K-ary Heap

Prerequisite – Binary Heap

K-ary heaps are a generalization of binary heap(K=2) in which each node have K children instead of 2. Just like binary heap, it follows two properties:

1) Nearly complete binary tree, with all levels having maximum number of nodes except the last, which is filled in left to right manner.

2) Like Binary Heap, it can be divided into two categories: (a) Max k-ary heap (key at root is greater than all descendants and same is recursively true for all nodes). (b) Min k-ary heap (key at root is greater than all descendants and same is recursively true for all nodes)

Examples:

3-ary max heap - root node is maximum
                 of all nodes
         10
     /    |   \
   7      9     8
 / | \   /
4  6  5 7


3-ary min heap -root node is minimum 
                of all nodes
         10
      /   |  \
    12    11  13
  / | \
14 15 18 

The height of a complete k-ary tree with n-nodes is given by logkn.

Applications of K-ary Heap:

  • K-ary heap when used in the implementation of priority queue allows faster decrease key operation as compared to binary heap ( O(log2n)) for binary heap vs O(logkn) for K-ary heap). Nevertheless, it causes the complexity of extractMin() operation to increase to O(k log kn) as compared to the complexity of O(log2n) when using binary heaps for priority queue. This allows K-ary heap to be more efficient in algorithms where decrease priority operations are more common than extractMin() operation.Example: Dijkstra’s algorithm for single source shortest path and Prim’s algorithm for minimum spanning tree
  • K-ary heap has better memory cache behaviour than a binary heap which allows them to run more quickly in practice, although it has a larger worst case running time of both extractMin() and delete() operation (both being O(k log kn) ).

Implementation
Assuming 0 based indexing of array, an array represents a K-ary heap such that for any node we consider:

  • Parent of the node at index i (except root node) is located at index (i-1)/k
  • Children of the node at index i are at indices (k*i)+1 , (k*i)+2 …. (k*i)+k
  • The last non-leaf node of a heap of size n is located at index (n-1)/k

buildHeap() : Builds a heap from an input array.
This function runs a loop starting from the last non-leaf node all the way upto the root node, calling a function restoreDown(also known as maHeapify) for each index that restores the passed index at the correct position of the heap by shifting the node down in the K-ary heap building it in a bottom up manner.
Why do we start the loop from the last non-leaf node ?
Because all the nodes after that are leaf nodes which will trivially satisfy the heap property as they don’t have any children and hence, are already roots of a K-ary max heap.

restoreDown() (or maxHeapify) : Used to maintain heap property.
It runs a loop where it finds the maximum of all the node’s children, compares it with its own value and swaps if the max(value of all children) > (value at node). It repeats this step until the node is restored into its original position in the heap.

extractMax() : Extracting the root node.
A k-ary max heap stores the largest element in its root. It returns the root node, copies last node to the first, calls restore down on the first node thus maintaining the heap property.

insert() : Inserting a node into the heap
This can be achieved by inserting the node at the last position and calling restoreUp() on the given index to restore the node at its proper position in the heap. restoreUp() iteratively compares a given node with its parent, since in a max heap the parent is always greater than or equal to its children nodes, the node is swapped with its parent only when its key is greater than the parent.

Combining the above, following is the C++ implementation of K-ary heap.

// C++ program to demonstrate all operations of
// k-ary Heap
#include<bits/stdc++.h>
using namespace std;

// function to heapify (or restore the max- heap
// property). This is used to build a k-ary heap
// and in extractMin()
// att[] -- Array that stores heap
// len   -- Size of array
// index -- index of element to be restored
//          (or heapified)
void restoreDown(int arr[], int len, int index,
                                        int k)
{
    // child array to store indexes of all
    // the children of given node
    int child[k+1];

    while (1)
    {
        // child[i]=-1 if the node is a leaf
        // children (no children)
        for (int i=1; i<=k; i++)
            child[i] = ((k*index + i) < len) ?
                    (k*index + i) : -1;

        // max_child stores the maximum child and
        // max_child_index holds its index
        int max_child = -1, max_child_index ;

        // loop to find the maximum of all
        // the children of a given node
        for (int i=1; i<=k; i++)
        {
            if (child[i] != -1 &&
                arr[child[i]] > max_child)
            {
                max_child_index = child[i];
                max_child = arr[child[i]];
            }
        }

        // leaf node
        if (max_child == -1)
            break;

        // swap only if the key of max_child_index
        // is greater than the key of node
        if (arr[index] < arr[max_child_index])
            swap(arr[index], arr[max_child_index]);

        index = max_child_index;
    }
}

// Restores a given node up in the heap. This is used
// in decreaseKey() and insert()
void restoreUp(int arr[], int index, int k)
{
    // parent stores the index of the parent variable
    // of the node
    int parent = (index-1)/k;

    // Loop should only run till root node in case the
    // element inserted is the maximum restore up will
    // send it to the root node
    while (parent>=0)
    {
        if (arr[index] > arr[parent])
        {
            swap(arr[index], arr[parent]);
            index = parent;
            parent = (index -1)/k;
        }

        // node has been restored at the correct position
        else
            break;
    }
}

// Function to build a heap of arr[0..n-1] and alue of k.
void buildHeap(int arr[], int n, int k)
{
    // Heapify all internal nodes starting from last
    // non-leaf node all the way upto the root node
    // and calling restore down on each
    for (int i= (n-1)/k; i>=0; i--)
        restoreDown(arr, n, i, k);
}

// Function to insert a value in a heap. Parameters are
// the array, size of heap, value k and the element to
// be inserted
void insert(int arr[], int* n, int k, int elem)
{
    // Put the new element in the last position
    arr[*n] = elem;

    // Increase heap size by 1
    *n = *n+1;

    // Call restoreUp on the last index
    restoreUp(arr, *n-1, k);
}

// Function that returns the key of root node of
// the heap and then restores the heap property
// of the remaining nodes
int extractMax(int arr[], int* n, int k)
{
    // Stores the key of root node to be returned
    int max = arr[0];

    // Copy the last node's key to the root node
    arr[0] = arr[*n-1];

    // Decrease heap size by 1
    *n = *n-1;

    // Call restoreDown on the root node to restore
    // it to the correct position in the heap
    restoreDown(arr, *n, 0, k);

    return max;
}

// Driver program
int main()
{
    const int capacity = 100;
    int arr[capacity] = {4, 5, 6, 7, 8, 9, 10};
    int n = 7;
    int k = 3;

    buildHeap(arr, n, k);

    printf("Built Heap : \n");
    for (int i=0; i<n; i++)
        printf("%d ", arr[i]);

    int element = 3;
    insert(arr, &n, k, element);

    printf("\n\nHeap after insertion of %d: \n",
            element);
    for (int i=0; i<n; i++)
        printf("%d ", arr[i]);

    printf("\n\nExtracted max is %d",
                   extractMax(arr, &n, k));

    printf("\n\nHeap after extract max: \n");
    for (int i=0; i<n; i++)
        printf("%d ", arr[i]);

    return 0;
}

Output

Built Heap : 
10 9 6 7 8 4 5 

Heap after insertion of 3: 
10 9 6 7 8 4 5 3 

Extracted max is 10

Heap after extract max: 
9 8 6 7 3 4 5 

Time Complexity Analysis

  • For a k-ary heap, with n nodes the maximum height of the given heap will be logkn. So restoreUp() run for maximum of logkn times (as at every iteration the node is shifted one level up is case of restoreUp() or one level down in case of restoreDown).
  • restoreDown() calls itself recursively for k children. So time complexity of this functions is O(k logkn).
  • Insert and decreaseKey() operations call restoreUp() once. So complexity is O(logkn).
  • Since extractMax() calls restoreDown() once, its complexity O(k logkn)
  • Time complexity of build heap is O(n) (Analysis is similar to binary heap)

This article is contributed by Anurag Rai. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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