9. Heap : Priority Queue

Transcription

1 9. Heap : Priority Queue

2 Where We Are? Array Linked list Stack Queue Tree Binary Tree Heap Binary Search Tree

3 Priority Queue Queue Queue operation is based on the order of arrivals of elements FIFO(First-In First-Out) or FCFS(First-Come First-Serve) Queue Priority Queue Insert Queue operation is based on specific priority values, rear front Delete regardless of the order of arrivals. For instance, priority includes grade, urgency, importance, etc. Applications: Job scheduling (OS), Emergency treatment in Hospital, etc.

4 Priority=1 Priority=5 Priority=3 Priority=6 Priority Queue Priority Queue (Cont.) Each element of Queue has its own priority value (called Key) Priority is an integer in general Max Priority Queue Priority Queue The element of the highest priority (i.e., key) is deleted first. Min Priority Queue Insert rear rear front Delete The element of the lowest priority (i.e., key) is deleted first Main issue of Priority Queue comes under how to search for the maximum or minimum value

5 Max Priority Queue: Implementation How to implement Max Priority Queue? Unordered Array Ordered Array Unordered Linked List Ordered Linked List Binary Search Tree Max Heap (Min Heap)

6 Use of Unordered Array Keys are not Sorted Delete operation is very inefficient; O(n) Search from the first to the last! Sequential search is used: O(n) Insert operation is very efficient; O(1) Insert a new Key into the last of array No (data) movement takes place [0] [1] [2] [3] [4] [5] [6] list i=0 [7] i=1 i=2 i=3 Insert directly By sequential search 2

7 Use of Ordered Array As to Priorities, Keys are sorted in ascending order Delete is very efficient; O(1) 2 Insert by binary search The highest key is always positioned at the last of array [0] [1] [2] [3] [4] [5] [6] list [7] Insert is very inefficient; O(n) Insert a new key into its appropriate position Binary search is used; O(log 2 n) After insertion, data should be moved; O(n) left list(mid) =2 right list(mid) =6 right Delete directly

8 Use of Linked Lists Unordered Linked List Deletion is very inefficient; O(n) Sequential search is used Insertion is very efficient; O(1) Always, insert a new key at the first position of linked list Ordered Linked List Deletion is very efficient; O(1) Insertion is very inefficient; O(n) Not possible to use Binary search

9 h : height Use of BST Keys are Stored in Binary Search Tree Delete operation is inefficient; To find the highest key, it visits continuously the right child from the root Time complexity is the height of BST which depends on the shape of BST Worst: O(n), Average: O(log 2 n) Insert operation is also inefficient; New key is always inserted into the leaf node Insert by Search in BST Delete by Search in BST Also, time complexity is the height of BST which depends on the shape of BST Worst: O(n), Average: O(log 2 n)

10 Worst Case Complexity Performance Comparison Data Structures Insertion Deletion Unsorted Array O(1) O(n) Unsorted Linked list O(1) O(n) Sorted Array O(n) O(1) Sorted Linked list O(n) O(1) Binary Search Tree O(n) O(n) Max (Min) Heap O(log 2 n) O(log 2 n) Q) When n = 1,000, what will become of time complexity of each method?

11 Why Heap? Heap is efficient for searching the largest (or smallest) value. It compromises on sorted and unsorted (data) structures Sorted structures (array or linked-list) Very efficient for finding largest (smallest) element (i.e., Deletion) Inefficient in insertion Unsorted structures (array or linked-list) Inefficient for finding largest (smallest) element (i.e., Deletion) Very efficient in insertion Thus, Heap is efficient for both insertion and deletion

12 What Max/Min Heap? Max Heap Heap is a Complete Binary Tree Key of each node is no smaller than its children s keys. K All K All K left subtree right subtree cf) Min Heap Key of each node is no larger than its children s keys.

13 Full Binary Tree Complete Binary Tree Complete Binary Tree (a) height = 1 (b) height = 2 (c) height = 3 A B C A B D E C F G (a) (b) (c) A B A B D E C F A B D E C F G H I A

14 Examples : Max Heap (a) [1] 14 [2] 12 7 [3] [4] [5] [6] (b) [2] [4] 5 [1] [3] (c) [2] 25 [1] 30 Root of a max heap always has the largest value

15 Examples : Not a Max Heap (a) [2] [1] [3] Problematic node! [4] [5] [6] (b) [2] [1] [5] 5 [3] [4] (c) [2] [1] [5] [6] [3]

16 Examples : Min Heap (a) [1] 2 [2] 7 4 [3] [4] [5] [6] (b) [2] [4] 50 [1] [3] (c) [2] 21 [1] 11 Root of a min heap always has the smallest value

17 Heap vs. BST Comparison: Max heap vs. BST K K All K All K All<K All>K left subtree right subtree left subtree right subtree Complete Binary Tree Max Heap Binary Tree Binary Search Tree

18 Heap Implementation: Use of Array Why Use Array for Implementing Heap? Max Heap Not waste (memory) space (since complete binary tree) Easy to find the positions of parent and its children Not need to use complex linked lists Finding a node location/position Left-Child(i) = 2i [2] [1] 14 [3] 12 7 [4] [5] [6] Right-Child(i) = 2i Parent(i) = i/ 2 (if i = 1, i is the root)

19 Max Heap Operation Insertion New key is inserted into Max heap Deletion After finding the largest key, it is deleted from Max heap Note: Two conditions of Max heap must be satisfied. 1) Complete Binary Tree 2) Key value of each node should be no smaller than values of its children

20 Insertion Inserting into Max heap (1) Insert a new key into Max heap To preserve 1 st condition, the new node should be added at the first empty position at the last leaf level 2 nd condition could be broken (2) Repair the structure so that 2 nd condition is satisfied Reheap-Up Not a heap! Heap!

21 Insertion a 1 b 2 c 3 Last leaf node d e f g h i j Procedures Find an empty location; This is a node that is right next of the last leaf node. If it violates any condition of (max or min) heap, repair it (Reheap).

22 Insertion Insert (4) Last leaf node A very simple case; Just insert it into the right next of the last leaf node. If It does not violate any heap condition, its insertion is done!

23 Insertion Insert (20) Last leaf node At first, insert 20 into the next of the last leaf node 2. Compare 20 with its parent, i.e., 7. Since 20>7, swap 20 and 7, i.e., move 20 up! 3. Compare 20 with its parent, i.e., 8. Since 20>8, swap 20 and 8, i.e., move 20 up! 4. Do this process until reaching the root or its parent is greater than 20.

24 Insertion Insert (20) Insertion Result!

25 Deletion Deleting from Max heap (1) Deletion occurs at the root (because the largest key is always at the root) 2 nd condition could be broken To preserve 1 st condition, the last node should move up to the root (2) Repair the structure so that 2 nd condition is satisfied Reheap-Down Not a heap! Heap!

26 Deletion Delete Last leaf node Delete the root since it always contains the largest key Put the last leaf node (i.e., 8) into the root Compare 8 with its child with larger value, i.e., 15. Since 8<15, swap 8 and 15, i.e., move 8 down! Do this process until reaching the leaf node or its child is smaller than 8.

27 Deletion Delete Deletion Result!

28 Building a Heap Building a Max Heap from Array Array Complexity=? Max Heap Heap To be Inserted

29 Time Complexity: Insertion/Deletion Insertion/deletion time is bounded by the Height of a Heap. What is a Height of Heap? Height of any complete binary tree is ceil(log 2 (n+1)) since the tree is Balanced Note: A Balanced tree is a binary tree whose height difference between left and right subtrees is at most 1 for all nodes. Thus, insertion & deletion time becomes O(log 2 n) ceil(log3)=2 h=ceil(log10)=4 h=ceil(log7)=3

30 Sorting! Finding k th Largest Element How to find k th largest element from an unsorted array? Sort and Select element at location k O(n 2 ) for sorting + O(1) for selection Build Heap and delete k times O(n log 2 n) for building heap + O(k log 2 n) for deletions By deletions 4 th largest [0] [1] [2] [3] [4] [5] [6] [0] [1] [2] [3] [4] [5] [6] th largest Heap Heap

31 Sorting Elements using Heap Is it Possible to Sort Elements (in ascending order) using Max Heap? Yes! because a deletion takes out the highest key from the Heap 1. At first, a deletion is performed and the delete key is stored (Generally, it is put into the end of array) 2. Second, the Heap is repaired to satisfying two (Max) Heap conditions 3. Iterate the two procedures until the Heap is empty. 1 st largest 2 nd largest Deletion Heap Heap

32 Sorting Elements using Heap 3 rd largest 4 th largest Deletion Deletion Heap Heap Deletion 5 th largest It is called Heap Sort Complexity=? Heap Sorted