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Questions and Answers
The time complexity to build a heap is O(n) for the ______ operation.
build
After performing n delete_min operations, the time complexity is O(n log n) for ______ operations.
delete
A full binary tree with height h has roughly ______ nodes.
2^(h+1) - 1
The maximum element in a heap is typically found at the ______ of the structure.
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When elements are deleted from the heap, the array representation needs to be ______ to maintain the heap property.
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A complete binary tree is completely filled except for the bottom level, which is filled from left to right, and the smallest element is at the ______.
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For any element in array[i], the left child is in position ______.
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To delete the minimum element from a heap, the strategy used is called ______.
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The process of inserting elements into a heap may involve the strategy of ______.
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The height of a complete binary tree with n nodes is approximately ______.
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Study Notes
Binary Heap Overview
- A binary heap is a complete binary tree, filled from left to right, where the smallest element is located at the root.
- For each node X, the key in the parent node is smaller than the key in X, maintaining the min-heap property.
Heap Operations
- Insert Operation: Has a time complexity of O(log n) and employs a strategy called percolate up, which maintains heap properties after adding a new element.
- Delete Minimum Operation: Also runs in O(log n) time and uses percolate down strategy to restore the heap properties after removing the root element.
Building a Heap
- To build a heap from n inputs, the process of percolating down is applied from the n/2 node to the root node.
- Time complexity for building a heap is O(n), utilizing the relationship between the heights of nodes and node positions.
Heap Properties
- The height of a complete binary tree is approximately log(n).
- Total number of nodes in a complete binary tree is calculated as 2^(h + 1) - 1, where h is the height.
Running Time Analysis
- The running time for building a heap utilizes a sum of heights formula, resulting in an overall complexity of O(n), contrary to the naive approach of O(n log n).
- Performing n delete_min operations after building a heap incurs a time complexity of O(n log n).
Example Heap Structure
- An example arrangement of elements might include:
- Before deletion of max: 97, 53, 59, 26, 41, 58, 31
- Maximum values in subsequent deletions show how the structure maintains order post-deletions.
Illustrative Node Positions
- Elements in the array are arranged such that:
- Left child is at position 2*i
- Right child is at position 2*i + 1
- Parent node position can be found at the integer division of i by 2.
HeapNode Class
- The HeapNode class is initialized with max_heap_size and current size of the heap set to zero, ready to hold elements.
Deletion Process in Action
- Example of consecutive delete operations shows how the heap restructures itself while maintaining the smallest element at the root after each deletion.
Importance of Heap Structures
- Binary heaps are widely used in priority queues, scheduling algorithms, and efficient sorting algorithms like heapsort due to their logarithmic performance on insert and delete operations.
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Description
This quiz explores the properties and operations of binary heaps, focusing on insertion and deletion complexities. Understand the organizational structure and array representation of binary trees. Test your knowledge on minimum elements and key relationships within a binary heap.
