Heapify

Description

Given an integer array, heapify it into a min-heap array.

For a heap array A, A[0] is the root of heap, and for each A[i], A[i * 2 + 1] is the left child of A[i] and A[i * 2 + 2] is the right child of A[i].

Clarification

What is heap?

  • Heap is a data structure, which usually have three methods: push, pop and top. where "push" add a new element the heap, "pop" delete the minimum/maximum element in the heap, "top" return the minimum/maximum element.

What is heapify?

  • Convert an unordered integer array into a heap array. If it is min-heap, for each element A[i], we will get A[i * 2 + 1] >= A[i] and A[i * 2 + 2] >= A[i].

What if there is a lot of solutions?

  • Return any of them.

Example

Given [3,2,1,4,5], return [1,2,3,4,5] or any legal heap array.

Challenge

O(n) time complexity

Analysis

Heapify一个Array,也就是对array中的元素进行siftup或者siftdown的操作。根据min heap定义进行操作即可。

这里值得注意的是,对于扫描整个array的情况下,siftupsiftdowncomplexity上的区别。

基本的原因在于:siftdown的complexity,实质上是node相对于bottom移动的次数,而根据binary heap本身的特性,决定了约靠近bottom的node越多;相对照的是siftup,是node相对于root节点的移动次数。

The number of operations required for each operation is proportional to the distance the node may have to move. For siftDown, it is the distance from the bottom of the tree, so siftDown is expensive for nodes at the top of the tree. With siftUp, the work is proportional to the distance from the top of the tree, so siftUp is expensive for nodes at the bottom of the tree. Although both operations are O(log n) in the worst case, in a heap, only one node is at the top whereas half the nodes lie in the bottom layer. So it shouldn't be too surprising that if we have to apply an operation to every node, we would prefer siftDown over siftUp

关于其中具体的理由以及分析方法,可以参考stackoverflow上一个问答:
http://stackoverflow.com/questions/9755721/how-can-building-a-heap-be-on-time-complexity

如果Heapify可以用O(n)实现,那么HeapSort所需的时间复杂度为何是O(nlogn)?因为HeapSort其实包含了两个步骤,第一步,Heapify,build (min) heap,所需时间复杂度O(n),第二步,依次删除最小值(min heap),对于Heap来说,删除操作的复杂度是O(logn), 而这个删除需要执行O(n),来得到最终sort的结果,于是总体时间复杂度是O(nlogn)。

Solution

Sift-up - Time O(nlogn)

public class Solution {
    /**
     * @param A: Given an integer array
     * @return: void
     */

    private void swap(int[] A, int i, int j) {
        int temp = A[i];
        A[i] = A[j];
        A[j] = temp;
    }

    private void siftup(int[] A, int k) {
        while (k != 0) {
            int parent = (k - 1) / 2;
            if (A[k] > A[parent]) {
                break;
            }
            swap(A, k, parent);
            k = parent;
        }
    }

    public void heapify(int[] A) {
        for (int i = 0; i < A.length; i++) {
            siftup(A, i);
        }
    }
}

Sift-down - Time O(n)

public class Solution {
    /**
     * @param A: Given an integer array
     * @return: void
     */
    private void siftdown(int[] A, int k) {
        while (k < A.length) {
            int smallest = k;
            if (k * 2 + 1 < A.length && A[k * 2 + 1] < A[smallest]) {
                smallest = k * 2 + 1;
            }
            if (k * 2 + 2 < A.length && A[k * 2 + 2] < A[smallest]) {
                smallest = k * 2 + 2;
            }
            if (smallest == k) {
                break;
            }
            int temp = A[smallest];
            A[smallest] = A[k];
            A[k] = temp;

            k = smallest;
        }
    }

    public void heapify(int[] A) {
        for (int i = A.length / 2; i >= 0; i--) {
            siftdown(A, i);
        }
    }
}

Reference

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