CS2420: Lecture 20 Vladimir Kulyukin Computer Science Department Utah State University

Outline HeapSort (Chapter 7)

The Heap Property A Heap is a Complete Binary Tree. All levels are filled except possibly the last one. The Heap Property: –Minimal heap property: for every node X with parent P, the key in P is smaller than or equal to the key in X. –Maximum heap property: for every node X with parent P, the key in P is greater than or equal to the key in X.

The Heap Property P X P X The minimum heap property The maximum heap property

Packing Heaps into Arrays We can convert this heap into a one dimensional array by doing a level-order traversal on it.

Packing Heaps into Arrays Note that the index count of the array starts with 1. We can convert this heap into an array by doing a level-order traversal on it.

Packing Heaps into Arrays

Interval Nodes and Leaves Internal NodesLeaves

Leaves vs. Non-Leaves

Maintaining the Heap Property A heap is a container so items can be inserted and deleted at any time. The problem is that if we insert/delete an item, the heap property may be broken after the insertion/deletion. We need to make sure that the heap property is restored after every insertion and deletion.

Maintaining the Property: Max Heap Example Heap Property is broken.

Maintaining the Property: Example We swap 4 with the maximum of 14 and 7.

Maintaining the Property: Example We swap 4 with the maximum of 2 and 8.

Maintaining the Property: Example The heap property is now restored at every node.

RestoreHeap: Maintaining The Heap Property RestoreHeap(A, i) { int Max = 0; int LCH = LeftChild(i); int RCH = RightChild(i); If ( LCH A[i] ) { Max = LCH; } Else { Max = i; } If ( RCH A[MAX] ) { Max = RCH; } If ( Max != i ) { Swap(A[i], A[Max]); RestoreHeap(A, Max); }

RestoreHeap: Asymptotic Analysis

Building a Heap Given the dimensions of the array A, discard all the leaves. The leaf indices are [N/2 + 1, N]. Start from the Rightmost Inner (Non- Leaf) Node and go up the tree restoring the heap property at every inner node until you reach and restore the heap property at the root node.

Building a Heap

Building a Max Heap: Example There are 7 nodes in the heap; so the inner node range is [1, 3]. Thus, we start with node 3.

Building a Max Heap: Example RestoreHeap(3)

Building a Max Heap: Example RestoreHeap(2)

Building a Max Heap: Example RestoreHeap(1)

Building a Max Heap: Example RestoreHeap(3)

Building a Max Heap: Example