1. CS223 Advanced Data Structures and Algorithms 1 Priority Queue and Binary Heap Neil Tang 02/09/2010

2. CS223 Advanced Data Structures and Algorithms 2 Class Overview  Priority queue  Binary heap  Heap operations: insert, deleteMin, de/increaseKey, delete, buildHeap  Application

3. CS223 Advanced Data Structures and Algorithms 3 Priority Queue A priority queue is a queue in which each element has a priority and elements with higher priorities are supposed to be removed before the elements with lower priorities.

4. CS223 Advanced Data Structures and Algorithms 4 Possible Solutions  Linked list: Insert at the front (O(1)) and traverse the list to delete (O(N)).  Linked list: Keep it always sorted. traverse the list to insert (O(N)) and delete the first element (O(1)).  Binary search tree

5. CS223 Advanced Data Structures and Algorithms 5 Binary Heap A binary heap is a binary tree that is completely filled, with possible exception of the bottom level and in which for every node X, the key in the parent of X is smaller than (or equal to) the key in X.

6. CS223 Advanced Data Structures and Algorithms 6 Binary Heap  A complete binary tree of height h has between 2 h and 2 h+1 -1 nodes. So h =  logN .  For any element in array position i, its left child in position 2i and the right child is in position (2i+1), and the parent is in  i/2 .

7. CS223 Advanced Data Structures and Algorithms 7 Insert 14

8. CS223 Advanced Data Structures and Algorithms 8 Insert (Percolate Up) Time complexity: O(logN)

9. CS223 Advanced Data Structures and Algorithms 9 deleteMin

10. 10 deleteMin (Percolate Down) Time complexity: O(logN)

11. CS223 Advanced Data Structures and Algorithms 11 Other Operations  decreaseKey(p,  )  increaseKey(p,  )  delete(p)?  delete(p)=decreaseKey(p,  )+deleteMin()

12. CS223 Advanced Data Structures and Algorithms 12 buildHeap

13. CS223 Advanced Data Structures and Algorithms 13 buildHeap

14. CS223 Advanced Data Structures and Algorithms 14 buildHeap

15. CS223 Advanced Data Structures and Algorithms 15 buildHeap  Theorem: For the perfect binary tree of height h with (2 h+1 - 1) nodes the sum of the heights of the nodes is (2 h+1 -1- (h+1)).  Time complexity: 2 h+1 -1-(h+1) = O(N).

16. CS223 Advanced Data Structures and Algorithms 16 Applications  Problem: find the kth smallest element.  Algorithm: buildHeap, then deleteMin k times.  Time complexity: O(N+klogN) = O(NlogN).

17. CS223 Advanced Data Structures and Algorithms 17 Applications  Problem: find the kth largest element.  Algorithm: buildHeap with the first k elements, check the rest one by one. In each step, if the new element is larger than the element in the root node, deleteMin and insert the new one.  Time complexity: O(k+(N-k)logk) = O(NlogN).