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
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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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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.
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.
Implement the three improvements to mergesort Add a cutoff for small subarrays, test whether the array is already in order, and avoid the copy by switching arguments in the recursive code.
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- Consider the following algorithm: int f(n) /* n is a positive integer */ if (n <=3) return n int sum = f(n-1) if (n is even) return sum + f(n-2) else return sum + f(n-3) Trace execution of f(6) by drawing the recursion tree. Show all function calls (callers and called functions), and intermediate results; and show what f(6) returns. Make sure that you draw all subtrees even if some are identical.arrow_forwardIn java and in O(logn) time Write a method that balances an existing BST, call it balance(). A BST is balanced if the height of its left and right sub-trees are different by at most one. Recursively applied. If the tree is balanced, then searching for keys will take act like binary search and require only logn comparisons. No performance requirements on your balancing algorithm. (Come up with a way yourself - don't skip to 3.3. That section is really complicated and meant for the harder case where you need to do it in log time.)arrow_forwardFor each, draw the recursion tree, find the height of the tree, the running time of each layer, and the sum of running times. Then use this info to find the explicit answer for T(n). a. T(n) = 2T(n/4) + √ n (n is a power of 4 (n = 4^k) for some positive integer k) b. T(n) = 9T(n/3) + n^2 (n is a power of 3 (n = 3^k) for some positive integer k) c. T(n) = T(n/2) + 1 (n is a power of 2 (n = 2^k) for some positive integer k)arrow_forward
- Draw 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_forwardshow the recursiontree for applying mergesort to the list 15 20 7 3 19 6 17 25 and determine the number of comparisons performed during the sortarrow_forwardProvide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-Θ by determining a formula that counts the number of nodes in the tree as a function of n.arrow_forward
- AvgCompares(), a recursive function that calculates the average number of comparisons needed by a random search hit in a given BST (the internal path length of the tree divided by its size plus one), should be added to the BST. Create two implementations: a recursive method that adds a field to each node in the tree and takes linear space and constant time every query, and a method similar to size() that takes linear space and constant time per query.arrow_forwardAdd a recursive function to BST called avgCompares() that computes the average number of comparisons required by a random search hit in a particular BST (the internal path length of the tree divided by its size plus one). Create two implementations: a recursive approach (which requires linear time and space proportionate to the height) and a way similar to size() that adds a field to each node in the tree (which requires linear space and constant time each query).arrow_forwardUsing 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.arrow_forward
- Create a bottom-up insertion technique based on the same recursive approach, a red-black representation, and balanced 2-3-4 trees as the underlying data structure for an implementation of the fundamental symbol-table API. Only the sequence of 4-nodes (if any) at the bottom of the search path should be split by your insertion technique.arrow_forwardSort the array {3,17,38,2,43,19,25,40} using Merge Sort, also showing the recursiontree.arrow_forwardBelow is given a header file and a source file of a Binary Search Tree. Inputs are 6 , 4, 2 , 5, 1, 3, 8, 7, 9 Simulate the recursion for in-order and post order traversal of BST. you have to draw the tree and queue for the given inputs which will be for in-order and post order traversal. You need to write the steps of simulation in paper. binarysearchtree.h#ifndef BINARYSEARCHTREE_H_INCLUDED#define BINARYSEARCHTREE_H_INCLUDED#include "quetype.h"templatestruct TreeNode{ItemType info;TreeNode* left;TreeNode* right;};enum OrderType {PRE_ORDER, IN_ORDER,POST_ORDER};templateclass TreeType{public:TreeType();~TreeType();void MakeEmpty();bool IsEmpty();bool IsFull();int LengthIs();void RetrieveItem(ItemType& item,bool& found);void InsertItem(ItemType item);void DeleteItem(ItemType item);void ResetTree(OrderType order);void GetNextItem(ItemType& item,OrderType order, bool& finished);void Print();private:TreeNode* root;QueType preQue;QueType inQue;QueType postQue;};#endif //…arrow_forward
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