"In computer science, a heap is a specialized tree-based data structure that satisfies the heap property."
A specialized tree-based data structure that satisfies the heap property, that is, any parent node is always greater than or equal to its child nodes. Heaps are commonly used in priority queue algorithms.
Binary Heaps: A binary heap is a complete binary tree in which the keys of parent nodes are always greater than or equal to those of its children.
Heap Operations: Heap operations include insertion, deletion, creation of a heap, and heap sort.
Priority Queue: A priority queue is a data structure that stores items with priorities associated with them. It allows the highest priority item to be accessed at any given time.
Heap Sort: Heap sort is a sorting algorithm that uses a heap data structure to sort elements in ascending or descending order.
Heapsort Complexity Analysis: The time complexity of the heap sort algorithm is O(n log n).
Binary Heap Implementation: Binary heaps can be implemented using arrays or linked lists.
Min Heap and Max Heap: Min heap is a binary heap where the parent nodes have a smaller key than the children. A max heap is the opposite: parent nodes have a larger key than the children.
Heapify: The process of converting an array into a heap is called heapifying.
Heap Data Structure: A heap data structure is a binary tree where the value of each node is greater than or equal to the values of its children.
D-ary Heaps: D-ary heaps are similar to binary heaps, but each node has up to 'd' children instead of two.
Binary Heap: A binary heap is a complete binary tree where the value of each node is greater than or equal to its children (in a max-heap) or less than or equal to its children (in a min-heap). It is commonly used as a priority queue for efficient implementation of algorithms like Heap Sort and Dijkstra’s Shortest Path Algorithm.
Binomial Heap: A binomial heap is a set of binomial trees, where each binomial tree is a complete binary tree that satisfies the heap property. It is commonly used in graph algorithms like Prim’s Minimum Spanning Tree Algorithm and Dijkstra’s Shortest Path Algorithm.
Fibonacci Heap: A Fibonacci heap is a collection of trees where each tree is a minimum heap ordered according to the key value of each node. It is commonly used in graph algorithms like Dijkstra’s Shortest Path Algorithm and Prim’s Minimum Spanning Tree Algorithm.
Pairing Heap: A pairing heap uses a tree structure to store nodes, where each node has a key value that determines its priority. This heap is commonly used in graph algorithms like Prim’s Minimum Spanning Tree Algorithm and Dijkstra’s Shortest Path Algorithm. It is a simple and efficient data structure that provides constant time complexity for most operations.
"In a max heap, for any given node C, if P is a parent node of C, then the key (the value) of P is greater than or equal to the key of C. In a min heap, the key of P is less than or equal to the key of C."
"The node at the 'top' of the heap (with no parents) is called the root node."
"The heap is one maximally efficient implementation of an abstract data type called a priority queue."
"In a heap, the highest (or lowest) priority element is always stored at the root."
"A heap is not a sorted structure; it can be regarded as being partially ordered."
"A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspersed with removals of the root node."
"The heap data structure, specifically the binary heap, was introduced by J. W. J. Williams in 1964."
"When a heap is a complete binary tree, it has the smallest possible height—a heap with N nodes and a branches for each node always has loga N height."
"Note that, as shown in the graphic, there is no implied ordering between siblings or cousins."
"Heaps are also crucial in several efficient graph algorithms such as Dijkstra's algorithm."
"The maximum number of children each node can have depends on the type of heap." Quotes for study questions 1-12 are covered in the initial response. Continuing with additional study questions:
"The heap data structure, specifically the binary heap, was introduced by J. W. J. Williams in 1964, as a data structure for the heapsort sorting algorithm."
"When a heap is a complete binary tree, it has the smallest possible height—a heap with N nodes and a branches for each node always has loga N height."
"...there is no implied sequence for an in-order traversal (as there would be in, e.g., a binary search tree)."
"The heap relation mentioned above applies only between nodes and their parents, grandparents, etc."
"priority queues are often referred to as 'heaps', regardless of how they may be implemented."
"A common implementation of a heap is the binary heap, in which the tree is an almost complete binary tree."
"A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspersed with removals of the root node."
"The heap data structure, specifically the binary heap, was introduced by J. W. J. Williams in 1964."