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

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Find a Recurrence Relation

I want to find a recurrence relation for number of decimal numbers with length n, (we called $a_0$ ) that not includes 0 and any combination of 11,12, 21. i see the result is: ...
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24 views

Geometric sum of recurrence relation

I am reading the textbook Algorithm Design by Kleignberg and Tardos and I am having trouble on page 216. $$T(n) \le \sum_{j=0}^{log_2n - 1} \left ( \frac{q}{2}\right )^j cn = cn \sum_{j=0}^{log_2n - ...
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2answers
63 views

Recurrence Relation from Old Exam

I see this challenging recurrence relation that has a solution of $T^2(n)=\theta (n^2)$. anyone could solve it for me? how get it? $$T(n) = \begin{cases} n,\quad &\text{ if n=1 or n=0 ...
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32 views

Need help checking my recurrence for a simple algorithm

All I'm writing to get a second opinion on the algorithm shown in this link. I'm pretty sure its supposed to be $T(n)=2T(n/2)+n$ but I can't see where I'm supposed to get the +n from. So far I'm ...
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47 views

recursive equation: divide and conquer, subtract and conquer problem in one

I have a recursive equation that does not apply to the master method, since there is subtraction in the equation. I'd like to use the substitution method, but I have no idea where to start. Could ...
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24 views

Proving the correctness of a program

So I have this program below SquareRootRecursion that I need to prove is correct. However i'm not sure what it's pre and post conditions would be and how I would use those to prove it's correctness. ...
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55 views

Proof of recursivity of Shannon's Entropy

Does anybody know a book where the proof of recursivity property of Shannon's Entropy can be found? I mean this: $$H(q_1,...,q_n)=H(q_1 + q_2, q_3,...,q_n) + (q_1 +q_2)H( \frac{q_1}{q_1+q_2} , ...
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41 views

How to solve a recurrence

Apologies for this is a really trivial question, but I cannot figure out what to do after understanding the pattern of the function. Say I have this: $\begin{align} T(n) & = T(n-1) + 1/n \\ ...
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163 views

What are algorithm? Can we relate algorithm using set theory concepts?

What really algorithm are? Can we define algorithm as functions or in terms of set theory Can we reconvert proof using algorithm in set theory concept.For example, Theorem: A function defined on ...
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43 views

Recurrence equation of $ T(n) = T(n/2 ) + dn\log_2(n)$

I have the following equation: $$T(n) = T\left({n \over 2}\right) + d n \log_2 n$$ A little investigation: $T(2^1) = 1 + 2d$ $T(2^2) = T(2^1) + 2^2d\times 2 = 1 + 10d$ $T(2^3) = T(2^2) + ...
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What's the next “recursion” here?

Plotting a single 3d helix is x = cos(t); y = sin(t); z = t; From this equation: x = [R + a cos(\omega t)] cos t y = [R + a cos(\omega t)] sin t z = h t + a sin(omega t) Comes the awesome ...
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86 views

Algorithm to calculate rating based on multiple reviews (using both review score and quantity)

First of all I must state that I am not a mathematician, so please correct me if I use wrong terminology. I am building a web application which needs to calculate the rating for each entity based on ...
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1answer
22 views

Finding a tight upper-bound on $T(n) = 3T(\frac{2}{3}n)$

Can the master theorem be used to prove a tight upper-bound on $T(n) = 3T(\frac{2}{3}n)$? I've drawn the tree for the recurrence and found a sequence: $n + 2n + ...
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How can I solve the particular solution of the following recurrence (recursive) relation?

Having $a_n = 3a_{n-1} + 2a_{n-2} + 3·2^{2n-1}$ $a_1 = 12$ $a_0 = 0$ I solved the homogeneous part and got: $a^{{h}}_n = 1/12·2^n - 1/12·1^n$ This is the particular solution that I need to ...
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Using a Recursion Tree to solve the recurrence $T(n) = \sqrt n T(\frac{n}{2}) + 10n$?

I am attempting to solve the above recurence by giving tight $\Theta$ bounds. Assume that the logs here are all base 2! To solve a recursion tree as far as I understand, I need two things. The ...
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Algorithms recurrence equation using unwinding / substitution

This is homework for introduction to Algorithms this is what I have below. (25 pts) Use the direct unwinding to prove that the recurrence equation T(n)= 2T(n/2)+ nlogn has a lower bound of ...
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1answer
35 views

Recurrence Relation

So I am just making sure I am on the right track with this. I have the recurrence: T(n) = 2T(n-2) + 1 I am trying to solve this recurrence to get the time complexity T(n) = 2(2T(n-4) + 1) + 1 T(n) ...
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1answer
32 views

Question about Growth Rates

I see in some notes from my instructor in Algorithm course that $\Sigma_{i=0}^{log n} (n/2^i)$ has growth bigger than $\Sigma_{i=1}^{n} (i log i)$. i couldn't understand why? any tutorial or hint?
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Is there a simplification for the coefficients generated with the Mandelbrot iteration rule?

The Mandelbrot Set is obtained using the equation $z_n=z_{n-1}^2+c$ for some constant $c \in \mathbb{C}$ with $z_0=c$. Therefore, $z_1=c^2+c$, $z_2=c^4+2c^3+c^2+c$, etc. I have a function $f(n,x)$ ...
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recursive sequence - Which approach can I take to solve this equation?

Having this recurrence relation $a_n = 5a_{n-1} - 6a_{n-2} + 4·3^n$ $a_1 = 36$ $a_0 = 0$ How can I solve this? I tried by characteristics roots and got stuck: *making $a_n=r^n$ $r^n = 5r^{n-1} - ...
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Can this recursive summation function be simplified?

I have the recursive function $$ a_n = \sum_{x=1}^n a_{n-x} a_{x-1} $$ where $a_0=1$ and $n$ is a positive integer. Looking at a graph of this function, it's very exponential in form, but it's not ...
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67 views

Tight asymptotic upper and lower bounds

I have a equation: $T(n) = 4T(n/3) + n\ln n$ In this equation, I have to give tight asymptotic upper and lower bounds. What does that mean? I know I can apply Master theorem (which gives me theta ...
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1answer
39 views

Big O complexity of the partition function derived from this code?

I am not able to calculate the Big O complexity of the partition function given in the code below. I tried to calculate it by estimating the number of nodes in the tree. So far, I've figured out that ...
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1answer
26 views

How to solve the recursive complexity?

I have such recursive complexity $T(n) = T(n-2) + log(n)$. The problem is that I cant use a recursion tree or the master theorem. The only way, which I know, is to guess and then proof the answer. ...
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40 views

Quicksort probabilistic analysis

Let us say that we randomly pick up a pivot element and partition the array around it. What is the probability that we always pick the pivots in subsequent recursive calls such that it partitions the ...
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27 views

limit of $f_n(a) = a^{f_{n-1}(a)}$ as $n$ approaches infinty for small values of $a$

So a friend started, in boredom, calculating values of what I have formalized as $f_n(a) = a^{f_{n-1}(a)}$ (also $f_0(a)=a$) for $a = 1.1$. He noticed that on his calculator it was not changing value ...
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Proving a summation

My exercise is to prove $\sum_{i=0}^n i = \frac{n(n+1)}{2} $. This is what I tried: Let $P(n) = \sum_{i=0}^n i$ Step 1: Show $P(1)$ is true $$P(1) = \sum_{i=0}^{1} i = 0 + 1 = 1$$ $$\frac{n(n + ...
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Divisible 4 for every different number?

My formula is for ABC 3 digits number; 100A + x*B + y*C. What should coefficient of B and C be for everytime different result for different number? (Result number ) mod 4 has to be zero. For ...
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28 views

Complexity of FFT Algorithm

OkayI am using iterative FFT algorithm and I have found that since there are 2N computation per stage and there are logN stages the complexity should be O(2NlogN) I can reduce the number of ...
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1answer
33 views

Recurrence relation of the following sequence?

This is the code: for (unsigned int i = 0; i < n; ++i) if (i % 2 == 0) ++k; And this is the output for when ...
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How do I create a function from this code? [closed]

Here is the code: for (int i = 1; i < n; i *= 2) ++k; I need to express this as a function. I don't know where to begin.
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Find Recurrence Relation of Code

Suppose A(n) be the number of stars that wrote with the following example. for n>=3, i want calculate the recurrence relation for this code. any idea or solution? ...
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1answer
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An inequality and random number and algorithm problem

I've been run into a problem which is absolutely not as simple as is looking, I just don't know how to describe it.I think it is a math problem eventually? Suppose a target number A and a mistake C ...
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How does this simplification work?

The following recursive function was given: $$T\left(n\right) = T\left(n - 1\right) + x$$ The author stated that by using repeated substitution we can solve the recurrence relation: The basic ...
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Time Complexity of one Challenging Example

Anyone would help me to calculate the order (time complexity) of this example ?
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172 views

Algorithm for adding n 1-bit numbers

suppose adding two numbers, (that first number has a bits and second number has b bits) can be done in ...
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68 views

Local informatics Olympiad and Algorithm

I see one of recent local informatics Olympiad question. i have a trouble to solve it. any idea? hint? or solutions? thanks to all creative man. We have two function $P_1, P_2$ and input an array $n$ ...
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1answer
154 views

Turing machines that compute $\pi$

For each $K > 0$ there is a brut force Turing machine $\pi_K$ that "computes" the first $K$ digits of $\pi$ starting on the blank tape (all $b$s) with $K+1$ states $S \in \mathsf{S} = ...
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Number of strings of size $k$ that do not have 'ab'

Consider $\Sigma = \{a,b,c\}$ and the language $L$, the set of all strings that do not contain 'ab' Find strings, of size $k$ is in $L$ ($L_k$) Consider $A_k$ (strings of size $k$ that end in $a$) ...
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1answer
62 views

Hashing With Chaining Collision

We have $1000$ elements with key=1 to 1000, and a hashing function $$ h(i)=i^3 \mbox{ mod } 10 $$ for an array with length $10$ (array index from $0$ to $9$) with chaining method. What is the ...
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115 views

Water Box with n Liter

I ran into a basic challenging problem. I see an high school local math Olympiad question. we have a box that keep n Liter water. each time we extract 1/k Water from box. how many times (minimum) we ...
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Polynomial Multiplication through Toom-Cook into Karatsubas

I'm trying to solve a polynomial multiplication problem recursively through using Toom-Cook (Toom-3) once and Karatsuba (Toom-2) five times, although I'm stuck after the first round of Karatsubas. ...
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Need an Algorithm Such that $\sum_{k-i}^{j}{A[k]}$

I need an algorithm for real application. Suppose we have array A (positive & negative ) numbers. we want to find index i, j such that $\sum_{k-i}^{j}{A[k]}$ has the lowest difference to zero. ...
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Can we sort 6 numbers with at most 9 comparison? [duplicate]

i know there is an algorithm to sort 5 numbers with 7 comparison. Can we sort 6 numbers with at most 9 comparison? thanks to all.
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3answers
102 views

2000 Olympiad in Informatics Question on Box

I have an old Olympiad question on informatics. There are 31 boxes. In each box there is one number. We know the number if and only if we open the box. We want to calculate the minimum number of ...
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58 views

Problematic Initial Condition of a Recurrence Relation

I encountered this equation, and tried to solve it: $T(n) = T(\sqrt{n})+log(n)$ Under the initial condition T(1)=1. Can someone tell me why is this initial condition helpful? I mean, of course ...
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110 views

Meaning of 'expected value' in the following problem

Ok, I have found an interesting probabilites problem on TopCoder. I have truncated the statement: "What is the expected number of dice throws needed to attain a value of at least n (candies, in this ...
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71 views

Recursive definition of the set of odd numbers

Show that the following is another recursive definition of the set ODD (keep in mind you’ll need say something about even numbers too): Rule 1: 1 and 3 are in ODD. Rule 2: If x is in ODD, then so is x ...
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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: ...
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164 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 ...