Chapter 6: Priority Queues Priority Queues Binary Heaps Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Lydia Sinapova, Simpson College.
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Chapter 6: Priority Queues Priority Queues Binary Heaps Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Lydia Sinapova, Simpson College
2 Priority Queues The model Implementations Binary heaps Basic operations Animation Animation
3 The Model A priority queue is a queue where: Requests are inserted in the order of arrival The request with highest priority is processed first (deleted from the queue) The priority is indicated by a number, the lower the number - the higher the priority.
4 Implementations Linked list: - Insert at the beginning - O(1) - Find the minimum - O(N) Binary search tree (BST): - Insert O(logN) - Find minimum - O(logN)
5 Implementations Binary heap Better than BST because it does not support links - Insert O(logN) - Find minimum O(logN) Deleting the minimal element takes a constant time, however after that the heap structure has to be adjusted, and this requires O(logN) time.
6 Binary Heap Heap-Structure Property: Complete Binary Tree - Each node has two children, except for the last two levels. The nodes at the last level do not have children. New nodes are inserted at the last level from left to right. Heap-Order Property: Each node has a higher priority than its children
7 6 10 12 15171823 201934 Binary Heap Next node to be inserted - right child of the yellow node
8 Binary heap implementation with an array Root - A(1) Left Child of A(i) - A(2i) Right child of A(i)- A(2i+1) Parent of A(I) - A([i/2]). The smallest element is always at the root, the access time to the element with highest priority is constant O(1).
9 Example 6 10 12 15 17 18 23 20 19 34 Consider 17: position in the array - 5. parent 10 is at position [5/2] = 2 left child is at position 5*2 = 10 (this is 34) right child - position 2*5 + 1 = 11 (empty.)
10 Problems Problem 1: 2 8 10 16 17 18 23 20 21 30 Reconstruct the binary heap
11 Problems Problem 2: Give the array representation for 3 10 16 13121823 151930 14
12 Basic Operations Insert a node – Percolate Up Delete a node – Percolate Down Decrease key, Increase key, Remove key Build the heap
13 Percolate Up – Insert a Node A hole is created at the bottom of the tree, in the next available position. 6 10 12 15171823 201934
23 Percolate Down 16 131823 201930 10 12 15 Complexity of deletion: O(logN)
24 Other Heap Operations 1. DecreaseKey(p,d) increase the priority of element p in the heap with a positive value d. percolate up. 2. IncreaseKey(p,d) decrease the priority of element p in the heap with a positive value d. percolate down.
25 Other Heap Operations 3. Remove(p) a. Assigning the highest priority to p - percolate p up to the root. b. Deleting the element in the root and filling the hole by percolating down and trying to insert the last element in the queue. 4. BuildHeap input N elements place them into an empty heap through successive inserts. The worst case running time is O(NlogN).
26 Build Heap - O(N) Given an array of elements to be inserted in the heap, treat the array as a heap with order property violated, and then do operations to fix the order property.