0
votes
2answers
36 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 ...
0
votes
1answer
34 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
58 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 ...
1
vote
1answer
38 views

Recurrence Relation - Merge Sort

We know the recurrence relation for normal merge sort. It is T(n) = 2T(n/2) + n. After solving it we can get T(n) = cnlogn. I ...
1
vote
0answers
76 views

Comment on this Pi Formula? [on hold]

I compute Pi in my own way as follows: $$\frac{\pi}{2}=1+\frac{1}{1+\frac{1}{\frac{1}{2}+\frac{1}{\frac{1}{3}+\frac{1}{\frac{1}{4}+\frac{1}{\vdots}}}}}$$ and use ...
1
vote
2answers
46 views

Solve $T(n) = T(n-1)+\log^2(n)$

I was trying to solve $T(n) = T(n-1)+\log^2(n)$ using substitution method and variables substitution but I can't find the correct answer. My attempt: Let $m = \log(n)$ then $T(2^m) = ...
4
votes
5answers
90 views

How to solve the recurrence relation $T(n) = T(\lceil n/2\rceil) + T(\lfloor n/2\rfloor) + 2$

I'm trying to solve a recurrence relation for the exact function (I need the exact number of comparisons for some algorithm). This is what i need to solve: $$\begin{aligned} T(1) &= 0 \\ T(2) ...
0
votes
1answer
23 views

Summation of infinte series

Sir, I have three infinite summation $A =J_1 \sum_{n=2}^\infty (n-1) f(n-2) \tag 1$ , $B =\sum_{n=0}^\infty f(n) \tag 2$ and $C =J_2\sum_{n=1}^\infty f(n-1) \tag 3$, with ...
3
votes
2answers
89 views

Fibonaaci Recurrence

This is an interesting question where we are trying to solve another recursion which has same tree structure as the given recursion and also has term similarities Given Data in question ...
0
votes
0answers
24 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
1
vote
0answers
50 views

Solving the recursion $F(n)=K_0F(n-1)/(n-1)+K_1F(n-2)/(n-2)$

Please help me in solving the recursion $F(n)=K_0\frac{F(n-1)}{n-1}+K_1\frac{F(n-2)}{n-2}$, preferably using power series for the values of $F(n)$ in terms of $n$. Here $K_1$ and $K_2$ are ...
5
votes
3answers
363 views

Recurrence with varying coefficient

Problem 1 $$ {\rm f}\left(n\right) = \frac{1}{n}\, \left[{\rm f}\left(n - 1\right)k_{0} + {\rm f}\left(n-2\right)k_{1}\right]\tag{1} $$ ( This can also be written as ${\rm Q}\left(n\right) = ...
7
votes
3answers
221 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
0
votes
0answers
21 views

Strassens Matrix Multiplication Algorithm to compute product of 2 4X4 Matrices

Im trying to learn starssens matrix multiplication Algorithm.So far i know that it uses 7 multiplications and replaces a multiplication by several additions and subtractions,to achieve better ...
3
votes
1answer
195 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
0
votes
0answers
19 views

Question about set and series notation in the GSP Algorithm

I tried this in programming forums with no luck and since my primary issue is with the notation, I'm asking here: I have the definition of the GSP Algorithm: Definition However I don't understand ...
0
votes
0answers
26 views

Produce a list of the most-similar units, given various correlations/relationships

I have a database full of units (U1 - U50, U51...) where every unit has the same standard attributes (A1 - A10) and where a % of each attribute defines the amount of that attribute for that particular ...
2
votes
0answers
79 views

Building Minimum warehouses

A big international retailer is setting up shop in India and plans to open stores in N towns (3 ≤ N ≤ 1000), denoted by 1, 2, . . . , N. There are direct routes connecting M pairs among these towns. ...
1
vote
1answer
53 views

How to prove a very basic algorithm by induction

I just studied proofs by induction on a math book here and everything is neat and funny: the general strategy is to assume the LHS to be true, and use it to prove the RHS (for the inductive step). Now ...
2
votes
1answer
64 views

Algorithm - iterative method

I'm stuck on an exercise on algorithms, can you help me with this exercise? Solve this recursion using iterative method. $$T(n) = \begin{cases}1 & n=1;\\ 2, & n=2;\\ T(n-2) + n/2,& ...
2
votes
0answers
44 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 ...
0
votes
1answer
36 views

Number Triangle pattern

I have a number triangle as follows: $$\begin{array}{|c|c|c|} \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ \hline 0 ...
0
votes
1answer
29 views

Why $T(n) = 2T(n-1) + O(1)$ is $\Omega(2^n)$?

I was told that the complexity of $T(n) = 2T(n-1) + O(1)$ is $\Omega(2^n)$; however, since I was not convinced, I searched in the Internet and all I found is that problem or very similar ones with ...
-2
votes
1answer
42 views

N balls of k colors on a cirlce, no two neighboring balls have same color - recursive algorithm

Suppose we have N balls of k different colors. What is the number of arrangements of these N balls on a circle with no two neighboring balls having the same color? Actually, the task is to make up a ...
2
votes
2answers
49 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: ...
4
votes
1answer
44 views

Problem understanding Master theorem

I'm studying the Master theorem (for the analysis of recursive algorithms) and I perfectly understand why a binary search is of order $\log_2(n)$. I also understand that if we formulate it as $T(n) ≤ ...
8
votes
6answers
619 views

Non-literal applications of “Shortest Path” algorithm?

It's obvious that it's used in stuff like Google Maps, but what are some more metaphorical applications where you're minimizing the path between nodes (which can represent anything)
1
vote
1answer
145 views

Solving the recurrence relation $T(n) = T(n-\sqrt n) + 1$

I have an algorithm that at each step can discard $\lceil\sqrt(n)\rceil$ possibilities at a cost $1$. The solution to the recurrence relation below is related to the question of complexity of such ...
2
votes
0answers
26 views

Verify that this recurrence relation is in O(log n)

For the recurrence $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ is in $\Theta(lg n)$ My attempt at a solution (mostly just wanting to verify its correct). Lower Bound: $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ ...
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 ...
1
vote
1answer
78 views

Cyclic tower of hanoi problem [duplicate]

If I have 3 rods in a circle and it is allowed to move the disks only in the clockwise direction. How many moves is necessary to move n disks from first rod to the third rod?
1
vote
2answers
44 views

Calculate a recursive equation in terms of theta

I am struggling with the following equation for one week! Please help me solve it. $$T(n)=T(\frac{n}{2})+\frac{n}{logn}$$ So far, I have come to the equation $T(n)=\Sigma \frac{2^x}{x}$
2
votes
1answer
194 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 ...
0
votes
1answer
25 views

Recursive Algorithm Analysis

$$T(n) = 2\cdot \sqrt{n} \cdot T(\sqrt{n}) + \Theta (\lg n)$$ I have been trying to solve this question but I could not find anything. My approach: $n = 2^k$ $S(k) = T(2^n)$ and $S(k/2) = ...
1
vote
1answer
22 views

What is the link between the quotient and the Bézout coefficients in the Extended Euclidean Algorithm?

Here is an image from the wikipedia page of the EEA for working out the Bézout coefficients for 240·sk + 46·tk = rk where rk = GCD(240,183): What I am trying to work out is what si and ti ...
0
votes
0answers
48 views

Why can I not generalize O(n^log5) for squaring matrice of size n

I have a question that is bugging me for around a 3 days, I first asked this question in stackoverflow but no one could answer it reasonably though they tried to help, so finally I found here as a ...
1
vote
1answer
29 views

Multiplication cannot be obtained from zero, successor, and identity by composition without recursion

The task is to show that multiplication cannot be obtained by zero, successor, or identity functions by composition without using recursion at least twice. I'm primarily confused because it doesn't ...
0
votes
1answer
98 views

Show that $gcd(x,y)$ and $z = lcm(x,y)$ is primitive recursive

For the $gcd(x,y)$ we note: $gcd(x,0) = x$ $gcd(x,succ(y)) = gcd(succ(y),mod(x,succ(y)))$ $succ(x)$ and $mod(x,y)$ are both primitive recursive, so $gcd(x,y)$ must be as well. $z = lcm(x,y)$ if ...
1
vote
1answer
70 views

Find coefficients of polynomial that has zeros at certain points

Given a list of values z0, z1, ..., zn-1 (possibly with repetitions), show how to find the coefficients of a polynomial P(x) of degree-bound n + 1 that has zeros only at z0, z1, ..., zn-1 (possibly ...
1
vote
0answers
18 views

Determining the optimally scoring move on a probabilistically represented 2D grid in real time

I'm posting this to StackOverflow, cstheory.stackexchange.com, and math.stackexchange.com because I'm not really sure where it fits best. I hope that's OK. I have a 2D grid (size varies per map, ...
1
vote
3answers
158 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) } } ...
0
votes
0answers
146 views

Find all possible paths in a Matrix

I'm looking for algorithms to find all paths in a 4 x 4 matrix. The rules are as follows You can move in any direction (up, down, left, right, and diagonally) The next square in the path must be a ...
2
votes
1answer
75 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.
1
vote
1answer
95 views

Help with Recursive Algorithm

We are to determine a recurrence relation for a recursive algorithm. Let us use the Josephus Problem for this: Given n people standing in a circle, every kth person is killed until one person ...
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 ...
1
vote
1answer
127 views

Proving by induction that a palindrome contains an even number of $b$s and $c$s

Suppose we want to construct palindromes that contain an $aa$ in the middle if the length is even and an $a$ in the middle if the length is odd. I'm trying to prove by induction that all of these ...
0
votes
1answer
44 views

A recursive definition of palidrome over {a,b}

Can someone please explain how this recursive definition produces palindromes over $\{a,b\}$? For example, how can we get the string $abaaaba$ as a valid string? Rule 1: $\epsilon$, $a$, and $b$ are ...
1
vote
1answer
41 views

A Shifted Sorted Array - finding the shift

I was given a problem concerning a sorted array that is shifted by some "number of spaces", k. For example, take the sorted array $1, 2, 3, 4, 5 ,6$ It is shifted 2 spaces, we get: $5, 6, 1, 2, 3 ...
1
vote
1answer
29 views

Correctness of complexity analysis of recursive algorithm

Given following recursive equation: $$T(n) = T(n-3) + \Theta(1)$$ Is it correct that this equation is O(1)?
0
votes
0answers
339 views

Algorithm to Generate Number with 0 and 9 [duplicate]

You are given an integer N. Can you find the least positive integer X made up of only 9’s and 0’s such that X is a multiple of N? Is there any Algorithm to generate the least number. Thank you.