Algorithms Write whether the following statements are true or false. If true, prove whyit is true. If false, give a counter example or proof. Assumen is an integerthat is greater than 0.(а) 11"(b) Given that the time to execute an algorithm with n elements is T (n),where T(n) = 3T(n/2) + n², then the complexity of the algorithm isO(n²)(c) The computational complexity of the code given below is O(n³/2). Thevalue of k = vn1.6 is divisible by 5 for all values of n > 1int sum (int n){ int sum=0;for (int i=0; i Sorting and Red-Black Trees(a) Consider a sequence of sorted numbers, for example 1, 2, 3, 6, 7, 8, 9.We rotate this sequence k times from left to right. For example fork = 1, the sequence will be 2, 3, 6, 7, 8, 9, 1, for k = 3 the sequence willbe 6, 7,8, 9, 1, 2, 3.Given a sequence of n numbers that has been sorted k times, developa O(n) algorithm to find the value of k. Note that the lowest numberneed not necessarily be 1. You can assume that all numbers are unique.Explain the steps of your algorithmProve that the complexity is O(n)iii. Using the cxample 6, 7, 8, 9, 2, 3, 1, show that your algorithm2.i.ii.finds the k=3(b) Write the properties of Red-black trees. Explain why red-black treesrequire less rotations than AVL trees(c) Show all the steps of creating a red-black tree for the following set ofnumbers; { 3,5,6,2,1,4}

Properties of red-black tree, red black trees require less rotation as compare to avl trees

Answered: (b) Write the properties of Red-black…

Engineering

Computer Science

(b) Write the properties of Red-black trees. Explain why red-black trees require less rotations than AVL trees

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Assume that for each number I n is not 2. How could the algorithm be modified to handle the situation where n is odd? I have two approaches: one that directly adjusts the recursive method and the other that mixes the iterative and recursive approaches. Just one of the two tasks must be completed (as long as it works and does not increase the BigOh of the running time.)
Consider the following recursive algorithm, where // denotes integer division: 3//2 = 1, 5//2 = 2, etc. F(n):if n <= 1: returnF(n//2)for i from 0 to n for j from 0 to n//2 print(i+j) Let function T(n) denote the running time of this algorithm. Derive T(n) and prove its worst case timecomplexity
What is the leading term for the following expressions and specify the lowest Big O Complexity for each algorithm?
Consider the recursive algorithm below for computing the sum of the first n cubes: S(n) = 13 + 23 + … + n3 ALGORITHM S(n) //n is a positive integer if (n==1) return 1 else return S(n – 1) + (n*n*n); Solve the recurrence relation in your answer for the previous question. The runtime efficiency of the algorithm is? A. O(n) B. O(n2) C. O(n3) D. None of the above
Consider the following non-recursive algorithm: Algorithm Euclid2(m, n)//Input: Integers m and n where m >=n//Output: Greatest Common Divisor of m and nwhile n ≠ 0 do r ← m mod n m← n n ← r return m a) What does this algorithm compute?b) What is the input size for this algorithm?c) What is the basic operation of this algorithm?d) How many times the basic operation is executed worst case scenario?e) How many times the basic operation is executed in best-case scenario?
I'm trying to understand how both of these algorithms, separated by a) and b), have a time complexity of O(n), could you explain how this is? a)int i = 0;while (i < n) { ... i++;}for(int j = 3; j < n * n; j += n) { ...}int k = 1;while (k < n) { ... k *= 2;} b)for(int i = 1; i < n; i += 3) { for(int j = 3; j < 10; j++) { for(int k = 5; k < n; k += n) { ... } }}
Consider the following non-recursive algorithm:Algorithm Euclid2(m, n)//Input: Integers m and n where m >=n//Output: Greatest Common Divisor of m and nwhile n ≠ 0 do r ← m mod n m← n n ← r return ma) What does this algorithm compute?b) What is the input size for this algorithm?c) What is the basic operation of this algorithm?d) How many times the basic operation is executed worst case scenario?e) How many times the basic operation is executed in best-case scenario?
The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. Answer only 2 question
: The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers.
The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. Answer 2 and 3 questions
The following questions are independent. 1. A palindrome is a string that reads the same forward and backward. Using our algorithmic language, propose an algorithm that determines whether a string of n characters is a palindrome. 2. Let x be a real number, and n be an integer. Devise an algorithm that computes xn. [Hint: First, give a procedure for computing xn when n is nonnegative by successive multiplication by x, starting with 1 until we reach n. Then, extend this procedure and use the fact that x–n = 1/xn to compute xn when n is negative.] 3. The median of a set of integers is the middle element in the list when these integers are listed in increasing order. The mean of a set of integers is the sum of integers divided by the number of integers in the set. Write an algorithm that produces the maximum, median, mean, and minimum of a set of three integers. Answer question 3 only with details
Determine whether the following is true or false. Please cite the brief explanation so that I can know why is it true or false. a) The sentence “If x > 0, then compute √π”, is as tatement. b) The compound statement p ∧(q ∨¬p )∧¬q is ac ontradiction. c) ¬( p ∨ q) ≡ ¬p ∧¬q d) 1 ∈ {{1},{2},{3}} e) Set B = {x | x ∈ ℝ and x4 =16} is finite. f) [2,4] U (3,5) is a disjoint union g) A\B = A ∩ Bc h) A X B = B X A I) {{Ø}} = 0 j) The contrapositive of p -> (p V ¬p) = (¬p∧ q) -> ¬p