Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 14.1, Problem 7E
Program Plan Intro
To count the number of inversions in an array of size n for an order-statistic tree in
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Use the Transform-and-Conquer algorithm design technique with Instance Simplification
variant to design an O(nlogn) algorithm for the problem below. Show the pseudocode.
Given a set S of n integers and another integer x, determine whether or not there exist two
elements in S whose sum is exactly x.
Problem 3.
Consider the following recurrence.
T(n) = {(n) = 37(n
T(n) = 3T(n/2) + n²
if n=1
otherwise.
(a) Solve this recurrence exactly by the method of substitution. You may assume n is a
power of 2.
(b) Solve it using the recursion tree method.
Suppose that f(n) satisfies the divide-and-conquer recurrence relation f(n) = 3f(n/4)+n2/8 with f(1) = 2. What is f(64)?
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- Algorithm FHL Version of Schreier-Sims is O ( I f) 16), provided that the size of the initial generating set is O ( I ~ 14).arrow_forwardWrite a recurrence relation describing the WORST CASE running time of each of the following algorithms. Justify your solution with either substitution, a recursion tree, or induction. Simplify your answer to O(nk) or O(nk log(n)) whenever possible. If the algorithm takes exponential time, then just give exponential lower bounds. (b) 1: function Func(A[],n) 2: if n < 10 then return A|1| 4: end if 5: X = 6: for i = 1 to n3/2 do 7: x = x + A[[i/n]] 8: end for 9: x = x + Func(A[],n – 5) 10: return x 11: end function 3:arrow_forwardfor the following problem we need to use a recursion tree. so we can determine an asymptotic upper bound on therecurrence T(n) = 3T(n/2) + n. the substitution method must be used to solve.arrow_forward
- Consider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.arrow_forwardFor the 8-queens problem, define a heuristic function, design a Best First Search algorithm in which the search process is guided by f(n) = g(n) + h(n), where g(n) is the depth of node n and h(n) is the heuristic function you define, and give the pseudo code description.arrow_forwardPlease help me with this recurrence relation: B(0) = 1, B(n) = B(n −1) + n + 3 for n > 0 Use the method of unraveling to find a closed form for B(n).arrow_forward
- Given the recurrence relation: • T(n) = 8 if n 6. Find the value of T(495). [Hint: Use a recursion tree to solve the recurrence exactly and plug the argument into the function you obtain.] Answer: 493 The correct answer is: 336 Xarrow_forwardPlease explain Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) and (b) using Master Theoremarrow_forwardSolving recurrences using the Substitution method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the substitution method. Assume that T(n) is constant n ≤ 2. Make your bounds as tight as possible and justify your answers. Hint: You may use the recursion trees or Master method to make an initial guess and prove it through induction a. T(n) = 2T(n-1) + 1 b. T(n) = 8T(n/2) + n^3arrow_forward
- For each of the following recurrences, verify the answer you get by applying the master method, by solving the recurrence algebraically OR applying the recursion tree method. T(N) = 2T(N-1) + 1 T(N) = 3T(N-1) + narrow_forwardPlease explain!! Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) andarrow_forwardUsing the recursion tree method, show to work to derive the runtime for the following recurrence relation: Hint:The resulting runtime should should be O(n*n^(1/log3(4/3))) or O(n^4.8188). need the process how to reach the answer and that was the whole question Note: will like the correct and detailed answer. Thank you!arrow_forward
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