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?

    <p>The element is swapped with its parent (D)</p> Signup and view all the answers

    What characterizes a complete binary tree?

    <p>All nodes are filled from left to right (B)</p> Signup and view all the answers

    What is the down-heap operation used for?

    <p>To remove the maximum or minimum element from the heap (B)</p> Signup and view all the answers

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

    <p>To ensure the heap property is maintained after insertion (B)</p> Signup and view all the answers

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

    <p>Sift-down (B)</p> Signup and view all the answers

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

    <p>To restore the heap condition after insertion. (C)</p> Signup and view all the answers

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

    <p>Complete binary tree (D)</p> Signup and view all the answers

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

    <p>Meld (B)</p> Signup and view all the answers

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

    <p>It returns and removes the minimum value. (B)</p> Signup and view all the answers

    How is a heap typically implemented?

    <p>Using an implicit data structure in an array (B)</p> Signup and view all the answers

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

    <p>Internal operations are executed to restore the heap property. (D)</p> Signup and view all the answers

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

    <p>Replace (A)</p> Signup and view all the answers

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

    <p>Updates a key to a new higher value. (D)</p> Signup and view all the answers

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

    <p>Shift-up (C)</p> 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)?

    <p>2n + 1 (C)</p> Signup and view all the answers

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

    <p>Removing the root and replacing it with the last element (A)</p> Signup and view all the answers

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

    <p>Cascade-down (A)</p> Signup and view all the answers

    Which of the following best describes a complete binary tree?

    <p>All levels are fully populated except possibly for the last level (B)</p> 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?

    <p>It must be swapped with its parent until the order is correct (D)</p> Signup and view all the answers

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

    <p>Replacement (C)</p> Signup and view all the answers

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

    <p>To sort an array in-place (B)</p> Signup and view all the answers

    Flashcards

    Heap

    A specialized tree-based data structure that satisfies the heap property: either a max-heap (largest element at the root) or a min-heap (smallest at the root).

    Binary Heap

    A complete binary tree-based implementation of a heap, that arranges the elements efficiently in an array.

    Heap Operations: Insert

    Adding a new element to the heap while maintaining the heap property (max or min).

    Heap Operations: Extract-Max/Min

    Removing and returning the maximum/minimum element from a max/min-heap respectively.

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    Heap Operations: Heapify

    Converting an array of elements into a valid heap.

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    Complete Binary Tree

    A binary tree where every level is completely filled, except possibly the last.

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    Heap Implementation

    Usually stored in an array; doesn't require pointers between elements.

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    Heap Property

    The property that ensures the largest or smallest element is at the root in max or min heaps respectively.

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    Binary Heap Insertion

    Adding a new element to a binary heap while maintaining the heap property (e.g., all parent nodes are greater than or equal to their children in a max-heap).

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    Heap Structure

    A complete binary tree stored in an array where the elements are arranged to satisfy the heap property.

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    Insert Algorithm (Step 1)

    Place the new element in the next available position at the bottom level of the heap.

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    Insert Algorithm (Step 2)

    Compare the newly added element with its parent; if the heap property is satisfied at this point, you're done.

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    Insert Algorithm (Step 3)

    If the heap property is violated, swap the element with its parent and repeat step 2.

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    Complexity of Insert (time)

    Insert operation on a binary heap takes logarithmic time (O(log n)),.

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    Down-heap / Extract-Max/Min

    The process of removing the root (maximum or minimum element) from a heap and restructuring it to maintain the heap property.

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    Complete Binary Heap Structure

    A heap structure where elements are arranged in a tree-like array, with each node having a maximum of two children. The root element is always the largest or smallest element (depending on the type of heap).

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    Heap Element Indexing

    In a complete binary heap, the parent of a node at position "n" (one-based indexing) is at position "≤ n/2". Children are at positions 2n and 2n + 1 (one-based), or 2n + 1 and 2n + 2 (zero-based).

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    Heap Balancing

    Maintaining the heap property after an insertion or deletion. This involves either shifting elements up (percolate up) or down (percolate down) in the array to restore the order.

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    Heap Insertion

    Adding a new element to the heap by placing it at the next available position and then repeatedly swapping it with its parent until the heap property is restored.

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    Heap Deletion (Max/Min)

    Removing the root element (max or min). Replacing it with the last element and then repeatedly swapping it with its larger or smaller children until the heap property is restored

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    Up-heap Operation

    Moving an element up the heap until it's in the correct order relative to its parent. Necessary after insertion to maintain heap ordering.

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    Heap Sort

    An in-place sorting algorithm that uses a heap structure to sort an array efficiently, typically using a max-heap or min-heap.

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    Heap Efficiency

    Heap operations (insertion, deletion) run in logarithmic time (O(log n)) due to the balanced structure and efficient rearranging.

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    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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    Related Documents

    Binary Heaps PDF

    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.

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