Questions tagged [asymptotics]

Questions about asymptotic notations and analysis

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33 views

Why is $T(n)=3T(n/4) + n\log n$ solvable with Master Method but $T(n)=2T(n/2) + n\log n$ is not?

I am having difficulties in understanding why the recurrence $$T(n)=3T(n/4) + n\log n$$ is solvable with Master Method but $$T(n)=2T(n/2) + n\log n$$ isn't? Despite they both look very similar ...
0
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1answer
26 views

Big-O notation for the given function whose runtime complexity grows faster than the input

I struggle to determine the runtime complexity of a function I thought of while trying to solve this quiz. The quiz itself goes like this: Write a program to find the n-th ugly number. Ugly numbers ...
1
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2answers
31 views

Simplify the asymptotic expressions $O(n^2 + n) + \Omega (n^2 + n \log n)$

How can it be shown that the expression $O(n^2 + n) + \Omega (n^2 + n \log n)$ simplifies to $\Omega (n^2)$? Why is it not $\Theta(n^2)$?
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1answer
24 views

In Big-O notation, what does it mean for T(n) to be upper bounded by something

I do not have much experience in mathematics but I would really like to grasp Big-O notation on its mathematical level. I already read What does the "big O complexity" of a function mean? ...
7
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5answers
162 views

Is $n^{1/\log \log n} = O(1)$?

Is $n^{1/\log \log n} = O(1)$ ? Suppose that $n^{1/\log \log n} = c$ where $c$ is constant. Taking logs of both sides, $$\frac{1}{\log \log n}\log n = \log c.$$ I am not able to spot an error. ...
12
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3answers
217 views

Is O((n^2)*log(n)) greater than O(n^(2.5))?

I know that $O(n^2\times \log(n))$ is greater than $O(n^2)$, but is $O(n^2\times \log(n))$ greater than $O(n^{2.5})$?
2
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1answer
65 views

Quickly obtaining sums of sets of numbers

We are given a set of $n$ bits, call them $a_1$, $a_2$,...,$a_n$. We are also given a set of $m$ sums, where the sums $s_1$, $s_2$,...,$s_k$,...,$s_m$ are given as sums of some of the bits. For ...
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2answers
36 views

Summation of asymptotic notation

How can we solve summation of asymptotic notations like given below: $$ \sum_{k=1}^{n-1} O(n). $$
4
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1answer
76 views

Asymptotics of $\frac{1}{\log(\frac{2^n}{2^n-1})}$

I am trying to understand the asymptotics of \begin{equation} f(n) = \frac{1}{\log(\frac{2^n}{2^n-1})} \end{equation} In particular, is there some $c \geq 1$ such that $f(n) = O(n^c)$?
1
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1answer
23 views

The space complexity of a function that allocates space based on the input value and not size

What is the space complexity of the following hyphotetical function: void function(int n) { int[] array = new int[n]; // allocate array of size n return; } ...
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2answers
55 views

How to determine time complexity with a simple way?

I'm learning about time complexity but all the cases we did in class were rather simple. Now I'm working on my home work and the cases our teacher let us have was: $$f(n) = 4n(n + 2 \log^2 n^2) + e^{−...
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2answers
73 views

Show that: $0.01n \log n - 2000n+6 = O(n \log n)$

Show that $0.01n \log n - 2000n+6 = O(n \log n)$. Starting from the definition: $O(g(n))=\{f:\mathbb{N}^* \to \mathbb{R}^*_{+} | \exists c \in \mathbb{R}^*_{+}, n_0\in\mathbb{N}^* s. t. f(n) \leq cg(...
2
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1answer
54 views

Show that the union of Θ and o is not O

Show that: $\Theta(n\log n)\cup o(n\log n)\neq O(n\log n)$ I tried to start this in many ways but I don't really know how... intuitively isn't $\Theta \cup o = o$? So that would mean that I would ...
2
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0answers
32 views

Difference Between $n^{\Omega{(1)}}$ and $\Omega{(n)}$ [closed]

I am not sure about the difference between $n^{\Omega(1)}$ and $\Omega(n)$. It seems to me that the only difference is that $n^{\Omega(1)}$ can contain some sublinear functions, i.e., $n^{\frac{1}{2}}$...
2
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1answer
45 views

Big theta of function with multiple types of n [duplicate]

I have the following function: $\displaystyle\frac{n \cdot 7^n+\frac{8}{n!}}{(n+7) \cdot 7^n}=\Theta(1)$ I don't how they come to this. What is the proper way to analyse a function to theta notation?...
1
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1answer
36 views

Is log(n) equivalent to (log(n))^x for big-O analysis?

My professor noted that we could treat any logarithmic function with an exponent as equivalent to log(n) for the purposes of big-O analysis. ie. $(n log(n) + 1)^2 + (log(n) + 1)(n^2 + 1)$ From the ...
4
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1answer
73 views

Solving $T(n) = 2T(n/2) + T(n-1)/\log n$

I am interesting in the asymptotic rate of growth of the following recursion: $$ T(n) = 2T(n/2) + \frac{T(n − 1)}{\log n}, $$ with base case $T(1) = 1$. I'm having trouble of solving this recurrence ...
2
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1answer
67 views

When are log complexities considered equivalent?

Would we consider $O(\log_2(n))$ to be the same complexity as $O(\log_2(n-1))$? Why or why not? I'm specifically wondering about how the number we take the log of affects the time complexity.
1
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3answers
58 views

How to mathematically prove that a relation T(n)=T($\sqrt{n}$)+c is O(log(log(n))?

following question, I understood the intuition behind how cutting down the size of input by square root on each iteration leads to O(log(log(n))) complexity. I tried to derive it on paper. Let T(n) =...
2
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1answer
27 views

Splitting summations?

From CLRS Introduction to Algorithms, Appendix A, page 1152. They discuss a method called "Splitting Summations", where they split the summation and bound each term separately. For example, ...
1
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1answer
54 views

Is $(\sqrt{n})!=Θ(\sqrt{n}^{\sqrt{n}})$?

I would like to express $(\sqrt{n})!$ in terms of $Θ()$ notation. My approach is the following: $$(\sqrt{n})!=f(n)\Leftrightarrow$$ $$\log(\sqrt{n})!=\log(f(n))$$ Now from Stirling's approximation ...
0
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1answer
46 views

Simplifying this expression with big O when several variables are involved

I have an algorithm which depends on three variables an where the running time is in $\mathcal{O}(m+2 m\cdot n\cdot p+p\cdot(n+m))$ and I would like to simplified it. I proceeded as follows : \begin{...
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0answers
13 views

Detailed explanation of Perlin Noise algorithmic complexity

I am doing a project in analysis of algorithm and I have been looking all over for something more complex than Perlin Noise is $O(n \cdot 2^n)$ because of the doubling in $n$ dimensions and array ...
2
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1answer
43 views

Showing that $n\log n - n$ is $\Omega(n)$

Prove that $n\log{n} − n$ is $\Omega(n)$. I do know the answer: $\log{n} ≥ 2, \forall n \ge 4$. Thus, $n\log{n}−n\ge n ,\forall n\ge 4 \implies n \log n − n ∈ Ω(n)$. But can someone please ...
4
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2answers
194 views

Is asymptotic ordering preserved when taking log of both functions?

In one of my exercise sheets I have the following question; Let $f,g\colon \mathbb{N}\longrightarrow\mathbb{R}$ be positive functions with $f(n) \in O(g(n))$. Prove or disprove; $\ln(f(n)) \in O(\ln(...
1
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1answer
35 views

What is the difference between Big(O) and small(o) notations in asymptotic analysis? [duplicate]

What is the difference between $O$ (big oh) and $o$ (small oh) notations in asymptotic analysis? Even though I understand that $o$ is used for a bound that is not tight, is it allowed to use $O$ ...
1
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1answer
50 views

Why is building a heap $\mathcal O(n)$ and not $\theta(n)$?

From what I see online, all seem to suggest that heapifying takes $\mathcal O (n)$ time, but it seems like it should always takes $\theta(n)$ time, even in the best case. Is something wrong with my ...
2
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1answer
46 views

Spotting the difference between two arrays using divide-and-conquer

Say we have two equal-sized arrays that contain a 1 or 0 at each of their indices. These two arrays are identical, except at one unique index. We want to find and output that particular index. For ...
2
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0answers
38 views

Why can't we use the Master Theorem on recurrences with floor or ceiling operations? [duplicate]

From my understanding, using such operators on large numbers doesn't have an impact on running time, since the integer rounding becomes negligible after a certain point. For example, the recurrence $$...
2
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1answer
37 views

Can we apply the Master Theorem to the following recurrence?

Our recurrence is $$ T(n)= \begin{cases} T(\lfloor{n/2}\rfloor)+(\log(n))^{2}, & \text{if $n>1$} \\ 1 & \text{if $n=1.$} \end{cases} $$ I have identified $a = 1 > 0$, and $b = 2 > 1$...
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2answers
43 views

How to show that every quadratic, asymptotically nonnegative function $\in \Theta(n^2)$

In the book CLRS the authors say that every quadratic, asymptotically nonnegative function $f(n) = an^2 + bn + c$ is an element of $\Theta(n^2)$. Using the following definition \begin{align*} \...
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1answer
41 views

Bubble sort: how to calculate amount of comparisons and swaps

For a given sequence 1, N ,2 ,N −1 ,3, N −2, ... I want to calculate the number of comparisons and swaps for bubble sort. How can I accomplish that using $\theta ()$ notation? I would know how to do ...
2
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1answer
41 views

Solving a recurrence relation involving square roots

Give an asymptotic upper bound for $$T(n) = \sqrt{n}·T(\sqrt{n})+n+n/\log n. $$ How can I solve this recurrence relation, which involves square roots?
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1answer
36 views

Find function that satisfy the relation

Can you find the function that satisfy the relation? $$f(n) = \Theta(g(n)), f(n) = o(g(n))$$
3
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1answer
26 views

Prove or disprove that $log^{k}(n) \in O(\sqrt{n}) \forall k > 0$

I'm trying to solve the problem described in the title. By using the free version of wolfram and testing some increasing values of $k$ I get that: $$\lim_{n \rightarrow \infty} \frac{log^{k}(n)}{\...
0
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0answers
12 views

Growth of n/logn vs n^(1+e) [duplicate]

Is $\Theta(\frac{n}{log(n)})$ faster growing function than $\Theta(n^{1+\epsilon})$, where $\epsilon > 0$?
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1answer
40 views

Derive a while loop (which seemingly have some logarithmic traits) runs in $\Theta(n)$

I know for a fact that algorithm A runs in $\Theta(n)$, but how does one derive that? Algorithm A ...
2
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3answers
94 views

How can $\Theta$ and $O$ complexities be different?

From the definition of the $\Theta$-notation, $$f(n)=\Theta(g(n))\\\implies \exists n_0, \exists c_1,c_2\gt 0, \forall n\gt n_0, c_1\cdot g(n)\le f(n)\le c_2\cdot g(n)$$ We can see that the ...
5
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2answers
93 views

What constant in the latest fast matrix multiplication is hidden by Big-O notation?

I'm evaluating the theoretical run time of matrix multiplication algorithms as it has improved within the last few decades. Algorithms to solve matrix multiplication run in O(n^w) time, where w has ...
3
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0answers
48 views

Does a function that belongs to θ(n) belong to O(n^2)? [closed]

My understanding is that the answer is yes. To be θ(n) implies O(n), which itself implies O(n^2). In other words, a function that is tightly bounded by n is trivially upper-bounded by n^2 as well. ...
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1answer
42 views

confused with Time Complexity [duplicate]

I was reading book related to Time Complexity, and came up with 4 lines of equations that I could not understand properly, could you please explain why are those true? 1) $n = o(n\log\log n)$ 2) $...
1
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1answer
66 views

Running time of algorithm (effect of j*j in for loops) - Theta Runtime

In Theta notation what are the running times of these algorithms? Algorithm 1 for i=1..n j=1 while j*j <= i: j = j + 1 Since the outer loop ...
2
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0answers
38 views

Asymtotic bound for recurrence of $T(n)=2T(n/2)+ \sum_{i=0}^{n} (i+2)^2$ using substitution

What can be an initial guess for finding the tight asymptotic bounds of $T(n)=2T(n/2)+ \sum_{i=0}^{n} (i+2)^2$ using substitution method?
0
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2answers
96 views

Runtime complexity of a brute force factoring algorithm? (in terms of bits)

Let N be an n bit number. A brute force algorithm factors N by trying to divide N by all of the numbers between 2 and sqrt(N). Given that dividng two n bit integers takes O(n^2) time, what is the ...
2
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1answer
26 views

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 ...
0
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0answers
23 views

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 ...
0
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1answer
72 views

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 $g(n)$. ...
0
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0answers
23 views

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 ...
4
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1answer
72 views

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 ...
2
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1answer
19 views

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. ...