31
votes
Accepted
How long does the Collatz recursion run?
This is Collatz conjecture - still open problem.
Conjecture is about proof that this sequence stops for any input, since this is unresolved, we do not know how to solve this runtime recurrence ...
- 9,325
22
votes
Accepted
Efficient algorithm to compute the $n$th Fibonacci number
You can use matrix powering and the identity
$$
\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{bmatrix}.
$$
In your model of ...
- 270k
16
votes
Solving a recurrence relation with √n as parameter
In your comment you mentioned that you tried substitution but got stuck. Here's a derivation that works. The motivation is that we'd like to get rid of the $\sqrt{n}$ multiplier on the right hand side,...
- 14.6k
16
votes
Accepted
How to solve a recurrence relation with a sum?
Here are several ways to solve your recurrence relation.
Guessing
Anyone with enough experience in computer science might recognize your recurrence as the one satisfied by $T(n) = 2^n$. Given this ...
- 270k
15
votes
How long does the Collatz recursion run?
You translated the code correctly. There are many methods for solving recurrences.
However, it is currently unknown if collatz even halts for all ...
- 70.9k
14
votes
Accepted
How to solve T(n) = T(n-1) + n^2?
Don't expand the squared terms; it'll just add confusion. Think of the recurrence as
$$
T(\fbox{foo}) = T(\fbox{foo}-1)+\fbox{foo}\;^2
$$
where you can replace foo with anything you like. Then from
$$
...
- 14.6k
13
votes
How long does the Collatz recursion run?
The time complexity function is
\begin{cases}
T(n)= O(1) \text{ for } n\le 1\\
T(n)=T(n/2) + O(1) \text{ for } n\text{ even}\\
T(n)=T(3n+1) + O(1)\text{ for } n\text{ odd}\\
\end{cases}
which can be ...
- 4,747
13
votes
Can we solve a "very" exponential recurrence?
It depends what you mean by "solve". This is tetration, and it has a number of "closed" forms. For example:
$$\begin{eqnarray*}T_0 & = & 1 \\ T_{n+1} & = & 2^{T_n}\...
- 18.9k
12
votes
Solving or approximating recurrence relations for sequences of numbers
Summations
Often one encounters a recurrence of the form
$$ T(n) = T(n-1) + f(n), $$
where $f(n)$ is monotone. In this case, we can expand
$$ T(n) = T(c) + \sum_{m=c+1}^n f(m), $$
and so given a ...
- 270k
12
votes
Accepted
Big-O proof for a recurrence relation?
As you pointed out, the reason for splitting the term into two pieces is to be able to cancel the $an$ term. If we go directly from $(8/9)cn^2 + an \leq cn^2 + an$, then we get stuck as we cannot do ...
- 1,103
10
votes
Why do these recurrences determine the number of ways of tiling a 3xN rectangle with 2x1 dominoes?
The picture should say more than words.
- 27.6k
9
votes
Solving or approximating recurrence relations for sequences of numbers
There may be times when you come across a strange recurrence like this:
$$T(n) = \begin{cases}
c & n < 7\\
2T\left(\frac{n}{5}\right) + 4T\left(\frac{n}{7}\right) + cn & n\geq 7
\end{...
- 4,371
9
votes
Accepted
Solving recurrence relation with two recursive calls
As you mention, the Akra–Bazzi theorem shows that the solution to the recurrence $T(n,p)$ is $O(n\log n)$ for all $p \in (0,1)$. However, this does not reveal the nature of the dependence on $p$....
- 270k
9
votes
How to solve T(n)=2T(√n)+log n with the master theorem?
Let us actually use the master theorem.
Define $S(n) = T(e^n)$ for all $n$. Then
$$S(n) = T(e^n) = 2T(\sqrt{e^n}) + \log(e^n) = 2T(e^{n/2}) + n = 2S(n/2) + n$$
Now we can apply the second case of ...
- 34.1k
8
votes
Accepted
Understanding an algorithm for the gas station problem
The problem is in the condition for the first argument to min() in Equation (4) on p. 7. It's currently
...
- 5,154
8
votes
Solving or approximating recurrence relations for sequences of numbers
After checking this post again, I'm surprised this isn't on here yet.
Domain Transformation / Change of Variables
When dealing with recurrences it's sometimes useful to be able to change your ...
- 4,371
8
votes
Accepted
What is the Big O of T(n)?
Yes, all functions $f(n)$ satisfy $f(n) \in O(f(n))$. The definitions are meaningful even if $f(n)$ isn't the running time of any function. Indeed, this notation comes from number theory, where $f(n)$ ...
- 270k
8
votes
Accepted
Why do these recurrences determine the number of ways of tiling a 3xN rectangle with 2x1 dominoes?
Suppose that the rectangle to be tiled has 3 rows and $n$ columns.
□□□...□
□□□...□
□□□...□
123...n
Consider a tiling of this rectangle using 2$\times$1 dominos. ...
- 270k
8
votes
Accepted
Solving or estimating the recurrence $T(n) = x + T(n-\log_2 n)$
Assuming an appropriate base case, it is easy to see that $T(n) \geq (n/\log_2 n) \cdot x$, since at each step we subtract at most $\log_2 n$, and thus it takes t least $n/\log_2 n$ steps to reach ...
- 270k
8
votes
Accepted
Recurrence relation of the coin change problem
For that recurrence to make sense, $V$ can only be the array that contains the coin values; that is, $V=\{C_1, C_2, ..., C_m\}$.
Whenever confronted with a new dynamic programming problem, you should ...
- 3,424
8
votes
Accepted
Solving recurrence relation with square root
The answer cannot be $O(\log\log n)$. Already without applying any recursion we have the inequality $T(n) = T(\sqrt{n}) + n \ge n$. So the complexity cannot be smaller than $O(n)$.
But now to your ...
- 1,336
7
votes
Accepted
Solving the recurrence T(n) = 3T(n-2) with iterative method
Note that $3^{n/2-1/2} = \frac{1}{\sqrt{3}} 3^{n/2} = \frac 1{\sqrt{3}} \sqrt{3^n}$. So the $-\frac 12$ indeed becomes a constant factor that is absorbed by the $O()$, but $\frac n2$ in the exponent ...
- 6,519
7
votes
Solving or approximating recurrence relations for sequences of numbers
Case 2 of the master theorem, as usually stated, handles only recurrences of the form $T(n) = aT(n/b) + f(n)$ in which $f(n) = \Theta(n^{\log_ab}\log^k n)$ for $k \geq 0$. The following theorem, taken ...
- 270k
7
votes
Changing variables in recurrence relations
What $S(m) = T(2^m)$ means is that $S$ and $T$ are two different functions which produce the same result while taking inputs as $m$ and $2^m$ respectively.
Function $S$ can be considered as an ...
7
votes
Accepted
Solution of complex recurrence relation
Everything becomes much simpler if when $n$ is odd we stop the sum at $\lfloor n/2 \rfloor$, and when $k$ is even we discount the term corresponding to $k/2$ by a half. In that case you can write
$$ ...
- 270k
7
votes
Accepted
Master Method to solve recurrences is 'a' related to 'b'?
No it's not always the case that $a=b$, since you might not necessarily use every sub-problem. Consider for example, the binary search algorithm. In the algorithm, you have a sorted array that you ...
- 1,103
7
votes
Accepted
What is the correct representation of Master Theorem?
By convention, the notation $log^p(x)$ is defined to be $(log(x))^p$, not $p$ iterations of the $log$ function. This is similar to the trigonometric functions, which gives us identities like $sin^2(x)+...
- 1,944
7
votes
Efficient algorithm to compute the $n$th Fibonacci number
You can read this mathematical article:
A fast algorithm for computing large Fibonacci numbers (Daisuke Takahashi):
PDF .
More simple, I implemented several Fibonacci's algorithms in C++ (without and ...
- 173
7
votes
Accepted
How to solve recurrence T(n) = 2T(n/2) + n/log(n) using substitution method
$$T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log n}$$
Would yield the following summation (assuming $n$ is a power of 2 and base case is $n=2$):
$$
\begin{align}
T(n) &= \frac{n}{\log n} + 2 \...
- 4,371
7
votes
Accepted
Reccurence $T(n) = \sqrt{n}T(\sqrt{n})+n$
You should really be asking a third question: what happens if $n$ isn't a perfect square. The answer to this question is that the actual recurrence should have $T(\lfloor \sqrt{n} \rfloor)$ or $T(\...
- 270k
Only top scored, non community-wiki answers of a minimum length are eligible