Full Binary Tree A full binary tree All

Full Binary Tree • A full binary tree: – All the leaves are at the bottom level – All nodes which are not at the bottom level have two children. – A full binary tree of height h has 2 h leaves and 2 h-1 internal nodes. 1 3 2 4 5 6 7 A full binary tree of height 2 Trees This is not a full binary tree. 1

Properties of Proper Binary Trees • Notation n number of nodes e number of external nodes i number of internal nodes h height 1 2 Properties for proper binary tree: n e = i + 1 n n = 2 e - 1 n h i n e 2 h 3 n h log 2 e 7 6 14 Trees 15 No need to remember. 2

Part-D 1 Priority Queues 3

Sequence-based Priority Queue • Implementation with an unsorted list 4 5 2 3 1 • Performance: – insert takes O(1) time since we can insert the item at the beginning or end of the sequence – remove. Min and min take O(n) time since we have to traverse the entire sequence to find the smallest key Priority Queues • Implementation with a sorted list 1 2 3 4 5 • Performance: – insert takes O(n) time since we have to find the place where to insert the item – remove. Min and min take O(1) time, since the smallest key is at the beginning 4

Heaps • A heap is a binary tree storing • The last node of a heap is keys at its nodes and the rightmost node of satisfying the following depth h properties: • Root has the smallest key – Heap-Order: for every node v other than the root, key(v) key(parent(v)) – Complete Binary Tree: let h be the height of the heap • for i = 0, … , h - 1, there are 2 i nodes of depth i • at depth h - 1, the internal nodes are to the left of the external nodes A full binary without the last few nodes at the bottom on the right. Priority Queues 2 5 9 6 7 last node 5

Complete Binary Trees Once the number of nodes in the complete binary tree is fixed, the tree is fixed. For example, a complete binary tree of 15 node is shown in the slide. A CBT of 14 nodes is the one without node 8. 1 2 3 4 5 6 7 8 A CBT of 13 node is the one without nodes 7 and 8. A binary tree of height 3 has the first p leaves at the bottom for p=1, 2, …, 8=23. Priority Queues 6

Height of a Heap • Theorem: A heap storing n keys has height O(log n) Proof: (we apply the complete binary tree property) – Let h be the height of a heap storing n keys – Since there are 2 i keys at depth i = 0, … , h - 1 and at least one key at depth h, we have n 1 + 2 + 4 + … + 2 h-1 + 1=2 h. – Thus, n 2 h , i. e. , h log n depth keys 0 1 1 2 h-1 2 h-1 h 1 Priority Queues 7

Heaps and Priority Queues • • We can use a heap to implement a priority queue We store a (key, element) item at each internal node We keep track of the position of the last node For simplicity, we show only the keys in the pictures (2, Sue) (5, Pat) (6, Mark) (9, Jeff) (7, Anna) Priority Queues 8

Insertion into a Heap • Method insert. Item of the priority queue ADT corresponds to the insertion of a key k to the heap • The insertion algorithm consists of three steps – Create the node z (the new last node). Store k at z – Put z as the last node in the complete binary tree. – Restore the heap-order property, i. e. , upheap (discussed next) Priority Queues 2 5 9 6 z 7 insertion node 2 5 9 6 7 z 1 9

Upheap • After the insertion of a new key k, the heap-order property may be violated • Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node • Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k • Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 9 1 7 z 6 Priority Queues 5 9 2 7 z 6 10

An example of upheap 1 2 4 9 5 10 2 3 11 6 14 15 7 16 17 2 2 3 7 Different entries may have the same key. Thus, a key may appear more than once. Priority Queues 11

Removal from a Heap • Method remove. Min of the priority queue ADT corresponds to the removal of the root key from the heap • The removal algorithm consists of three steps – Replace the root key with the key of the last node w – Remove w – Restore the heap-order property. e. , downheap(discussed next) Priority Queues 2 5 9 6 7 w last node 7 5 w 6 9 new last node 12

Downheap • After replacing the root key with the key k of the last node, the heaporder property may be violated • Algorithm downheap restores the heap-order property by swapping key k along a downward path (always use the child with smaller key) from the root • Downheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k • Since a heap has height O(log n), downheap runs in O(log n) time 7 5 5 6 9 7 6 9 Priority Queues 13

An example of downheap 2 4 17 2 3 9 17 4 17 9 17 5 10 11 6 14 15 7 16 Priority Queues 14

Priority Queue ADT using a heap • A priority queue stores a collection of entries • Each entry is a pair (key, value) • Main methods of the Priority Queue ADT – insert(k, x) inserts an entry with key k and value x O(log n) – remove. Min() removes and returns the entry with smallest key O(log n) Priority Queues • Additional methods – min() returns, but does not remove, an entry with smallest key O(1) – size(), is. Empty() O(1) Running time of Size(): when constructing the heap, we keep the size in a variable. When inserting or removing a node, we update the value of the variable in O(1) time. is. Empty(): takes O(1) time using size(). (if size==0 then …) 15

Vector-based Heap Implementation • We can represent a heap with n keys by means of an array of length n + 1 • For the node at rank i 2 – the left child is at rank 2 i – the right child is at rank 2 i + 1 • Links between nodes are not explicitly stored • The cell of at rank 0 is not used • Operation insert corresponds to inserting at rank n + 1 • Operation remove. Min corresponds to removing at rank n • Yields in-place heap-sort • The parent of node at rank i is i/2 Priority Queues 5 6 9 0 7 2 5 6 9 7 1 2 3 4 5 16