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Inserting an element into a heap

In this article we examine the idea laying in the foundation of the heap data structure. We call it sifting, but you also may meet another terms, like "trickle", "heapify", "bubble" or "percolate".

Insertion algorithm

Now, let us phrase general algorithm to insert a new element into a heap.

  1. Add a new element to the end of an array;
  2. Sift up the new element, while heap property is broken. Sifting is done as following: compare node's value with parent's value. If they are in wrong order, swap them.

Example

Insert -2 into a following heap:

Insert a new element to the end of the array:

In the general case, after insertion, heap property near the new node is broken:

To restore heap property, algorithm sifts up the new element, by swapping it with its parent:

Now heap property is broken at the root node:

Keep sifting:

Heap property is fulfilled, sifting is over.

Source heap
After -2 insertion

Complexity analysis

Complexity of the insertion operation is O(h), where h is heap's height. Taking into account completeness of the tree, O(h) = O(log n), where n is number of elements in a heap.

Code snippets

Java implementation

public class BinaryMinHeap {

     

     

     

public void insert(int value) {

            if (heapSize == data.length)

                  throw new HeapException("Heap's underlying storage is overflow");

            else {

                  heapSize++;

                  data[heapSize - 1] = value;

                  siftUp(heapSize - 1);

            }

      }    

 

     

     

private void siftUp(int nodeIndex) {

            int parentIndex, tmp;

            if (nodeIndex != 0) {

                  parentIndex = getParentIndex(nodeIndex);

                  if (data[parentIndex] > data[nodeIndex]) {

                        tmp = data[parentIndex];

                        data[parentIndex] = data[nodeIndex];

                        data[nodeIndex] = tmp;

                        siftUp(parentIndex);

                  }

            }

      }

}

C++ implementation

void BinaryMinHeap::siftUp(int nodeIndex) {

      int parentIndex, tmp;

      if (nodeIndex != 0) {

            parentIndex = getParentIndex(nodeIndex);

            if (data[parentIndex] > data[nodeIndex]) {

                  tmp = data[parentIndex];

                  data[parentIndex] = data[nodeIndex];

                  data[nodeIndex] = tmp;

                  siftUp(parentIndex);

            }

      }

}

 

void BinaryMinHeap::insert(int value) {

      if (heapSize == arraySize)

            throw string("Heap's underlying storage is overflow");

      else {

            heapSize++;

            data[heapSize - 1] = value;

            siftUp(heapSize - 1);

      }

}


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