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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Question
Chapter 6, Problem 1P
(a)
Program Plan Intro
To show the difference between the output of the array when it is passed as an array for BUILD-MAX-HEAP and BUILD-MAX-HEAP'.
(a)
Expert Solution
Explanation of Solution
Given Information: A heap that has n-elements.
Explanation:
Pseudo code for BUILD-MAX-HEAP is given below:
| BUILD-MAX-HEAP( A ) |
| A.heapsize = A.length |
| for i = A.length/2 downto 1 |
| MAX-HEAPIFY( A,i ) |
Consider an array A= {1, 2, 3, 4, 5, 6}. If the above
The for-loop will iterate from 3 to 1.
At i=3 , MAX-HEAPIFY will be called with arguments as array A and i=3 . The following pseudo code for MAX-HEAPIFY is as follows:
| MAX-HEAPIFY( A,i ) |
| l = LEFT( i ) |
| r = RIGHT( i ) |
| ifl <= A.heapsize and A[l]> A[i] |
| largest = l |
| Else |
| largest = i |
| ifr <= A.heapsize and A[r]> A[largest] |
| largest=r |
| iflargest != i |
| exchange A [ i ] with A [ largest ] |
| MAX-HEAPIFY( A,largest ) |
Currently at i=3 , A [3]=3 and have the following tree:
At the end of the iteration:
At i=2 , A [2]=2 and after executing the algorithm, the following tree is evolved:
At i=1 , A [1]=1 and after executing the algorithm, the following tree is evolved:
Resultant array after BUILD-MAX-HEAP is A= {6, 5, 3, 4, 2, 1}
However, according to the question, the algorithm for BUILD-MAX-HEAP ', there is an alteration in A.heapsize, which is by default 1. Moreover, the value of i starts from 2 and goes on till A.length that is 6.
At i=2 , A [2]=2. The tree is given below-
Look at the algorithm of BUILD-MAX-HEAP' . It is using MAX-HEAP-INSERT. So, it becomes-
At i=3 , A [3]=3 and after using MAX-HEAP-INSERT
At i=4 , A [4]=4 and after using MAX-HEAP-INSERT
At i=5 , A [5]=5 and after using MAX-HEAP-INSERT
Finally at i=6,A [6]=6 and after using MAX-HEAP-INSERT
Resultant array after BUILD-MAX-HEAP' is A= {6, 4, 5, 1, 3, 2}
Hence, the resultant array after BUILD-MAX-HEAP and BUILD-MAX-HEAP' is not the same.
(b)
Program Plan Intro
To determine the worst case scenario for BUILD-MAX-HEAP' while entering n-elements.
(b)
Expert Solution
Explanation of Solution
The BUILD-MAX-HEAP' has MAX-HEAP-INSERT in it, which has worst case of Θ(log n). The call to this function is done for n-1 times, since it is not considering the parent node here.
The worst case for MAX-HEAP-INSERT happens when an array is sorted in ascending order.
Each insert step takes maximum Θ(log n ) steps, and since it is done for n times, it takes a worst case of Θ(nlog n ). Each iteration in the new block is actually taking more time as the older version as it has more iteration in it. The new block will work in even work in even worse case as it has the usage of other algorithm as well.
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Students have asked these similar questions
A 4-ary max heap is like a binary max heap, but instead of 2 children, nodes have 4 children. A 4-ary heap can be represented by an array as shown in Figure (up to level 2). Write down MAX_HEAPIFY(A,i) function for a 4-ary max-heap that restores the heap property for ith node.
Suppose we have p max heaps, with q elements in each. We wish to combine these into a single heap. Each of the following sub-parts describes one approach merging the heaps, and you are asked to derive the big O running time. MUST show derivation
We create a new, empty heap H. From each of the input heaps, P, until P is empty we repeatedly remove the highest priority element using the heap delete method, and insert it into H using the heap insert method. What is the worst-case big O running time? Show derivation.
We create a new, empty linked list L. For each of the input heaps, P, we iterate over its internal array and insert each element at the front of L. We then transfer the elements of L into an array, and heapify the array using repeated sift downs. What is the worst-case running time? Show derivation.
We divide the p input heaps into p/2 pairs, and apply the algorithm from part (b) on each pair, resulting in p/2 heaps. We then repeat this process until we are left with a…
Suppose we want to use the Heapsort algorithm to sort a large list of numbers.
Our first step is to convert the input list to a heap, and then run BUILD-MAX-HEAP, which applies MAX-HEAPIFY on all the nodes in the heap, starting at the bottom and moving towards the top.
For example, if A = [4,1,3,2,16,9,10,14,8,7], then BUILD-MAX-HEAP(A) makes a total of 7 swaps, and returns the array [16, 14, 10, 8, 7, 9, 3, 2, 4, 1].
The swaps are made in this order: (14, 2), (10, 3), (16, 1), (7, 1), (16, 4), (14, 4), (8, 4). Suppose A = [2,4,6,8,10,12,14,1,3,5,7,9,11,13,15].Determine the total number of swaps made by BUILD-MAX-HEAP(A).
Chapter 6 Solutions
Introduction to Algorithms
Ch. 6.1 - Prob. 1ECh. 6.1 - Prob. 2ECh. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.2 - Prob. 1ECh. 6.2 - Prob. 2ECh. 6.2 - Prob. 3E
Ch. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6 - Prob. 1PCh. 6 - Prob. 2PCh. 6 - Prob. 3P
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