MEDIUM
DATA STRUCTURES AND ALGORITHMS

How to solve the Build a Max Heap Problem

HEAPSTREES
Written By
tom-wagner.png
Tom Wagner
Christopher Ang

Introduction Build a Max Heap

The Build Max Heap problem involves constructing a max heap from an array of integers. A max heap is a type of binary tree stored in an array that is organized such that the maximum value is always at the root of the tree. The solution to this problem must be able to rearrange the elements of the input array into a new array representing a max heap.

Problem Build a Max Heap

Given an array of integers, transform the array in-place to a max heap.

Example Inputs and Outputs

Example 1

Input: [1,2,3,4,5,6,7]

image9.png

Output:

  • [7,4,6,1,3,2,5] (sift up)

OR

  • [7,5,6,4,2,1,3] (sift down)

image3.png

Example 2

Input: [4,6,52,10,17,28,39,43,11]

image8.png

Output:

  • [52,43,39,17,10,6,28,4,11] (sift up)

OR

  • [52,43,39,11,17,28,4,10,6] (sift down)

image6.png

Constraints

  • 1 <= len(arr) <= 10,000
  • Every integer in the array will be <= 64-bits

Solution Build a Max Heap

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Show Transcript

Max (or min) heaps are complete binary trees which satisfy the heap property, meaning each node is at least as large (or small) as each of its children. Max heaps can be encoded in arrays, where:

  1. The leftmost element of the array is the largest in the array
  2. The children of the node at index i are located at 2*i+1 and 2*i+2

To demonstrate this, consider the following max heap: [7, 5, 6, 4, 2, 1, 3].

image1.png

Within this array, you can see the two children of 7 (the 0th index element) at the following indexes: 2 * 0 + 1 = 1, and 2 * 0 + 2 = 2.

Additionally, for 5 (the 1st index element) you can use the equation again to find its children at indexes: 2 * 1 + 1 = 3, and 2 * 1 + 2 = 4.

Given an arbitrary array, it is possible to reorder the elements such that they satisfy the above properties.

Note: Given a collection of integers, there may be many valid orderings that satisfy the min- or max-heap property, which will depend on the algorithm used to “heapify” the array.

Two common approaches to “heapifying” an array are to rely on sifting nodes up the tree, and sifting nodes down the tree.

Problem Approaches

Approach 1: Sift Up

To sift a node up the tree, we swap a node that is larger than its parent with its parent, and continue doing this (via e.g. recursion) until it is no larger than its parent.

To solve this problem using the “sift up” approach, we can use algebra to modify the equation we use to find a node’s children. Consider the following equation and note how it can be adjusted to solve for the parent node’s index:

child_one_idx = parent_idx * 2 + 1
child_two_idx = parent_idx * 2 + 2

Using algebra to isolate parent_idx yields the following:

(child_one_idx + 1) / 2 = parent_idx
(child_two_idx + 2) / 2 = parent_idx 

Note we now have two separate equations to find the parent index. Specifically, if we use the formula for child one on a node which is the second child, we will end up with a value that is 0.5 larger than the actual parent index. So we need to find one equation that will work in all circumstances, and to accomplish this we can modify our parent_idx equation slightly by using Math.floor():

parent_index = floor((i - 1 )/ 2)

image5.png image4.png image10.png

To build the heap via sifting up involves starting at the beginning of the tree (head of the array), and iteratively sifting up each node. As the algorithm proceeds, nodes to the left of the current index will form a max heap, until finally all nodes have been sifted up.

from math import floor

def build_heap_siftup(arr):
    # Skip the first element, since it has no parent
    for i in range(1, len(arr)):
        siftup(arr, i)

def siftup(arr, index):            
    if index < 1:
        return
    parent_idx = floor((index-1)/2)
    # is the current node larger than its parent? 
    if arr[parent_idx] < arr[index]:
        # we're larger, swap and check if we're larger than the next parent
        swap(arr, parent_idx, index)
        siftup(arr, parent_idx)

def swap(arr, first, second):
    tmp = arr[first]
    arr[first] = arr[second]
    arr[second] = tmp

Time/Space Complexity

  • Time complexity: O(n * log(n)). This algorithm needs to call siftUp() on all but the root node (the n term), and needs to move each node to the top of the heap (which takes worst-case log(n) time). Hence the O(n * log(n)) time complexity.
  • Space complexity: O(n)

Approach 2: Sift Down

Conversely, we can also “heapify” an array by shifting smaller nodes down the tree.

To sift a node down the tree, we swap a node that is smaller than its children with its largest child, and continue doing so (via e.g. recursion) until it is no smaller than its children.

image2.png

image7.png

To build the heap via sifting down we start at the end and iteratively sifting down each node with the larger of its two children assuming at least one of them is bigger than the node itself. As the algorithm proceeds, nodes to the right of the current index will form max heaps.

Note this approach can also be applied to min heaps by sifting down nodes with children that are smaller than itself.

from math import floor

def build_heap_siftdown(arr):
    # iterating right to left in the array
    # we don't need to start strictly at the end, since those nodes have no children
    # floor(len(arr)/2) gets the first node with children in a complete binary tree
    for i in range(floor(len(arr)/2), -1, -1):
        siftdown(arr, i)

def siftdown(arr, level=0):
    if len(arr) <= 1:
        return

    left_idx = 2*level + 1
    right_idx = 2*level + 2
    largest = level

    # See if the left child is larger
    if left_idx < len(arr) and arr[left_idx] > arr[largest]:
        largest = left_idx

    # See if the right child is larger
    if right_idx < len(arr) and arr[right_idx] > arr[largest]:
        largest = right_idx

    if largest != level:
        # There's a child that's larger; switch with the largest child, and
        # see if we're smaller than its children
        swap(arr, largest, level)
        siftdown(arr, largest)

def swap(arr, first, second):
    tmp = arr[first]
    arr[first] = arr[second]
    arr[second] = tmp

Time/Space Complexity

  • Time complexity: O(n)
  • Space complexity: O(n)

To see why the time complexity differs between these two approaches, see this stack overflow post: https://stackoverflow.com/a/18742428

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