Questions dealing with recursive algorithms. Their analysis often involves recurrence relations, which have their own tag.

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0
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1answer
719 views

Is this formula for the number of nodes for a complete tree or a full and complete tree?

In a lecture it was said that "How many nodes are there in a complete k-ary tree with height h?" and this was the answer: $$ \sum^{h}_{i = 0}k^i $$ where h is the height and k is the max number of ...
2
votes
1answer
195 views

Basic recurrence problem, not sure if solution is correct (solution included)

I have the following exercise: We know that the solution of $T(n) = 2T(n/2) + n$ is $\mathcal{O}(n \log_2 n)$. Show that the solution of this recurrence is also $\mathcal{\Omega}(n \log_2 n)$. ...
0
votes
2answers
98 views

Explanation needed on this rather basic recurrence solution

We are studying about recurrences in our analysis of algorithms class. As an example of the substitution method (with induction) we are given the following: $$T(n) = \lbrace 2T\left(\frac{n}{2}\right) ...
6
votes
1answer
5k views

Show that the solution to $T(n) = T(n - 1) + n$ is $O(n^2)$

Hello and thanks for taking the time to answer my question. The question is really the title itself. We're studying about solving recurrences using the method of substitution and induction. How can I ...
0
votes
1answer
42 views

segment a line according to possibly overlapped points

There are finite number of possibly overlapped points (say 100) on a finite length line, how to partition the line into finite number of segments (say 12), each has as same number of points as ...
0
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3answers
116 views

How can $n \lg n = O(n^{log_3 4 - r})$?

How can I understand this bound, for me it is not true. $$n\lg n = O(n^{\log_3 4 - r})$$ where $\lg n = \log_2 n$ and $r > 0$ I'm trying to solve this recurrence $T(n) = 4T(n/3) + n\lg n$ using ...
3
votes
3answers
261 views

Recurrence relation by substitution

I have an exercise where I need to prove by using the substitution method the following $$T(n) = 4T(n/3)+n = \Theta(n^{\log_3 4})$$ using as guess like the one below will fail, I cannot see why, ...
0
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2answers
333 views

How does one rewrite a recursive function to be strictly non-recursive?

Given the recursive function: $$f(0) = \frac{x^2}{2} + \frac{x}{2}, f(n) = \frac{f(n-1)}{2} + \frac{x}{2}$$ where $x$ = some integer How would one rewrite this function to be strictly ...
1
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5answers
642 views

How do we deal with recurrence relation characteristic equations that are not quadratic or have imaginary roots?

Suppose we have $$H(n) = H(n-1)-H(n-2) \rightarrow x^2-x+1 \rightarrow r_1 = \frac{1+\sqrt{-3}}{2}, r_2 = \frac{1-\sqrt{-3}}{2}$$ or $$H(n) = H(n-1)+H(n-2)+H(n-3) \rightarrow x^3-x^2-x-1=0$$ In ...
4
votes
3answers
402 views

Can someone intuitively explain the towers of Hanoi and how a proof by induction can be used?

I'm having trouble understanding precisely how the proof by induction is used to show that it takes at most $2^n-1$ moves to get all the disks from one peg to another. Are there any helpful sources ...
0
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2answers
70 views

Properties of algorithms

I have 2 questions. 1.Let's have an algorithm input a; x ← -7; y ← a; while x $<$ y do x ← x+5; y ← 2·x+y-6; done Question: What is the greatest "stopping" ...
1
vote
1answer
319 views

number of recursive calls

How to estimate the number of recursive calls that would be used by the code ...
2
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2answers
198 views

Theta bound about $\sum \lfloor {\sqrt{n}}\rfloor$

$$S_k=\sum_{n=1}^{k^2-1}\lfloor\sqrt{n}\rfloor $$ Can somebody give me an idea about the steps I should follow? Initially I thought $$n^{1/2}\log(n) \leq n^{1/2}\leq n^{3/2}$$ so $\Theta(f(n))=S_k ...
4
votes
1answer
85 views

Is every context free language equivalent to one whose grammar has only one non-terminal symbol?

A context free language is generated by a context free grammar, which can be expressed in the Backus-Naur form e.g. I believe that if we only allow one nonterminal symbol in the grammar, the resulting ...
1
vote
1answer
1k views

Which approach to follow: greedy, divide-n-conquer or dynamic programming?

Given any problem say we have to pick few objects out of N so that the total weight is below W considering all objects of SAME value, A variation for this problem can be to have values assigned to ...
2
votes
2answers
136 views

Combination/Permutation Question

I'm trying to solve a programming challenge, and I have narrowed down all the challenge to a combination/permutation problem. I ended up with 5 possible scenarios, and I need to find all possible ...
2
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2answers
638 views

How to prove that a dynamic programming algorithm is a monotonic function

I've written an algorithm, which is based on the Needleman-Wunsch algorithm for matching sequences of proteins. This algorithm is a dynamic programming approach, where the optimal matching of two ...
1
vote
4answers
91 views

Solving a simple recurrence.

This isn't a homework question, but it is a problem in my textbook. Given $T(n) = T(n-1) + n$, show that $T(n) = O(n^2)$ My approach: Given $T(n) = T(n-1)$ Need to show $T(n) = cn^2$, where $c ...
1
vote
1answer
80 views

Recursive function

For a sequence $a$, $a_1=2$, $$a_n=\frac{n-1}{a_{n-1}}+n-1$$Express $a_n$ in terms of $n$. I tried keep expanding, got many level of fraction until $n=1$, but I still can't see the pattern. ...
3
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0answers
258 views

Question about recursive defined functions.

This question is about counting functions. With counting functions $F$ I mean functions from the positive integers to the positive integers that are strictly nondecreasing and can grow no faster than ...
0
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1answer
515 views

Help in understanding search of Vantage-Point tree

This is my reference: http://stevehanov.ca/blog/index.php?id=130 A vantage-point tree is a way of organizing a set of points so that finding the n-nearest neighbors is as efficient as possible. It ...
7
votes
2answers
2k views

Numerical method for finding the square-root.

I found a picture of Evan O'Dorney's winning project that gained him first place in the Intel Science talent search. He proposed a numerical method to find the square root, that gained him $100,000 ...
4
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1answer
1k views

Karatsuba multiplication with integers of size 3

I understand how to apply Karatsuba multiplication in 2 digit integers. $$\begin{array} \quad & \quad & x & y \\ \times & \quad & z & w \\ \hline \quad ...
0
votes
1answer
72 views

EFA and recursive algorithm

1) Is EFA stronger than recursive algorithm? (This can be in term of proof theoretic ordinal, or whatsoever - to rephrase the question, are all problems that can be solved(and halt) by recursive ...
1
vote
2answers
157 views

implicit equation for covering a $3\times{n}$ face with $2\times{1}$ mosaics with recursion solution?

This is my university homework. I was on it for a day but I couldn't solve the problem. I found a implicitness solution for this problem: $f(2)=3$ $f(4)=11$ $f(n)=3f(n-2)+2f(n-4)$ I thought that ...
1
vote
1answer
358 views

Derive a General formula for each term of this periodic sequence?

I have a sequence $a_0 = 1, a_1, a_2, a_3, \dots$ such that $a_4 = a_{24}$ which implies that period repeats after $a_{24}$ to $a_{43}$. Each $a_n$ depends on $a_{n-1}$ only. I need general term for ...
1
vote
1answer
521 views

Sum of series with log in each term

I was solving recurrence relation of Introduction to Algorithms by {Cormen, Leiserson, Rivest, Stein}, 3rd. edition. Problem 4-3 (i) $$ T(n) = T(n-2) + \frac{1}{lg \; n} $$ I tried few ways, like ...
14
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6answers
1k views

How to solve this recurrence relation? $f_n = 3f_{n-1} + 12(-1)^n$

How to solve this particular recurrence relation ? $$f_n = 3f_{n-1} + 12(-1)^n,\quad f_1 = 0$$ such that $f_2 = 12, f_3 = 24$ and so on. I tried out a lot but due to $(-1)^n$ I am not able to ...
2
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0answers
78 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
0
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1answer
1k views

How to solve the recurrence $T(n)=3T(n/2)+n$

The exercise stated that i have to solve the recurrence using the Recursion-Tree Method. I have already finished the base part, which is $\Theta(n^{\lg3})$ But for the recursive part I'm having ...
2
votes
1answer
167 views

In-place inverse of DFT?

I'm trying to understand (by implementing) the Cooley Tukey algorithm for an array $[x_0, \dotsc, x_{2^N-1}]$ of real valued data. Since the input data is real valued, the spectrum will have ...
3
votes
2answers
228 views

Finding the nearest integers to real numbers defined implicitly

I was trying to bound the maximum cost of top-down merge sort: $$ f(0) = f(1) = 0,\quad f(n) = n\lceil{\lg n}\rceil - 2^{\lceil\lg n\rceil} + 1, $$ where $\lg n$ is the binary logarithm of $n$ and ...
2
votes
0answers
202 views

Who invented the breadth-first permutation algorithm?

My initial problem was solved here. It is about enumerating all n-tuples of a permutation in a specific order. The solution algorithm is very simple and I'm sure has been used before. However, I did ...
7
votes
1answer
231 views

Finding $a_n$ for very large $n$ where $a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $

I have a recurrence relation, $$ a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $$ for $n>3$ and $a_1 = 0, a_2 = 0, a_3 = 1$ I have to find the value of $a_n$ for very large values of n. I tried ...
2
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0answers
77 views

Proving that an effective procedure is correct

I will start with definitions, theorems, and a few solved exercises which I am taking as theorems now. My actual question will be last, if you want to scroll ahead to see it. Definitions: (1) The ...
1
vote
1answer
1k views

What is the lower bound and upper bound on time for inserting n nodes into a binary search tree?

So given a $n$ array of few numbers(say $n$) we can sort them using the binary search tree (BST) as a black box . In order to that we first build a BST out of the array taking all the elements in ...
2
votes
2answers
4k views

Worst case analysis of MAX-HEAPIFY procedure .

From CLRS book for MAX-HEAPIFY procedure : The children's subtrees each have size at most 2n/3 - the worst case occurs when the last row of the tree is exactly half full I fail to see this ...
3
votes
1answer
352 views

Karatsuba Multiplication

Karatsuba's equation to reduce the amount of time it takes in brute force multiplication is as follows (I believe this is a divide-and-conquer algorithm): $$ x y = 10^n(ac) + 10^{n/2}(ad + bc) + bd ...
6
votes
1answer
559 views

Why does Strassen's algorithm work for $2\times 2$ matrices only when the number of multiplications is $7$?

I have been reading Introduction to Algorithms by Cormen et al. Before explaining Strassen algorithm the book says this: Strassen’s algorithm is not at all obvious. (This might be the biggest ...
2
votes
3answers
4k views

What will the recursion tree of Fibonacci series look like?

I am watching the Introduction to algorithm video, and the professor talks about finding a Fibonacci number in $\Theta(n)$ time at point 23.30 mins in the video. How is it $\Theta(n)$ time? Which ...
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2answers
153 views

In mathematics, what is meant by induction?

I was going through MIT video lectures on "Introduction to Algorithms " . In order to solve recurrences by substitution the professor says that we can solve them by induction. What is actually the ...
1
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1answer
77 views

Does this sequence of operators in Hilbert space, given by an algorithm, terminate

Let $H$ be an infinitedimensional Hilbert space and $T$ a compact selfadjoint operator in it. Consider the following Algorithm: Let $$ H_{1}=H,\ T_{1}=T $$ and let $\lambda_{1}$ be that ...
1
vote
1answer
1k views

Ulam's problem - guessing a chosen number in a set

I tried to solve the following problem, which I found in the book "Discrete Mathematics and Its Applications", by Kenneth Rosen (Problem 28 of the section 7.3 of the 6th Edition): Suppose someone ...
2
votes
0answers
207 views

google page rank algorithm with range of values

According to the wikipedia article, the iterative google page rank algorithm is defined as follows: Can this be modified to include a range of values, not just binary, and would it look something ...
5
votes
1answer
191 views

Solving Recurrence $T_n = T_{n-1}*T_{n-2} + T_{n-3}$

I have a series of numbers called the Foo numbers, where $F_0 = 1, F_1=1, F_2 = 1 $ then the general equation looks like the following: $$ F_n = F_{n-1}(F_{n-2}) + F_{n-3} $$ So far I have got the ...
3
votes
1answer
198 views

$\mu$-recursive definition of ulam (3n+1) function

$\newcommand{\ulam}{\operatorname{ulam}}$ The ulam function is defined as $$ \ulam(x) = \begin{cases} 1 & x = 1 \\ \ulam\left( \frac{x}{2}\right) & x \text{ even}\\ \ulam(3x+1) & ...
2
votes
0answers
134 views

Help on Generating function - Analysis of Median of three - Quick Select

I am trying to find out the average case analysis of median of 3 - quick select. The recurrence relation is \begin{equation} C_{n,j} = 1 + \sum_{k=1}^{j-1}\pi_{n,k}C_{n-k,j-k} + \sum_{k=j+1}^{n} ...
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5answers
248 views

Any fastest algorithm for $f(n) = f(n-1) \cdot f(n-2)$ where $3 \leq n \leq 1000000$

Any fastest algorithm for $$ f(n) = f(n-1)\cdot f(n-2)\quad\text{ where }\quad f(1) = 1,\quad f(2) = 2 $$ for $3 \leq n \leq 1000000 $.
5
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1answer
2k views

Quick sort algorithm average case complexity analysis

This is for self-study. This question is from Kenneth Rosen's "Discrete Mathematics and Its Applications". The quick sort is an efficient algorithm. To sort $a_1,a_2,\ldots,a_n$, this algorithm ...
0
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1answer
51 views

Algorithms and generalisation of functions

I admit I'm little bit poor in functions in mathematics. But I'm in real urge to get this riddle out. How to express $$x(n)=x(n-1)+x(n-2)+1,$$ where $n>1$ and $x(0)=0$ and $x(1)=1$, in terms of ...