1. Priority Queue and Binary Heap Neil Tang 02/12/2008
CS223 Advanced Data Structures and Algorithms

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

3. CS223 Advanced Data Structures and Algorithms
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.
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
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
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
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.
CS223 Advanced Data Structures and Algorithms

6. CS223 Advanced Data Structures and Algorithms
Binary Heap
A complete binary tree of height h has between 2h and 2h+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.
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Insert 14
CS223 Advanced Data Structures and Algorithms

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

9. CS223 Advanced Data Structures and Algorithms
deleteMin
CS223 Advanced Data Structures and Algorithms

10. deleteMin (Percolate Down)
Time complexity: O(logN)
CS223 Advanced Data Structures and Algorithms

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

12. CS223 Advanced Data Structures and Algorithms
buildHeap
CS223 Advanced Data Structures and Algorithms

13. CS223 Advanced Data Structures and Algorithms
buildHeap
CS223 Advanced Data Structures and Algorithms

14. CS223 Advanced Data Structures and Algorithms
buildHeap
CS223 Advanced Data Structures and Algorithms

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

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

17. CS223 Advanced Data Structures and Algorithms
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, deleteMin and insert the new one.
Time complexity: O(k+(N-k)logk) = O(NlogN).
CS223 Advanced Data Structures and Algorithms