Recall that the height of a node in a binary tree is the number of edges (links) from the node to the deepest leaf in its subtree (e.g., the height of leaves is 0). To answer the following question, consider a complete binary tree T of size n in which the last level contains all possible nodes from left to right. (a) Specify the number of nodes at height h in T for any h≥ 0 (no proof is needed). (b) Specify the asymptotic running time of Bubble-Down for a node of height h as a function of h. Then show the total time complexity to Bubble-Down all nodes at level h of T when we apply the Heapify operation on T. Justify your answer in one or two sentences. (c) Use your answer in part to prove that the time complexity of the Heapify operation is O(n). You need to sum the total work over all levels. Show your work. =0(1). Hint: Σ x=0
Recall that the height of a node in a binary tree is the number of edges (links) from the node to the deepest leaf in its subtree (e.g., the height of leaves is 0). To answer the following question, consider a complete binary tree T of size n in which the last level contains all possible nodes from left to right. (a) Specify the number of nodes at height h in T for any h≥ 0 (no proof is needed). (b) Specify the asymptotic running time of Bubble-Down for a node of height h as a function of h. Then show the total time complexity to Bubble-Down all nodes at level h of T when we apply the Heapify operation on T. Justify your answer in one or two sentences. (c) Use your answer in part to prove that the time complexity of the Heapify operation is O(n). You need to sum the total work over all levels. Show your work. =0(1). Hint: Σ x=0
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
Related questions
Question
![Recall that the height of a node in a binary tree is the number of edges (links) from the node
to the deepest leaf in its subtree (e.g., the height of leaves is 0).
To answer the following question, consider a complete binary tree T of size n in which
the last level contains all possible nodes from left to right.
(a) Specify the number of nodes at height h in T for any h≥ 0 (no proof is needed).
(b) Specify the asymptotic running time of Bubble-Down for a node of height ʼn as a
function of h. Then show the total time complexity to Bubble-Down all nodes at level
h of T when we apply the Heapify operation on T. Justify your answer in one or two
sentences.
(c) Use your answer in part to prove that the time complexity of the Heapify operation is
O(n). You need to sum the total work over all levels. Show your work.
Hint: 0(1).
x=0
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4575c695-56bc-4a6a-843f-ec886ca258f2%2F85ad882c-d49d-4a05-bf71-1d6f457db8fb%2Fxzpnzah_processed.png&w=3840&q=75)
Transcribed Image Text:Recall that the height of a node in a binary tree is the number of edges (links) from the node
to the deepest leaf in its subtree (e.g., the height of leaves is 0).
To answer the following question, consider a complete binary tree T of size n in which
the last level contains all possible nodes from left to right.
(a) Specify the number of nodes at height h in T for any h≥ 0 (no proof is needed).
(b) Specify the asymptotic running time of Bubble-Down for a node of height ʼn as a
function of h. Then show the total time complexity to Bubble-Down all nodes at level
h of T when we apply the Heapify operation on T. Justify your answer in one or two
sentences.
(c) Use your answer in part to prove that the time complexity of the Heapify operation is
O(n). You need to sum the total work over all levels. Show your work.
Hint: 0(1).
x=0
=
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