream of inserts/deletes for constant-time access... My question though ...
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### runtime of 2 dependent nested for loops [duplicate]

for (i=1; i<=n ;i=i*2){ for (j=1; j<=i ;j++){ basic_step; } } Regarding the above nested loops, I can't seem to understand why is the following ...
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### Is there a unit of measurement that can express code execution speed in absolute terms?

I've always seen code execution speed measured either in units of time (e.g. t milliseconds), or using asymptotic analysis (e.g. O(n log n)). Execution speed will vary depending on hardware ...
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### Time complexity of rotation array m times using temporary array

I am new to asymptotic analysis, on solving the array rotation problem on geeksforgeeks the first solution provided was using a temporary array, I tried implementing this logic and found that the ...
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### How to find an algorithm's complexity from actual running times

I have a certain algorithm which I can run, but I do not have access to its code. Thus, it works as a black box. I would like to now the order of complexity of this algorithm on a certain set of ...
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### Approaches for analyzing the work, critical path length and parallelism

I'd like to know where to find references and approaches on how to analyze the work, critical path length and parallelism of algorithms. In particular, for solving the type of homework problems below:...
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### Deducing $3^f = o(3^g)$ from $f = o(g)$

I really need help solving the following question: Given: $$f(n) = o(g(n))$$ Prove: $$3^{f(n)} = o(3^{g(n)})$$ My attempt: I know that $\frac{f(n)}{g(n)} \xrightarrow{} 0$. I need to ...
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### Average case analysis of linear search

Based on CLRS question 2.2: Consider linear search again. How many elements of the input sequence need to be checked on the average, assuming that the element being searched for is equally likely to ...
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### The role of asymptotic notation in $e^x=1+𝑥+Θ(𝑥^2)$?

I'm reading CLRS and there is the following: When x→0, the approximation of $e^x$ by $1+x$ is quite good: $$e^x=1+𝑥+Θ(𝑥^2)$$ I suppose I understand what means this equation from math ...
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### Are there any problems that get easier as they increase in size?

This may be a ridiculous question, but is it possible to have a problem that actually gets easier as the inputs grow in size? I doubt any practical problems are like this, but maybe we can invent a ...
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### Prove that $T(n) \leq 8n^2$ or find value of $n$ when statement is not true (recurrence relation)

We have a function $T: \mathbb{N}\to\mathbb{N}$ defined recurrently: $$T(n)=\begin{cases} 0 &\text{ if } n=0,\\ 3T(\lfloor{n/2}\rfloor) + 2n^2 &\text{otherwise.} \end{cases}$$ Prove that for ...
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### How to use Master Theorem with strange format of $b$ parameter?

I have a funcion $T: \mathbb{N}\to\mathbb{N}$ defined as: $$T(n)=\begin{cases} 6 &\text{ if } n=0,\\ T(n-1) + 6n + 6 &\text{otherwise.} \end{cases}$$ How can I apply the Master Theorem to ...
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### which rule can conduct this formula $\log n = O(n^{0.000001})$? [duplicate]

i am learning this post about Big O, which gives this formula $$\log n = O(n^{0.000001})$$ why is that?
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### 3 * O(n^2) vs O(n^3)

Currently while I was coding, I got a doubt. While I was solving a particular type of problem I found it to be solved in $O(n^3)$. I have broken that problem and solved it in $O(n^2)$. But to ...
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### Asymptotics of a sinusoid

Consider the function $$f(n) = 2n^2 |\sin(\pi \cdot n/2)|.$$ Which of the following classes does $f(n)$ belong to? $$O(n^2), \Omega(n^2), \Theta(n^2), \omega(n^2), o(n^2).$$ I'm working in this ...
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### Proving that there exists a distance $d$-dominating set of size $O(n/\delta)$

Let $d > 1$, and consider a graph $G = (V,E)$ on $n$ vertices. A distance $d$-dominating set of $G$ is a set $D \subseteq V$ with the property that for any $v \in V$, either $v \in D$ or $v$ is at ...
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### How can I prove the linear time search algorithm takes O(n) time? [duplicate]

The recurrence relation for the algorithm is an eccentric form that has an additional term: $T(n) = T[\frac{n}{2}] + T[\frac{7n}{10} + 6] + n$. Exactly how can I prove that this recurrence relation ...
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### Mathematically calculating time complexity

This is a thread about mathematically calculating time complexity of nonlinear functions. I know that those questions were asked a lot, but it didn't make me understand fully the subject. Also I ...
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### Asymptotics question

Is $\frac {n!} {2!\cdot 4!\cdot 8!\dots (n/2)!}=O(4^n)$? I am really stuck and I tend to believe it's true, but I don't know how to prove it. Any help would be appreciated!
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### What is the asymptotic bound for $1n + 2(n-1) + 3(n-2) + … + (n-1)2 + n$?

My best guess is that the series $$\sum_{i=1}^n i(n-(i-1))$$ becomes $$2 \Bigg[ n + 2(n-1) + ... + \frac{n}{2} \bigg(n-\bigg(\frac{n}{2}-1\bigg)\Bigg)\Bigg]$$ So the highest term is $n^2$ and ...
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### Traveling Salesman Problem: Big O Complexity of Algorithm

I'm trying to figure out how to do this problem in my intro algorithm class, but I'm a little confused. The Traveling Salesman problem (TSP) is famous. Given a list of cities and the distances ...
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### Asymptotic Question [duplicate]

Hi how do I find the asymptotic bound for the recurrence T(n) = T(n/2) + T(n/4) + T(n/8) + n? Can I use master theorem or Substitution method only? If so, I need some help for substitution method. My ...
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### How can i prove this asymptotic comparison? [duplicate]

This is an exercise that's part of my assignment, but it is optional and flagged as a "challenge". I would like to discuss its solution: Prove that: $$27\log{n} + \sqrt{n} = \theta(\sqrt{n})$$ ...
### Lower bound on $1^k+2^k+\dots+n^k$
Assumptions Let's say we have any recurrence relation (however this is perhaps more applicable to "unpredictable" recurrence relations): $$T(n) = \;?$$ For example: T(n) = aT\left(\frac{n}{b}\...