1. Week 10: Heap and Priority queue

2. Any feature here?

4. Heap No particular relationship among nodes on any given level, even among the siblings When a heap is a complete binary tree, it has a smallest possible height—a heap with N nodes always has O(log N) height. A heap is a useful data structure when you need to remove the object with the highest (or lowest) priority.heap

5. Heaps For a maximum heap, the value of a parent is greater than or equal to the value of each of its children. For a minimum heap, the value of the parent is less than or equal to the value of each of its children. We assume that a heap is a maximum heap.

6. Vectors and complete binary trees In practice, heaps are usually implemented as arrays A complete binary tree of depth d contains all possible nodes through level d - 1 and that nodes at level d occupy the leftmost positions in the tree.

7. How to figure out parent-child from indices?

8. Heap = Array-based trees With some simple rules, we can view a direct-access container, such as an array or vector, as a binary tree. These trees are referred to as array-based trees and form the basis for a new container type, called a heap Heaps are designed to provide efficient access to the maximum element or the minimum element in the collection. In this way, a heap acts like a priority queue. The highest (or lowest) priority element is always stored at the root, hence the name heap.

9. Heapify Heapify() maintain the heap property Given: a node i in the heap with children l and r (two subtrees rooted at l and r ) Problem: The subtree rooted at i violate the heap property Action: let the value of the parent node recursively “float down” so subtree rooted a i satisfy the heap property

20. Analyzing Heapify() When Heapify is called, the running time depends on how far an element might move down in tree. In other words it depends on the height h (depth d) of node. In the worst case the element might go down all the way to the leaf level, which is O(logN).

21. Heapify all subtrees in a tree Wiki – Building a heap: Wiki – Building a heap

22. Insertion A heap is an array-based tree with a vector as the underlying storage structure. What makes the tree a heap is the ordering of the elements. 1.push_back -- O(1) 2.Re-heapify -- O(logN) 1.push_back -- O(1) 2.Re-heapify -- O(logN)

23. Deletion Deletion from a heap is restricted to the root only. Hence, the operation specifically removes the largest element. 1.Exchange -- O(1) 2.pop_back -- O(1) 3.Re-heapify -- O(logN) 1.Exchange -- O(1) 2.pop_back -- O(1) 3.Re-heapify -- O(logN)

24. Time complexity - Heap Find max/min: O(1) Delete max/min: O(logN) Insert max/min: O(logN) A heap is a useful data structure when you need to remove the object with the highest (or lowest) priority.

25. Still remember the Priority Queue? We discussed in week 5 on “Queues”

26. Main Index Contents 26 Priority Queue A Special form of queue from which items are removed according to their designated priority and not the order in which they entered. Items entered the queue in sequential order but will be removed in the order #2, #1, #4, #3.

27. Heaps and priority queues Heap delete() operation removes the optimal element from the heap, much like Priority-queue pop() operation remove the highest priority element from priority queue. Heap insertion() simply adds an element, much like the Priority-queue push() operation. Both the heap insertion and deletion operations update the underlying tree storage structure in such a way that it remains a heap; that is, they maintain the integrity of the storage structure.

28. Heaps and priority queues The standard container adaptor priority_queue priority_queue calls make_heap, push_heap and pop_heap automatically to maintain heap properties for a container.make_heappush_heappop_heap

29. 29 Heap Priority queue The storage structure should be selected so that the container operations can be implemented efficiently

30. Reading Chapter 6

31. Reminder: HW4 Due Wednesday 11/5/2014 Submission before Thursday 10/30/2014 will receive 30 extra points