💡 Problem Formulation: Heaps are specialized tree-based data structures that satisfy heap properties, where the parent nodes are compared with their children according to a specific condition. In a max heap, every parent node is larger than or equal to its children. This article discusses how to check if a given heap adheres to the max heap property or not. We want to input an array representation of a heap and output a boolean indicating whether it is a max heap.
Method 1: Iterative Comparison
This method involves iteratively comparing each parent node’s value with its children’s values in the array. The function is_max_heap takes an array as input, iterates through the array, and checks whether the max heap property is maintained throughout the heap. If the property fails for any node, the function returns False; otherwise, it returns True.
Here’s an example:
def is_max_heap(heap):
n = len(heap)
for i in range(n//2, -1, -1):
left = 2 * i + 1
right = 2 * i + 2
if (left < n and heap[i] < heap[left]) or (right < n and heap[i] < heap[right]):
return False
return True
print(is_max_heap([90, 15, 10, 7, 12, 2, 7, 3]))Output:
True
This code snippet defines a function is_max_heap which iterates over the internal nodes of the heap (up to index n//2 as leaf nodes have no children) and checks if a parent is smaller than its left or right child. If such a case is found, it breaks and returns False, indicating it’s not a max heap.
Method 2: Recursive Verification
A recursive approach to verify max heap property exists in a data structure. We recursively check for every node that it is larger than its children, and that its children are, in turn, max heaps. This concise verification stops at the first instance where the heap property is violated.
Here’s an example:
def is_max_heap_recursive(heap, i=0):
if i >= len(heap):
return True
left = 2 * i + 1
right = 2 * i + 2
if ((left < len(heap) and heap[i] < heap[left]) or
(right < len(heap) and heap[i] < heap[right])):
return False
return is_max_heap_recursive(heap, left) and is_max_heap_recursive(heap, right)
print(is_max_heap_recursive([90, 15, 10, 7, 12, 2, 7, 3]))Output:
True
The recursive function is_max_heap_recursive verifies the max heap property by comparing the current node with its children and then calling itself recursively to check the heap property for its children. It returns True if every recursive check is satisfied, indicating that the heap is a max heap.
Method 3: Using Heapq Module
Python’s heapq module provides functions for implementing heaps. We can directly use these to re-heapify the array and check if the re-heapified array is the same as the input array. If the input heap remains unchanged after applying heapq._heapify_max, it is a max heap.
Here’s an example:
import heapq
def is_max_heap_with_heapq(heap):
copied_heap = list(heap) # Create a copy to avoid modifying the original heap
heapq._heapify_max(copied_heap)
return copied_heap == heap
print(is_max_heap_with_heapq([90, 15, 10, 7, 12, 2, 7, 3]))Output:
True
This code snippet demonstrates how to verify a max heap using Python’s built-in heapq module. A copy of the heap is heapified using heapq._heapify_max, and then the function returns the result of checking the equality of the heapified copy with the original heap array.
Method 4: Validate Parent-Child Relationships
Another way to check max heap property is to validate the parent-child relationships for all parent nodes in the heap. This method explicitly compares parents with left and right children without the overhead of reconstructing or copying the heap.
Here’s an example:
def is_valid_max_heap(heap):
for i in range(len(heap)):
left = 2 * i + 1
right = 2 * i + 2
if ((left < len(heap) and heap[i] < heap[left]) or (right < len(heap) and heap[i] < heap[right])):
return False
return True
print(is_valid_max_heap([90, 15, 10, 7, 12, 2, 7, 3]))Output:
True
The function is_valid_max_heap directly inspects the parent-child relationship by iterating through each node and comparing its value with those of its children. Should a parent be smaller than any child node, the function acknowledges the breach of the max heap condition by returning False.
Bonus One-Liner Method 5: List Comprehension
For Python enthusiasts who prefer one-liners, here’s a succinct way to check for max heap property using list comprehension. This method compresses the iterative comparison logic into a single expression and is best suited for those comfortable with Python’s more advanced syntax features.
Here’s an example:
def is_max_heap_oneliner(heap):
return all(heap[i] >= heap[2 * i + 1] for i in range(len(heap)//2)) and all(heap[i] >= heap[2 * i + 2] for i in range(len(heap)//2) if 2 * i + 2 < len(heap))
print(is_max_heap_oneliner([90, 15, 10, 7, 12, 2, 7, 3]))Output:
True
The one-liner is_max_heap_oneliner function uses two list comprehensions to verify the max heap property for left and right children separately. It returns True only if all evaluations in both comprehensions are true, indicating a valid max heap.
Summary/Discussion
- Method 1: Iterative Comparison. Efficient for larger heaps. Easy to understand. May be slower than recursive approaches for smaller heaps.
- Method 2: Recursive Verification. Elegant and concise. Avoids explicit loop constructs. Performance degradation for very large heaps due to recursion limits.
- Method 3: Using Heapq Module. Utilizes Python’s standard library. Easy to implement. Involves extra space and overhead for copying the heap.
- Method 4: Validate Parent-Child Relationships. Directly operates on the heap’s array representation. Very efficient. Less Pythonic than other methods.
- Method 5: List Comprehension One-Liner. Concise and Pythonic. Readability may be reduced for those not accustomed to list comprehensions. Offers a balance between brevity and performance.