索引从1到n的HeapSort算法，实际代码必须从0到n-1

Question

I'm kind of new to algorithms and wanted to implement heap sort algorithm. The algorithm is given as follows:

Parent(i) return Math.floor(i/2)

Left(i) return 2i

Right(i) return 2i+1

Then there is HEAPIFY method that restores the heep property. Algorithm is as follows:

HEAPIFY(A, i)
 l = Left(i)
 r = Right(i)
 if (l <= heap-size[A] and A[l] > A[i]
  then largest = l
 else largest = i
 if r <= heap-size[A] and A[r] > A[largest]
  then largest = r
 if largest != i
  then exchange A[i] <-> A[largest]
       HEAPIFY(A, largest)

My Code that implements this method is:

public static void HEAPIFY(int[] A, int i) {
    int l = LEFT(i);
    int r = RIGHT(i);
    int largest = 0;
    if (l < A.length && A[l] > A[i]) {
        largest = l;
    } else {
        largest = i;
    }

    if (r < A.length && A[r] > A[largest]) {
        largest = r;
    }

    if (largest != i) {
        int temp = A[i];
        A[i] = A[largest];
        A[largest] = temp;
        HEAPIFY(A, largest);
    }
}

Now My question is in the book algorithm is shown by drawing the tree of heap and array so for example array is: [16,14,10,8,7,9,3,2,4,1] and for the tree and also for array it is indexed starting from 1 to n, so Array[1] = 16 and in coding Array[0] = 16. Now i can not adjust the heapify method to start either from index 1 and go up to 1 or somehow make it start from 0 and let the heap be indexed from 0 to n-1.

Sorry if its kind of confusing i'm still confused but i would really appreciate some help.

Thank you guys

Now HEAPIFY works and the following code is code to build the heap:

public static void BUILD_HEAP(int[] A) {
    heapSize = A.length;
    for (int i = (int) Math.floor(A.length / 2.0); i >= 0; i--) {
        HEAPIFY(A, i);
    }
}

build heap also works and the only method that doesnot work is heapsort.

 public static void HEAPSORT(int[] A) {
    BUILD_HEAP(A);
    for (int i = A.length-1; i >= 1; i--) {
        int temp = A[0];
        A[0] = A[i];
        A[i] = temp;
        heapSize = heapSize-1;
        HEAPIFY(A,0);
    }
}

this has to sort but when i try to traverse the array after the call of heapsort it does not give the sorted array. any ideas how to fix heapsort?

Answer 1

Parent(i) return Math.floor(i/2)

=> Parent(i) return Math.floor((i - 1) / 2)

Left(i) return 2i

=> Left(i) return 2i + 1

Right(i) return 2i+1

=> Right(i) return 2i + 2

You can work this out either by fiddling around (which is what I actually did) or considering j = i - 1.

If i' = 2 i and j = i - 1 so i = j + 1

j' = i' - 1 = (2i) - 1 = (2(j + 1)) - 1 = 2j + 1

Answer 2

if you want to start form index 1,then you can initialize the array like this: [-x,16,14,10,8,7,9,3,2,4,1] -x is the array[0],in other words,you can ignore the element which is in array[0].

if you want to start form index 0,then you have to modify the function LEFT(i) and RIGHT(i).

LEFT(i) return 2*i+1;

RIGHT(i) return 2*i+2;