Priority Queues What is a Priority Queue Container

What is a Priority Queue? • Container of elements where each element has an

Priority Queue ADT Operations • • • size() is. Empty() insert. Item(k, e) –

Example • • insert. Item(5, A) insert. Item(9, C) insert. Item(3, B) insert. Item(7,

The Comparator Pattern • Suppose we want a priority queue of “Appointment” objects •

Implementation • Using an array? • Using a linked list? • Running time of

Sorting • Can you sort a list of numbers using a priority queue? •

Heaps • A heap is a priority queue implemented with a binary tree

Heaps • Heap-Order Property: In a heap T, for every node v other than

Heaps • Complete Binary Tree Property: A heap T with height h is a

Heap Implementation • Insertion algorithm new_node 5 root 4 9 7 15 11 56

Up-Heap Bubbling while new node is smaller than its parent swap parent and node

Heap Implementation • Deletion algorithm delete 4 root 4 9 5 7 15 11

Down-Heap Bubbling if right child is null if left child is smaller than current

Vector Implementation 0 1 2 3 4 5 6 7 • Children are positions

Heap Sort • Algorithm to use a heap to sort a list of numbers

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Priority Queues

What is a Priority Queue? • Container of elements where each element has an associated key A key is an attribute that can identify rank or weight of an element • • Examples – passenger, todo list

Priority Queue ADT Operations • • • size() is. Empty() insert. Item(k, e) – insert element e with key k min. Element() – return ref to min element min. Key() – return const ref to smallest key remove. Min() – remove and return element with smallest key

Example • • insert. Item(5, A) insert. Item(9, C) insert. Item(3, B) insert. Item(7, D) min. Element() min. Key() remove. Min() • • size() min. Element() remove. Min() is. Empty()

The Comparator Pattern • Suppose we want a priority queue of “Appointment” objects • How do we determine that one appointment is before another? • Need to compare Appointment objects – operator overloading versus standard function

Implementation • Using an array? • Using a linked list? • Running time of insert. Item/remove. Min?

Sorting • Can you sort a list of numbers using a priority queue? • Algorithm?

Heaps • A heap is a priority queue implemented with a binary tree

Heaps • Heap-Order Property: In a heap T, for every node v other than the root, the key stored at v is greater than or equal to the key stored at v’s parent

Heaps • Complete Binary Tree Property: A heap T with height h is a complete binary tree, that is, levels 0, 1, 2, …, h-1 of T have the maximum number of nodes possible and all the internal nodes are to the left of the external nodes in level h-1. • What does this give us?

Example Heap root 4 9 7 15 11 56 10

Heap Implementation • Insertion algorithm new_node 5 root 4 9 7 15 11 56 10

Up-Heap Bubbling while new node is smaller than its parent swap parent and node • Running time?

Heap Implementation • Deletion algorithm delete 4 root 4 9 5 7 15 11 56 10

Down-Heap Bubbling if right child is null if left child is smaller than current swap left and current if both children not null swap current with smallest child

Vector Implementation 0 1 2 3 4 5 6 7 • Children are positions index*2 and index*2+1 • Implementation of insert and remove?

Heap Sort • Algorithm to use a heap to sort a list of numbers • Running time?