1. Binary Heaps What is a Binary Heap? Array representation of a Binary Heap MinHeap implementation Operations on MinHeap: Insert Delete Converting an array into a binary heap Heap Applications: Heap Sort Heap as a priority queue

2. What is a Binary Heap? A binary heap is a complete binary tree with one or both of the following heap order properties: MinHeap property: Each node must have a key that is less or equal to the key of each of its children. MaxHeap property: Each node must have a key that is greater or equal to the key of each of its children. A binary heap satisfying the MinHeap property is called a MinHeap. A binary heap satisfying the MaxHeap property is called a MaxHeap. A binary heap with all keys equal is both a MinHeap and a MaxHeap. Recall: A complete binary tree may have missing nodes only on the right side of the lowest level. All levels except the bottom one must be fully populated with nodes All missing nodes, if any, must be on the right side of the lowest level

3. MinHeap and non-MinHeap examples 21 24 6526 32 3119 16 68 13 A MinHeap Violates MinHeap property 21>6 21 6 6526 32 3119 16 68 13 Not a Heap 21 24 6526 32 31 19 16 68 13 Violates heap structural property Not a Heap 21 24 6526 32 31 19 16 13 Violates heap structural property Not a Heap

4. MaxHeap and non-MaxHeap examples 65 24 1520 31 3223 46 25 68 A MaxHeap Violates MaxHeap property 65 < 67 Violates heap structural property Not a Heap 65 67 1520 31 3223 46 25 68 Not a Heap 50 24 1520 25 3119 40 38 70 21 19 25 15 18 10 16 30 Not a Heap

5. Array Representation of a Binary Heap A heap is a dynamic data structure that is represented and manipulated more efficiently using an array. Since a heap is a complete binary tree, its node values can be stored in an array, without any gaps, in a breadth-first order, where: Value(node i+1 ) array[ i ], for i > 0 21 24 6526 32 3119 16 68 13 32266568193124162113 9876543210 The root is array[0] The parent of array[i] is array[(i – 1)/2], where i > 0 The left child, if any, of array[i] is array[2i+1]. The right child, if any, of array[i] is array[2i+2].

6. Array Representation of a Binary Heap (contd.) We shall use an implementation in which the heap elements are stored in an array starting at index 1. Value(node i )array[i], for i > 1 21 24 6526 32 3119 16 68 13 32266568193124162113 9876543210 10 The root is array[1]. The parent of array[i] is array[i/2], where i > 1 The left child, if any, of array[i] is array[2i]. The right child, if any, of array[i] is array[2i+1].

7. MinHeap Implementation The class hierarchy for BinaryHeap is: MyComparable Container PriorityQueue AbstractObject AbstractContainer BinaryHeap The PriorityQueue interface is: 1 public interface PriorityQueue extends Container 2 { 3 public abstract void enqueue(MyComparable comparable) ; 4 public abstract MyComparable findMin() ; 5 public abstract MyComparable dequeueMin() ; 6 }

8. MinHeap Implementation (contd.) 1 public class BinaryHeap extends AbstractContainer implements PriorityQueue 2 { 3 protected MyComparable array[] ; 4 // count is inherited from AbstractContainer 5 public BinaryHeap(int size) 6 { 7 array = new MyComparable[size + 1] ; // array[0] is not used 8 } 9 10 public BinaryHeap(MyComparable[ ] comparable) 11 { 12 this(comparable.length) ; 13 14 for(int i = 0 ; i < comparable.length ; i++) 15 array[i + 1] = comparable[i] ; 16 count = comparable.length ; 17 buildHeapBottomUp() ; 18 } 19 //... 20 }

9. MinHeap Insertion The pseudo code algorithm for inserting in a MinHeap is: 1 minHeapInsert(e1) 2 { 3 if(Heap is full) throw an exception ; 4 insert e1 at the end of the heap ; 5 while(e1 is not in the root node and e1 < parent(e1)) 6 swap(e1, parent(e1)) ; 7} The process of swapping an element with its parent, in order to restore the heap order property after an insertion is called percolate up, sift up, or reheapification upward. Thus, the steps for insertion are: 1.Insert at the end of the heap. 2.As long as the heap order property is violated, percolate up.

10. Percolate-Up Via a Hole Instead of repeated swappings, it is more efficient to use a hole (i.e., an empty node) to percolate up: To insert an element e, insert a hole at the end of the heap. While the heap order property is violated, percolate the hole up. Insert e in the hole. p hole e e p p e p < e p > e

11. MinHeap Insertion Example 21 24 6526 32 3119 16 68 13 Insert 18 21 24 6526 32 31 19 16 68 13 18 Percolate up 21 24 6526 32 18 19 16 68 13 31 Percolate up 18 24 6526 32 2119 16 68 13 31

12. MinHeap Insertion implementation 1public void enqueue(MyComparable comparable) 2{ 3 if(isFull()) throw new ContainerFullException() ; 4 // percolate up via a hole 5 int hole = ++count ; 6 while(hole > 1 && array[hole / 2].isGT(comparable)) 7 { 8 array[hole] = array[hole / 2] ; 9 hole = hole / 2 ; 10 } 11 array[hole] = comparable ; 12 } 1public boolean isFull() 2{ 3 return count == array.length - 1 ; 4 }

13. MinHeap deletion The pseudo code algorithm for deleting from a MinHeap is: 1 minHeapDelete(){ 2 if(Heap is empty) throw an exception ; 3 extract the element from the root ; 4 if(root is a leaf node){ delete root ; return; } 5 copy the element from the last leaf to the root ; 6 delete last leaf ; 7 p = root ; 8 while(p is not a leaf node and p > any of its children) 9 swap p with the smaller child ; 10 return ; 11} The process of swapping an element with its smaller child, in order to restore the heap order property after a deletion is called percolate down, sift down, or reheapification downward. Thus, the steps for deletion are: 1.Replace the value at the root by the last leaf value. 2.Delete the last leaf. 3.As long as the heap order property is violated, percolate down.

14. Percolate-Down Via a Hole Instead of repeated swappings, it is more efficient to use a hole (i.e., an empty node) to percolate down: A hole is created in the root by deleting the root element. Copy the value in the last node in a temporary variable. Delete the last node. While the heap order property is violated, percolate the hole down. Insert the value from the temporary variable into the hole. l r r l l r l r e e < min(l, r) e > l < r e > l > r

15. MinHeap Deletion Example delete last node 31 24 6526 32 21 23 19 68 18 21 24 6526 32 3123 19 68 18 24 6526 32 2123 19 68 13 31 Delete min element Replace by value of last node Percolate down 18 24 6526 32 2123 19 68 31 Percolate down

16. MinHeap Deletion Implementation 1 public MyComparable dequeueMin(){ 2 if(isEmpty()) throw new ContainerEmptyException(); 3 MyComparable minItem = array[1]; 4 array[1] = array[count]; 5 count--; 6 percolateDown(1); 7 return minItem; 8} 1 private void percolateDown(int hole) { 2 int child ; MyComparable temp = array[hole] ; 3 while(hole * 2 <= count) { 4 child = hole * 2 ; 5 if(child + 1 <= count && array[child + 1].isLT(array[child]) ) 6 child++ ; 7 if(array[child].isLT(temp)) 8 array[hole] = array[child] ; 9 else 10 break ; 11 hole = child ; 12 } 13 }

17. Converting an array into a Binary heap The algorithm to convert an array into a binary heap is: 1.Start at the level containing the last non-leaf node (i.e., array[n/2], where n is the array size). 2.Make the subtree rooted at the last non-leaf node into a heap by invoking percolateDown. 3.Move in the current level from right to left, making each subtree, rooted at each encountered node, into a heap by invoking percolateDown. 4.If the levels are not finished, move to a lower level then go to step 3. 5.Stop. The above algorithm can be refined to the following method of the BinaryHeap class: 1private void buildHeapBottomUp() 2{ 3 for(int i=count/2; i>=1; i--) 4 { 5 // restore heap property for subtree rooted at array[i] 6 percolateDown(i) ; 7 } 8 }

18. Converting an array into a MinHeap (Example) 29 65 1326 31 3219 68 16 70 31261316193265682970 At each stage convert the highlighted tree into a MinHeap by percolating down starting at the root of the highlighted tree. 29 65 1326 32 3119 68 16 70 29 13 6526 32 3119 68 16 70 29 13 6526 32 3119 16 68 70 13 26 6529 32 3119 16 68 70 26 29 6570 32 3119 16 68 13

19. Heap Application: Heap Sort A MinHeap or a MaxHeap can be used to implement an efficient sorting algorithm called Heap Sort. The following algorithm uses a MinHeap: 1 public static void heapSort(MyComparable[] array) 2 { 3 BinaryHeap heap = new BinaryHeap(array) ; 4 for(int i = 0; i < array.length; i++) 5 array[i] = heap.dequeueMin() ; 6 } Because the dequeueMin algorithm is O(log n), heapSort is an O(n log n) algorithm. Apart from needing the extra storage for the heap, heapSort is among efficient sorting algorithms.

20. Heap Applications: Priority Queue A heap can be used as the underlying implementation of a priority queue. A priority queue is a data structure in which the items to be inserted have associated priorities. Items are withdrawn from a priority queue in order of their priorities, starting with the highest priority item first. Priority queues are often used in resource management, simulations, and in the implementation of some algorithms (e.g., some graph algorithms, some backtracking algorithms). Several data structures can be used to implement priority queues. Below is a comparison of some: Dequeue MinFind MinEnqueueData structure O(n) O(1)Unsorted List O(1) O(n)Sorted List O(log n) AVL Tree O(log n)O(1)O(log n)MinHeap

21. Priority Queue (Contd.) 1 priorityQueueEnque(e1) 2 { 3 if(priorityQueue is full) throw an exception; 4 insert e1 at the end of the priorityQueue; 5 while(e1 is not in the root node and e1 < parent(e1)) 6 swap(e1, parent(e1)); 7 } 1 priorityQueueDequeue(){ 2 if(priorityQueue is empty) throw an exception; 3 extract the highest priority element from the root; 4 if(root is a leaf node){ delete root ; return; } 5 copy the element from the last leaf to the root; 6 delete last leaf; 7 p = root; 8 while(p is not a leaf node and p > any of its children) 9 swap p with the smaller child; 10 return; 11 } X Heap X is the element with highest priority