1. We can build a heap by repeatedly calling MAX-HEAP-INSERT to insert the ele- ments into the heap. Consider the following variation on the BUILD-MAX-HEAP procedure: BUILD MAX HEAP' (A) 1 A. heap_size=1 2 for i=2 to A. length 3 MAX HEAP INSERT (A, A[i]) (a) (10 points) Do the procedures BUILD-MAX-HEAP and BUILD-MAX-HEAP' always create the same heap when run on the same input array? Prove that they do, or provide a counterexample. (b) (10 points) Show that in the worst case, BUILD-MAX-HEAP' requires (nlgn) time to build an n-element heap.

1. We can build a heap by repeatedly calling MAX-HEAP-INSERT to insert the ele- ments into the heap. Consider the following variation on the BUILD-MAX-HEAP procedure: BUILD MAX HEAP' (A) 1 A. heap_size=1 2 for i=2 to A. length 3 MAX HEAP INSERT (A, A[i]) (a) (10 points) Do the procedures BUILD-MAX-HEAP and BUILD-MAX-HEAP' always create the same heap when run on the same input array? Prove that they do, or provide a counterexample. (b) (10 points) Show that in the worst case, BUILD-MAX-HEAP' requires (nlgn) time to build an n-element heap.

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

See similar textbooks

Similar questions

1. Assume we build a binary max-on-top heap from keys 10, 12, 1, 14, 6, 5, 8, 15, 3, 9, 7, 4, 11, 13, 2. a. Show the result of building this heap by inserting the above keys one at a time in the order given (from left to right), into an initially empty binary heap. Please show key steps and short illustrations if necessary. b. Show the heap from problem (a) above after executing three deleteMaximum operations on this heap. Show each step. 2. Use the same key set given Q1, write a program to output the median using heap structure. The median is the value separating the higher half from the lower half of a data sample. For example, the median of input is 8. 3. Use the same input stream of Q1, implement HeapSort in ascending order.
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.
Consider a min heap with 1023 distinct integers. If root is at Oth level then fill the blank given in belowsentence. The 10th minimum of the heap might be at _____ level. A:1 B:3 C:8 D.9
What does the initially empty max-heap h contain after the following sequence of pseudo-code operations? Show at least 5 binary heaps as stated in Figure 1 to show the changes after a sequence of heap operations.
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).
Given an initial sequence of 9 integers < 53, 66, 392, 62, 32, 41, 22, 36, 26 >, answer the following:a) Construct an initial min-heap from the given initial sequence above, based on the HeapInitialization with Sink technique learned in our course. Draw this initial min-heap. NOsteps of construction are required. b) With heap sorting, a second min-heap can be reconstructed after removing the root of theinitial min-heap above. A third min-heap can then be reconstructed after removing theroot of the second min-heap. Represent these second and third min-heaps with array (list)representation in the table form below. NO steps of construction required.
A d-heap is similar to a binary heap, except that nodes can have up to d children. Suppose d = 3, insert the following items into a max 3-heap: 10, 2, 6, 20, 9, 25, 12, 19. Show all the steps.
Design and implement a data structure Median − Heap to maintain a collection of numbers S that supports Build(S), Insert(x), Extract(), and Peek() operations, defined as follows: Build(S): Produces, in linear time, a data structure Median − Heap from an unordered input array S. For implementing Build(S), you can assume access to the procedure Find_Median(S), which finds the median of S in linear time. Insert(x): Insert element x into Median − Heap in O(log n) time. Peek(): Returns, in O(1) time, the value of the median of Median − Heap. Extract(): Remove and return, in O(log n) time, the value of the median element in Median − Heap.
A = ⟨4,33,6,90,33,32,31,91,90,89,50,33⟩ A is a min heap, extract TWO minimum values and then rearrange the array to satisfy the min-heap property pls show step by STEP HOW TO DO THIS PLS HELP!!!
Algorithm Demonstrate, step by step, how to construct a max-heap for the list of integers A[1..8] = (12, 8, 15, 5, 6, 14, 1, 10) (in the given order), following Example 4.14.
As we already studied different algorithms associated with binary heap. In place of binary heap,now consider an n-ary heap. Re-write following algorithms and their time complexities based on theproperties of n-ary heap. (i) Parent (i)(ii) index of kth child of node i(iii) indices of leaf nodes(iv) Max-Heapify (A, i) (v) Build-Max-Heap (A)
We can build a heap by top-down (iterated insertion), or by bottom-up. Given the numbers 10, 9, 12, 13, 14, 8 a. run the bottom-up algorithm to create a heap and show the resulting heap and ALLsteps b. run the top-down algorithm to create heap and show the resulting heap and ALLsteps