# Questions tagged [asymptotics]

Questions about asymptotic notations and analysis

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### Using the definition of Θ-notation proof

Using the definition of Θ-notation, prove that $(3 n+13)(7 n+2)\left(\log \left(1024 n^{2}+100\right)\right) \in \Theta\left(n^{2} \log n\right)$ I also found a useful example in Stack exchange: ...
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### Using limits to find which is the most appropriate symbol (O, o, Ω , θ , ω ) to describe 𝑓(𝑛)∈ __ 𝑔(𝑛)? [on hold]

I know you can use limits but got confused with the description in the textbook explaining the technicalities. Can someone explain how you would use limits to address problems like the above, in terms ...
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### Proving that $S_1+S_2 \leq f^{-\omega(1)}$

I am trying to show for every c, there exists $M\text{ such that }(x,y,z)\geq M$ then $S_1(x,y,z) + S_2(x,y,z) \leq ( f (x,y,z))^{-c}$ . For a particular $S_1,S_2,f$. Does it suffice to prove there ...
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### Is this computational complexity of the k-NN (custom distance) correct?

I read on a book that in general k-NN (no optimizations), given $d$ dimensions $n$ examples every computation of distance is $O(d)$. Since every example has to be compared with all the other ones, ...
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### How to justify $f(n) = O(g(n))$ [duplicate]

The following question is in my homework: Is the statement $f(n) = O(g(n))$ true, when $f(n) = n/2 + 4$ and $g(n) = \sqrt{n} + 2\log_2 n + 3$? I understand how $f(n)$ is the upper bound of $g(n)$. ...
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### Time complexity of simple function related to bits

I am wondering about correct answer to this task from a yesterday's test: A function Pow which calculates $y = a^k$ is given, where $k$ is an integer of length ...
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### Asymptotics and logarithms/exponents

We have four categories: additive constants, multiplicative constants, polynomials, and exponentials When determining the growth order of functions, we only care about polynomials and ...
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### Analyzing time complexity of solution in tutorial

Could someone explain time complexity of solution of in this tutorial? I'm having hard time figuring out, how asymptotic bounds for first solution is $O(3^k k)$. What I figured so far is, for ...
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### Exact meaning of $2^{\mathcal{O}(f(n))}$

In Sipser's Introduction to the Theory of Computation he uses the notation $2^{\mathcal{O}(f(n))}$ to denote some asymptotic running time. For example he says that the running time of a single-tape ...
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### Interpretation of an asymptotic notation

Assume that we measure the complexity of an algorithm (for some problem) by two parameters $n$ and $m$ (where $m \le n$). What is the formal interpretation of the following claim: there is no ...
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### Asymptotic Relationship from Limit

F(n) = n-100 G(n) = n-200 I am trying to show the asymptotic relationship between these two functions using limits. I take the limit n->∞ f(n) / g(n) and I get the result 1 which is constant c. ...
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### Help with Big-O homework [duplicate]

"er" is the Danish equivalent of "is" in English. I need some help with the square root one. Additionally, it would be nice to know if the other ones are correct.
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### Why is $\sum_{i=1}^n O(i)$ not the same as $O(1)+O(2)+\dots+O(n)$?

The well-known textbook Introduction to Algorithms ("CLRS", 3rd edition, chapter 3.1) claims the following: $$\sum_{i=1}^n O(i)$$ is not the same as (I'm not using DNE because the book explicitly ...
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### O(1) distinct elements in an array implies?

Could someone explain the following question - Given the following statement viz. Consider an input array a[1..n] of arbitrary numbers. It is given that the array has only O(1) distinct elements. ...
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### When the data size and processor speed are both multiplied by 10, then a linearithmic algorithm takes double the time to finish?

Robert Sedgewick mentioned, if a computer can handle 10x data and the processor is also 10x as fast, then a $O(n^2)$ algorithm actually runs slower than before. Is this the correct idea when a ...
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### Why does Union-Find have time complexity O(N + M lg* N) with the “log star N”?

The time complexity of Weighted Union-Find with Path Compression, for M union-find ops and N objects is said to be $$O(N + M \lg^*N)$$ and the $lg^*N$ is "log star N" and is iterated logarithm. ...
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### Basic Theta-notation question

Let $T$ be a function. Is it true that if $\exists f\forall n,m> 0.\\ \frac m {f(n)} \leq T(n,m)\leq m$ Then $\exists g.T(n,m)=\Theta(m\cdot g(n))$? In words: is such a case, is there a function ...
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### How to prove any polynomial of degree $k$ is in $\Theta(n^k)$?

I want to prove that any polynomial of degree $k$ is in $\Theta(n^k)$. The coefficient of $n^k$, $a_{k}$, is positive. I know I need $0 \leq c_{1}n^k \leq a_{k}n^k + ... + a_{0} \leq c_{2}n^k$ for ...
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### Recursion Time Complexity (Half n' Half)

This is my solution for Leetcode 395, and I'm wondering how I can come up with its time complexity: Input: string $s = s_1,\ldots,s_n$, integer $k$ Go over all symbols $s_1,\ldots,s_n$, one by one ...
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### Is there a data structure that can find the kth smallest in constant time with logarithmic add and delete operations?

I'm looking for a single or a conjunction of data structures that can find the kth smallest element in constant time, delete the kth smallest element in logarithmic time, and add a new element in ...
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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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### Solving recurrence relation with square root

I am trying to solve the following recurrence relation :- $T(n) = T(\sqrt{n}) + n$ using masters theorem. We can substitute $n = 2 ^ m$ $T(2^m) = T(2 ^ {\frac{m}{2}}) + 2^m$ Now we can rewrite 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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### Can an algorithm with $\Theta(n^2)$ run time be faster than an algorithm with $\Theta(n\log n)$ run time?

This is a question posted for extra practice (i.e., not for credit): Can an algorithm with $\Theta(n^2)$ run time be faster than an algorithm with $\Theta(n\log n)$ run time? Explain. I'm not sure ...
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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 ...
### 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 ...