Comp202: Principles of Object-Oriented Programming II
Fall 2007 -- Lecture #37: Heaps and Heap Sort   

Heap Sort

How is split() sped up?

What is a Heap?

Implementing a Heap?

Heap Sort Basics

Heap Sort: split()

public int split(int[] A, int lo, int hi) { 
    // Swap A[hi] and A[lo]. 
    int temp = A[hi]; 
    A[hi] = A[lo]; 
    A[lo] = temp;     

    // Restore the heap property by ``sifting down'' 
    //  the element at A[lo]. 
    Heapifier.Singleton.siftDown(A, lo, lo, hi - 1); 
    return hi; 

Example of siftDown()

siftDown(): The Implementation

public void siftDown(int[] A, int lo, int cur, int hi) { 
    int dat = A[cur];             // hold on to data. 
    int child = 2 * cur + 1 - lo; // index of left child of A[cur]. 
    boolean done = hi < child;     
    while (!done) { 
        if (child < hi && A[child + 1] < A[child]) { 
        } // child is the index of the smaller of the two children. 
        if (A[child] < dat) { 
            A[cur] = A[child]; 
            cur = child; 
            child = 2 * cur + 1 - lo; 
            done = hi < child; 
        }                     // A[cur] is less than its children. 
        else {                // A[cur] <= A[child]. 
            done = true;        // heap condition is satisfied. 
        }                     // A[cur] is less than its children. 
    }                       // location found for temp. 
    A[cur] = dat; 

Initializing the Heap: HeapSorter()

In order to sort an array, using the ordering capabilities of a heap, you must first transform the randomly placed data in the array into a heap.  This is called "heapifying" the array.  This can be accomplished by sifting down the elements.  Luckily, even though this operation takes place in O(n log(n)) time, it only occurs once, so in the end, it has no impact on the overall complexity of the algorithm.

Note that we only really have to sift down half the array, i.e. half the "tree".  This is because a single-element array (tree) is already a heap, so we can bypass all the leaves and immediately start working on the layer right above the leaves.

public class HeapSorter extends ASorter { 
    public HeapSorter(int[] A, int lo, int hi)  { 
        for (int cur = (hi + lo + 1) / 2; cur >= lo; cur--) { 
            Heapifier.Singleton.siftDown(A, lo, cur, hi); 
  // etc. . .

Inserting into an existing Heap: siftUp()

To insert a data element into an existing heap, we are forced to initially insert the element at the bottom of the tree, which is at the end of the array.  Since this may break the heap property, we need "sift up" the data through the tree to find a spot for it where the overall heap property will be restored.  When sifting up, we are essentially taking a data element, (starting with the one being inserted) comparing it to its parent and then taking the largest (or smallest) of the pair, leaving the other in the parent's position.   The process with the left-over data element and the next higher parent until the top of the heap is reached.

    * "Sifts" A[cur] up the array A to maintain the heap property.
    * @param A A[lo:cur-1] is a heap.
    * @param lo the low index of A.
    * @param cur lo <= cur <= the high index of A.
    public void siftUp(int[] A, int lo, int cur) {
        int dat = A[cur];
        int parent = (cur - lo - 1) / 2 + lo;  // index of parent.
        while (0 < (cur - lo) && dat < A[parent]) {
            A[cur] = A[parent];
            cur = parent;
            parent = (cur - lo - 1) / 2 + lo;
        A[cur] = dat;

Sample Code

Summary of Sorting (see also SortSummary.pdf)

Best-case Cost Worst-case Cost
Selection O(n2) O(n2)
Insertion O(n) O(n2)
Heap O(n log n) O(n log n)
Merge O(n log n) O(n log n)
Quick O(n log n) O(n2)


Last Revised Thursday, 03-Jun-2010 09:52:35 CDT

©2007 Stephen Wong and Dung Nguyen