2
votes
1answer
29 views

Recurrence relation from code

My Friends, Hi, I see an old book in mathematics for computer science. everyone could help me, for example how we calculate the order (Time complexity) of following code: ...
1
vote
2answers
44 views

Solving a recurrence relation with square root

I ran into a bad recurrence relation. Anyone would calculate T(n) or add some hint? $$T(n) = \begin{cases} n,\quad &\text{ if n=1 or n=0 }\\ \sqrt{1/2[T^2(n-1)+T^2(n-2)]+}n ,\quad ...
2
votes
1answer
47 views

Height of Recursion Tree for $T(n, k) = T(n/2, k) + T(n, k/4) + kn$

If we use a recursion tree for solving $$\begin{cases} T(*,1)=T(1,*)=a \\ T(n,k)=T(n/2,k)+T(n,k/4)+kn \end{cases}$$ What is the height of the recursion tree? Any idea or solution highly ...
0
votes
1answer
63 views

Solving the recurrence relation $T(n)=4T\left(\frac{\sqrt{n}}{3}\right)+ \log^2n$

How we calculate the answer of following recurrence? $$T(n)=4T\left(\frac{\sqrt{n}}{3}\right)+ \log^2n.$$ Any nice solution would be highly appreciated. My solution is: $n=3^m \to ...
0
votes
0answers
26 views

Writing convergence acceleration algorithm as a recursion type formula

Cohen et al. describes an algorithm for speeding up the convergence of an alternating series as follows: Initialize: $$d=\left(3+ \sqrt{8}\right)^n; \quad d=\frac{1}{2} \left(d+ \frac{1}{d}\right)$$ ...
2
votes
0answers
49 views

Dynamic Programming: Stock Exercise

I'm having a trouble dealing with this problem: Since future market prices, and the effect of large sales on these prices, are very hard to predict, brokerage firms use models of the market to ...
2
votes
2answers
53 views

Master theorem - why the log factor?

I think I finally managed to fully understand the master theorem but there's one thing left in the second clause (I'm following here: ...
0
votes
0answers
25 views

How to prove that in optimal strategie of tower of Hanoi none of the disks can't be placed on a disk of same parity

How to prove that in optimal strategie of tower of Hanoi none of the disks can't be placed on a disk of same parity if we have 3 rods. So for example disk 2 can't be placed on disk 4, or disk 1 can't ...
2
votes
1answer
197 views

How to solve this recurrence relation with Sigma notation (f(n, m) = f(n - 1, m) + f(n, m- 1) + c?

This recurrence relation was inferred from the function $f(n, m) = f(n - 1, m) + f(n, m-1) + c$. After expanding the latter, I ended up with the following: $$f(n,m)=\begin{cases} 0,&\text{if ...
3
votes
1answer
72 views

Recursively deleting every second element in a list

This question got me thinking. If you have a list of length n and recursively delete every other element from the list until only one element remains, is there any ...
1
vote
3answers
160 views

Recurrence relations (Big-O notation)

Say I'm given a recursive function such as: function(n) { if (n <= 1) return; int i; for(i = 0; i < n; i++) { function(0.8n) } } ...
2
votes
1answer
76 views

Solve the recurrence relation:$T(n)=\sqrt{n}T\left(\sqrt{n}\right)+\sqrt{n}$ [closed]

I have doubt in solving the following questions: $T(n)=2T(\sqrt{n})+n$ $T(n)=\sqrt{n}T(\sqrt{n})+c$ $T(n)=\sqrt{n}T(\sqrt{n})+\sqrt{n}$ T(2)=1 for all the problems Atleast give the final answer.
2
votes
3answers
94 views

Can I use the master theorem for this?

this is a HW question so please don't just give me the answer right away. Basically, I'm working on solving the running time of this recurrence method: $$T(n) = 4T(n/3) + n \log \log n$$ I want to ...
2
votes
1answer
94 views

Generalization of the Tower of Hanoi?

What is the least possible number of steps for the Tower of Hanoi with $n$ discs and an arbitrary number $k$ of towers? For example, Tower of Hanoi with $4$ towers, $5$ towers, etc.
0
votes
0answers
37 views

How many number of ways are there for getting a special prime?

Definition of special prime : Any integer (+ve, -ve or 0) that is divisible by at least one of the single digit primes (2, 3, 5, 7) is a special prime. Thus -21, -30, 0, 5, 14 etc are special ...
0
votes
1answer
66 views

How do I go about manipulating this summation equation to solve it?

In my textbook, Introduction to Algorithms, the following is shown: And I believe I understand that. However, I have a similar equation to the one on the first line, but instead of ...
1
vote
2answers
69 views

How does my textbook solve this summation equation for the answer?

Summations have always been my weakness in mathematics, and it's showing here as I'm very confused how my textbook, Introduction to Algorithms, goes from basically the second half of the following ...
1
vote
1answer
54 views

How does my textbook come up with this statement? I don't believe it to be true.

My textbook (Introduction to Algorithms) states the following: When polynomially comparing $n^\epsilon$ and $lgn$, it states that $n^\epsilon$ is polynomially greater for any positive $\epsilon$. ...
1
vote
3answers
79 views

Where is the value of the variable $\epsilon$ obtained in the following explanation my professor gave?

My professor gave us this explanation from the textbook Introduction to Algorithms regarding the Master Method/Theorem: As a first example, consider $$T(n)=9T(n/3)+n.$$ For this recurrence, we ...
0
votes
0answers
69 views

How is the algorithm for recursively printing permutations of a set of numbers this equation our professor gave us?

I'm having a great amount of trouble understanding where my prof got $T(m, n) = n(T(m+1, n-1) + m+1 + n)$ if $n > 1$ as the recursive formula for the algorithm for recursively printing the ...
2
votes
1answer
95 views

Algorithm analysis

Consider a recursive Mergesort implementation that calls Insertion Sort on sublists smaller than some threshold. If there are n calls to Mergesort, how many calls will there be to Insertion Sort? ...
2
votes
2answers
110 views

Purchasing Order: An Analysis on A ($\textrm{C++}$)-Approached Recurrence Relation

Suppose that we have $n$ dollars and that each day we buy either tape at a dollar, paper at a dollar, pens at two dollars, pencils at two dollars, or binders at three dollars. If $R_n$ is the number ...
1
vote
1answer
85 views

Orange Juice, Milk, or Beer

Suppose that we have $n$ dollars and that each day we buy either orange juice for a dollar, milk for two dollars, or beer for two dollars. If $R_n$ is the number of ways of spending all the money, ...
0
votes
3answers
168 views

(CHECK) $n$-bit Strings Containing a Pattern

$$\text{$\bf{PLEASE~~~CHECK~~~AUTHOR'S~~~ANSWER}$}$$ If $S_n$ denotes the number of $n$-bit strings that do not contain the pattern $00$, then what is the underlying recurrence relation and ...
4
votes
1answer
72 views

Number of times $g(p_1)$ occurs in $\sum_{d\mid n}g(d)$

$$ g(n)=\begin{cases} 1 & \text{if }n=1 \\[10pt] \sum_{d\mid n,\ d\ne n} g(d) & \text{else} \end{cases} $$ How can I calculate $g(n)$ efficiently? I was trying to collect all the $g(p)$ ...
1
vote
1answer
51 views

Recursive algorithm

I am trying to understand how this works. My instructor is teaching his first class, in summer on top of that, and only had 3 slides on this and had to rush over it. His example is: ...
2
votes
2answers
123 views

Recursive algorithm correctness: problem.

Considering that to prove a recursive algorithm we should refer to mathematical induction. Given the following algorithm (which sort an Array of size r) I found that base cases are for array size of 0 ...
3
votes
1answer
155 views

Issue while applying Master Theorem

I've read about the master theorem for solving recurrences in Introduction to Algorithms, but have a problem (probably, due to misunderstanding) while applying it in some cases. For example, having ...
1
vote
1answer
77 views

What kind of series / recursion is this?

I'm trying to find the explicit solution / sum of first n elements for the following sequence: d(2) = 2 d(n) = d(n/2) + n*log2(n) Can you help me to find out ...
4
votes
3answers
1k views

How can I explain this integer partitions function recursion?

How to explain how this algorithm works? I need to write an article about this but I can't explain why this recursion works fine. It defines the number of partitions of a given integer ...