Heaps in Data Structures

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Questions and Answers

What must be tracked to insert efficiently into a binary heap?

The total number of elements in the array

The maximum size of the heap

The location of the next open space in the tree (correct)

The value of the root element only

What is the first action taken when inserting an element into a max-heap?

Add the element at the bottom level of the heap (correct)

Remove the current maximum from the heap

Swap the new element with the minimum element

Place the element in the correct position immediately

What is the complexity of the insertion operation in a binary heap?

O(log n) (correct)

O(1)

O(n)

O(n^2)

What happens when the heap property is violated after adding an element?

The element is swapped with its parent Signup and view all the answers

What characterizes a complete binary tree?

All nodes are filled from left to right Signup and view all the answers

What is the down-heap operation used for?

To remove the maximum or minimum element from the heap Signup and view all the answers

What is the main purpose of the bubble-up operation in a binary heap?

To ensure the heap property is maintained after insertion Signup and view all the answers

Which of the following terms is synonymous with the down-heap operation?

Sift-down Signup and view all the answers

What is the primary purpose of the shift-up operation in a heap?

To restore the heap condition after insertion. Signup and view all the answers

In which type of binary tree are all levels completely filled except possibly the last?

Complete binary tree Signup and view all the answers

Which operation combines two heaps into a new heap while preserving the original heaps?

Meld Signup and view all the answers

What is the result of the extract-min operation in a min-heap?

It returns and removes the minimum value. Signup and view all the answers

How is a heap typically implemented?

Using an implicit data structure in an array Signup and view all the answers

What occurs when an element is inserted into a heap and the heap property is violated?

Internal operations are executed to restore the heap property. Signup and view all the answers

Which heap operation is more efficient than performing pop followed by push?

Replace Signup and view all the answers

What does the increase-key operation do in a max-heap?

Updates a key to a new higher value. Signup and view all the answers

What is the primary operation performed to restore the heap property after an insertion?

Shift-up Signup and view all the answers

In a complete binary tree represented in an array, which index contains the left child of an element at index n (using zero-based indexing)?

2n + 1 Signup and view all the answers

What strategy is employed when deleting an element from a heap?

Removing the root and replacing it with the last element Signup and view all the answers

What is NOT a typical type of operation involved in balancing a heap?

Cascade-down Signup and view all the answers

Which of the following best describes a complete binary tree?

All levels are fully populated except possibly for the last level Signup and view all the answers

When inserting an element into a heap, what must happen if the added element is out of order with respect to its parent?

It must be swapped with its parent until the order is correct Signup and view all the answers

Which operation involves deleting the root and placing a new element at the root in a heap?

Replacement Signup and view all the answers

What is the purpose of heapsort when used with a heap structure?

To sort an array in-place Signup and view all the answers

Study Notes

Heaps

Heaps are specialized tree-based data structures that follow the heap property.
If node A is a parent of node B, then the key of node A is ordered with respect to the key of node B, maintaining the same ordering across the heap.
Invented by J.J. Williams in 1964.
Heaps can be either max-heaps or min-heaps.

Max Heap

In a max-heap, parent nodes always have keys greater than or equal to their children's keys, and the highest key is in the root node.

Min Heap

In a min-heap, parent nodes always have keys less than or equal to their children's keys, and the lowest key is in the root node.

Example

For a min heap, the first element is the smallest.
Each parent node's value is less than or equal to its children's values.

Binary Tree

To efficiently use heaps, represent them using a binary tree.
Maintaining a balanced binary tree is crucial for logarithmic performance. This means the left and right subtrees of the root node generally have roughly the same number of nodes.
A complete binary tree is a specific type used in heap implementation. Each level in the tree has all of its nodes, except possibly the last level, which fills from left to right.

Heap Order Property

Heaps store items according to a heap order property.
In a heap, for any given node x with a parent node p, the key in p is smaller than or equal to the key in x.

Implementation as Arrays

Heaps are often implemented as arrays, which are very compact.
Elements in the array represent the nodes of the complete binary tree in a level-order traversal.
There are no empty cells, and no pointers are needed.
The array storage of a binary tree relies on how children and parent are represented in indexes in the array.

Heaps Implementation Details

The root node is often stored at index 1, however, starting at index 0 is also used.
The children of a node at index i are at indices 2i and 2i + 1 in a one-based array; or 2i + 1 and 2i + 2 in a zero-based array.
The parent of a node at index i is at index i / 2.

Common Operations

Find-max/min: Returns the maximum/minimum item in the heap.
Insert: Adds a new element to the heap.
Extract-max/min: Removes and returns the maximum/minimum item.
Delete-max/min: Removes the maximum/minimum, keeping heap properties intact
Increase/decrease key: Updates a key value within the heap.

Create Operations

Create-heap: Creates an empty heap.
Heapify: Builds a heap from an array of elements.
Merge/Union: Joins two heaps into a single valid heap, preserving the original heaps.
Meld: Joins two heaps into a new valid heap, discarding the original heaps.

Common Internal Functions

Shift-up: Used to maintain the heap condition after insertion. The node moving up the tree.
Shift-down: Used after deletion/replacement to adjust heap properties. The node moving down the tree.

Heap Complexity (In terms of Big O notation)

Time complexity of finding max/min is usually O(1)
Time complexity of inserting a node is usually O(log n)
Time complexity of extract-max/min is usually O(log n)
Time complexity of delete-max/min is usually O(log n)
Time complexity of increase/decrease key is usually O(log n)

Uses

Heaps are crucial in efficient graph algorithms (e.g., Dijkstra's algorithm) and sorting (e.g., heapsort).
They are good for priority queues.

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Description

This quiz covers the fundamentals of heaps, including max-heaps and min-heaps. You'll learn about their properties, examples, and how they are represented using binary trees. Test your understanding of how heaps function within data structures.