Heap Sort

Heap(An array visualized as a complete binary tree):

Heap Operations:

Insert, 　　　　　　　　　　　　　　　　　　　　　　　　O(log(n))
　　Add it to the end of the tree and bubble it up.

Find Max(return the element with maximum key), 　　O(1)
Extract Max(find max and remove it from heap), 　　　O(log(n))
　　Replace the root with the last entry in the tree.
Increase Key Value,　　　　　　　　　　　　　　　　　  O(log(n))

Tree_Heapify,　　　　    　　　　　　　　　　　　　　　O(nlog(n))
　　Bad: Inserting each element, one at a time, into an empty heap

Tree_Heapify,　　　　　　　　　　　　　　　　　 　　　O(n)
　　Good: Bottom-up

　　Starting from the last level of non-leaf nodes ( for i=n/2...1 )(Walk backward from the last internal node to the root), and heapify the node.
　　Assuming both sub-trees are heaps already, heapifying the root node takes O(log(n)) time given n children for the tree. So to heapify the whole tree, it takes:
　　1*n/4 + 2*n/8 + 3*n/16 + ... + log(n)*1
=n/4*(1+2/2 + 3/2^2 + 4/2^3 + ... + (k+1)/2^k)
=O(n)

parent = child/2
child = parent * 2, parent * 2 +1

Leaves in the array, A[n/2+1, ..., n]

 

Heap Sort:

bad for linked list

public static void main(String[] args){Random r = new Random();        int[] a = new int[]{r.nextInt(100), r.nextInt(100), r.nextInt(100),r.nextInt(100),r.nextInt(100),r.nextInt(100),r.nextInt(100),r.nextInt(100),r.nextInt(100),r.nextInt(100),r.nextInt(100)};        System.out.println(Arrays.toString(a));int[] aCopy1 = Arrays.copyOf(a, a.length);heapSort(aCopy1);System.out.println(Arrays.toString(aCopy1));            }private static void heapSort(int[] a){    //Bottom-up heapify.
        maxHeapify(a);int i = a.length - 1;while(i>0){//swap the root(the largest value) to the end of the arrayint temp = a[i];a[i] = a[0];a[0] = temp;--i;sinkMinDown(a, 0, i);}}private static void maxHeapify(int[] a){//find the last internal node in a 0-based arrayint iStart = (a.length-2)/2;while(iStart>=0){sinkMinDown(a, iStart, a.length-1);--iStart;}        }// [root, iEnd]private static void sinkMinDown(int[] a, int root, int iEnd){//left child 2i+1 for 0-based array//right child 2i+2 for 0-based arraywhile(root*2+1<=iEnd){int max = root;int leftChild = root*2+1;if(a[max]<a[leftChild])max = leftChild;int rightChild = leftChild+1;if(rightChild<=iEnd && a[max]<a[rightChild])max = rightChild;if(max == root)return; // a valid heapint temp = a[root];a[root] = a[max];a[max] = temp;root = max;}}