1. Class No.26 Data Structures http://ecomputernotes.com

2. Heap code in C++ template <class eType> void Heap<eType>::deleteMin( eType & minItem ) { if( isEmpty( ) ) { cout << &quot; heap is empty. “ << endl ; return; } minItem = array[ 1 ]; array[ 1 ] = array[ currentSize-- ]; percolateDown( 1 ); } http://ecomputernotes.com

3. Heap code in C++ // hole is the index at which the percolate begins. template <class eType> void Heap<eType>::percolateDown( int hole ) { int child; eType tmp = array[ hole ]; for( ; hole * 2 <= currentSize; hole = child ) { child = hole * 2; if( child != currentSize && array[child+1] < array[ child ] ) child++; // right child is smaller if( array[ child ] < tmp ) array[ hole ] = array[ child ]; else break; } array[ hole ] = tmp; } http://ecomputernotes.com

4. Heap code in C++ template <class eType> const eType& Heap<eType>::getMin( ) { if( !isEmpty( ) ) return array[ 1 ]; } template <class eType> void Heap<eType>::buildHeap(eType* anArray, int n ) { for(int i = 1; i <= n; i++) array[i] = anArray[i-1]; currentSize = n; for( int i = currentSize / 2; i > 0; i-- ) percolateDown( i ); } http://ecomputernotes.com

5. Heap code in C++ template <class eType> bool Heap<eType>::isEmpty( ) { return currentSize == 0; } template <class eType> bool Heap<eType>::isFull( ) { return currentSize == capacity; } template <class eType> int Heap<eType>::getSize( ) { return currentSize; } http://ecomputernotes.com

7. BuildHeap in Linear Time It is easy to see that this tree consists of (2 0 ) node at height h , 2 1 nodes at height h –1, 2 2 at h- 2 and, in general, 2 i nodes at h – i. http://ecomputernotes.com

8. Complete Binary Tree A B h : 2 0 nodes H D I E J K C L F M G N O h -1: 2 1 nodes h -2: 2 2 nodes h -3: 2 3 nodes http://ecomputernotes.com

9. BuildHeap in Linear Time The sum of the heights of all the nodes is then S =  2 i ( h – i ), for i = 0 to h-1 = h + 2(h-1) + 4(h-2) + 8(h-3)+ ….. + 2 h-1 (1) Multiplying by 2 gives the equation 2S = 2 h + 4(h-1) + 8(h-2) + 16(h-3)+ ….. + 2 h (2) Subtract the two equations to get S = - h + 2 + 4 + 8 + 16+ ….. + 2 h-1 +2 h = (2 h+1 – 1) - (h+1) Which proves the theorem. http://ecomputernotes.com

10. BuildHeap in Linear Time Since a complete binary tree has between 2 h and 2 h+1 nodes S = (2 h+1 – 1) - (h+1)  N - log 2 (N+1) Clearly, as N gets larger, the log 2 ( N +1) term becomes insignificant and S becomes a function of N . http://ecomputernotes.com

12. Height 1 Nodes Marking the left edges for height 1 nodes http://ecomputernotes.com

13. Height 2 Nodes Marking the first left edge and the subsequent right edge for height 2 nodes http://ecomputernotes.com

14. Height 3 Nodes Marking the first left edge and the subsequent two right edges for height 3 nodes http://ecomputernotes.com

15. Height 4 Nodes Marking the first left edge and the subsequent three right edges for height 4 nodes http://ecomputernotes.com

16. Theorem N=31, treeEdges=30, H=4, dottedEdges=4 (H). Darkened Edges = 26 = N-H-1 (31-4-1) http://ecomputernotes.com