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 2.3, Problem 2E
Program Plan Intro
To write the merge sort procedure so that it does not use sentinels and also stopping left or right of array and copied back to array.
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function(A[1.n])
{
if n = 1 then return;
for i:= 1 to n/2:
A[) := A[i]+i;
function (A[1.n/2);
Find the closed form of recurrence with substitution, iteration method or recursion tree method.
Draw the recursion tree for the merge sort procedure on an array of 16 elements. Explain why memoization fails to speed up a good divide-and-conquer algorithm such as merge sort.
Using the recursion tree method find the upper and lower bounds for the following recurrence (if they are the same, find the tight bound).
T (n) = T (n/2) + 2T (n/3) + n.
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Introduction to Algorithms
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- Create a recursion tree for T(n)=3T(n/2)+2n^2, with T(1)=c2, for n=8 . calculates the total number of comparisons.arrow_forwardCreate a recursion tree like for T(n)=3T(n/2)+2n^2, with T(1)=2, for n=8 . Like is done in the figure that calculates the total number of comparisons.arrow_forwardJava program to return the third smallest element in the Binary Search Tree in O(n) time. VERY IMPORTANT: The value of the third smallest element is ”3”. You may choose either a recursive or an iterative solution, but you are NOT allowed to maintain an extra array to store values of the tree elements for a linear scan later to return the required value. Also, you can’t just return 3 :)arrow_forward
- 2. method. Assume that T(1) (1). Solve the following recurrences using the recursion tree d. T(n) = 4T(n/2) + n e. T(n) = 2T(n-2) +1 f. T(n)= T(n/2) +T(n/3) + narrow_forwardUse the recursion tree method to guess tight asymptotic bounds for the recurrence T(n)=4T(n/2)+n. Use substitution method to prove it.arrow_forwardDraw the recursion tree for n = 12 (array length). sumSquares (array, first, last): if (first == last) return array [first] array[first]; int mid = (first + last)/2; return sumSquares (array, first, mid) + sumSquares (array, mid + 1, last); Recursion tree node count formula. Big-C runtime? • Formulate tree height. Big-C memory? Recursive vs iterative.arrow_forward
- Draw the recursion tree:arrow_forwardWrite a binary search tree method that takes two keys, low and high, and prints all elements X that are in the range specified by low and high. Your program should run in O(K + log N) average time, where K is the number of keys printed. Thus, if K is small, you should be examining only a small part of the tree. Use a hidden recursive method and do not use an in-order iterator. Bound the running time of your algorithmarrow_forwardUse the recursive strategy described in the chapter to implement a binary tree. Each node in this method is a binary tree. Thus, a binary tree includes references to its left and right subtrees in addition to the element stored at its root.You could also wish to make mention of its progenitor.arrow_forward
- Answer the given question with a proper explanation and step-by-step solution. Write a recursive method(Code ) static void mirrorTree(node root) This will take a tree as input and then change the tree such that it becomes a mirror of itself. We have the following methods implemented(you can use them and assume we coded somewhere else): getLeft(), getRight(), setLeft(), setRight(), getValue(). All the values in a node are integers. [Hint: Use them, and think how you would write when the tree only has 2 children ] Example:arrow_forwardSuppose 1,000 integer elements are generated at random and are inserted into a sorted linked list and a binary search tree (BST) separately. Considering the efficiency of searching for an element in the two structures, which of the following statements is true? The search operation on the BST takes shorter time because it is relatively balanced. O None of these The search operation on the list takes longer time because the numbers are not sorted. The search operation on the BST longer time because the numbers are not sorted. O The search operation will take the same time in both structures.arrow_forwardJav ADDCOMMENT in code, please Do TEST-CASES address your test cases in the form of code and not prose :) In today's Lab we will explore a specific way to perform a Depth First Search (DFS) of a given Graph [Ref : Figure 1]. You will implement the traversal using one of the two ways stated below : [1] With Recursion. When you follow the traversal process as specified - the complexity of the solution will be linear as shown below. Time Complexity: O(V + E), where V is the number of Vertices and E is the number of Edges respectively. Space Complexity: O(V )arrow_forward
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