While understanding the properties and operations of heaps is important, implementing them efficiently from scratch can be complex. Fortunately, Python's standard library provides the heapq module, offering a highly optimized implementation of the heap queue algorithm, also known as the priority queue algorithm.

The heapq module works directly with standard Python lists, treating them as min-heaps. This means the smallest element is always at the root (index 0). Let's explore its primary functions.

Basic Heap Operations

To use the heapq functions, you first import the module:

import heapq

You then operate on a standard Python list.

Adding Elements: `heapq.heappush`

To add an element to the heap while maintaining the heap property, use heapq.heappush(heap, item). The function modifies the list in-place.

# Start with an empty list (our heap)
min_heap = []

# Push items onto the heap
heapq.heappush(min_heap, 50)
heapq.heappush(min_heap, 20)
heapq.heappush(min_heap, 80)
heapq.heappush(min_heap, 10)
heapq.heappush(min_heap, 30)

print(f"Heap after pushes: {min_heap}")
# Expected output (order might vary slightly depending on implementation details,
# but 10 will be the first element):
# Heap after pushes: [10, 20, 80, 50, 30]

Internally, heappush adds the element to the end of the list and then "sifts it up" to its correct position to maintain the min-heap property. This operation typically takes $O(\log n)$ time, where $n$ is the number of elements in the heap.

A diagram representing the min-heap [10, 20, 80, 50, 30]. The root (10) is the smallest element.

Removing the Smallest Element: `heapq.heappop`

To remove and return the smallest element (the root) from the heap, use heapq.heappop(heap). This also modifies the list in-place and maintains the heap property.

# Continuing from the previous example
smallest_item = heapq.heappop(min_heap)

print(f"Popped item: {smallest_item}")
print(f"Heap after pop: {min_heap}")
# Expected output:
# Popped item: 10
# Heap after pop: [20, 30, 80, 50]

heappop replaces the root with the last element in the list and then "sifts it down" to restore the heap property. Like heappush, this operation typically takes $O(\log n)$ time.

The heap structure after popping the smallest element (10). 20 is now the root.

Accessing the Smallest Element

Since heapq uses a list where the first element heap[0] is always the smallest, you can peek at the minimum value without removing it simply by accessing the element at index 0. Remember to check if the heap is empty first.

if min_heap:
    print(f"Smallest item currently in heap: {min_heap[0]}")
else:
    print("Heap is empty.")
# Expected output:
# Smallest item currently in heap: 20

Building a Heap Efficiently: `heapq.heapify`

If you already have a list of items and want to turn it into a valid heap, you could repeatedly call heappush. However, a more efficient method is heapq.heapify(list). This function rearranges the list elements in-place to satisfy the heap property.

data = [50, 20, 80, 10, 30, 5, 90, 45]
print(f"Original list: {data}")

heapq.heapify(data) # Transform list into a heap in-place

print(f"Heapified list: {data}")
print(f"Smallest item: {data[0]}")
# Expected output (heapified list order might vary, smallest is first):
# Original list: [50, 20, 80, 10, 30, 5, 90, 45]
# Heapified list: [5, 10, 80, 20, 30, 50, 90, 45] # Example output
# Smallest item: 5

The advantage of heapify is its time complexity. It runs in linear time, $O(n)$ , which is generally faster than inserting $n$ elements one by one using heappush, which would take $O(n \log n)$ time in total.

Combining Operations

heapq also provides functions that combine push and pop operations efficiently.

heapq.heappushpop(heap, item): Pushes item onto the heap and then pops and returns the smallest item. It's more efficient than calling heappush followed by heappop separately. This is useful for maintaining a fixed-size heap (e.g., always keeping the 'k' largest elements).
heapq.heapreplace(heap, item): Pops and returns the smallest item from the heap, and then pushes the new item. It differs from heappushpop as it pops first. This can be slightly more efficient but raises an IndexError if the heap is initially empty.

# Example using heappushpop (maintaining top 3 elements)
top_k_heap = [10, 20, 30] # Assume this is already a heap
heapq.heapify(top_k_heap) # Ensure it's a heap

# Process new items, keeping only the top 3 largest (using min-heap for smallest)
# If we model this as keeping the 'k' largest using a min-heap of size 'k',
# we push a new item and pop the smallest if the heap size exceeds 'k'.
# heappushpop is useful if the heap is already at size 'k'.

new_item = 5
if len(top_k_heap) < 3:
    heapq.heappush(top_k_heap, new_item)
elif new_item > top_k_heap[0]: # Only consider if larger than the smallest in heap
    smallest = heapq.heappushpop(top_k_heap, new_item)
    # smallest variable now holds the element that was pushed out (10 in this case)

print(f"Heap after considering {new_item}: {top_k_heap}") # Output: [20, 30, 5] -> needs heapify logic

# Example using heapreplace
data_heap = [10, 20, 30]
heapq.heapify(data_heap)

smallest = heapq.heapreplace(data_heap, 5) # Replaces 10 with 5, returns 10
print(f"Replaced item: {smallest}")
print(f"Heap after replace: {data_heap}")
# Expected output:
# Replaced item: 10
# Heap after replace: [5, 20, 30] # 5 is pushed, 10 popped. Heap property maintained.

Implementing a Max-Heap

Since heapq directly implements only a min-heap, what if you need a max-heap (where the largest element is always at the root)? The standard techniques involve modifying the values stored in the heap:

Negate Values: Store the negative of each numeric value. The smallest negated value corresponds to the largest original value. When you pop, negate the result back.
Use Tuples: Store tuples where the first element is the negated priority (or priority multiplied by -1) and the second element is the actual item. Python's tuple comparison compares elements lexicographically, so it will prioritize based on the first element (the negated priority).

# Method 1: Negating values for a max-heap
max_heap_negated = []
data = [10, 50, 20, 80, 30]

for item in data:
    heapq.heappush(max_heap_negated, -item) # Push negative value

print(f"Internal min-heap (negated): {max_heap_negated}")

largest_item = -heapq.heappop(max_heap_negated) # Pop smallest negated, negate back
print(f"Largest item popped: {largest_item}") # Should be 80
print(f"Internal heap after pop: {max_heap_negated}")

# Method 2: Using tuples (priority, item) for a max-heap
max_heap_tuples = []
data_with_ids = [('task1', 10), ('task2', 50), ('task3', 20)]

for task_id, priority in data_with_ids:
    heapq.heappush(max_heap_tuples, (-priority, task_id)) # Push (-priority, item)

print(f"Internal min-heap (tuples): {max_heap_tuples}")

neg_priority, highest_priority_task = heapq.heappop(max_heap_tuples)
print(f"Highest priority task popped: {highest_priority_task} (Priority: {-neg_priority})") # task2, 50
print(f"Internal heap after pop: {max_heap_tuples}")

Utilities for Selection Problems

heapq also provides convenient functions nlargest(k, iterable) and nsmallest(k, iterable), which efficiently find the $k$ largest or smallest items from any iterable. These functions often use heaps internally, especially when $k$ is small relative to the total number of items $n$ .

scores = [34, 78, 12, 45, 99, 67, 23, 88, 50]

top_3 = heapq.nlargest(3, scores)
bottom_3 = heapq.nsmallest(3, scores)

print(f"Scores: {scores}")
print(f"Top 3 scores: {top_3}")     # Output: [99, 88, 78]
print(f"Bottom 3 scores: {bottom_3}") # Output: [12, 23, 34]

Using heapq allows you to leverage highly efficient, built-in implementations for priority queue and heap-related tasks within your machine learning workflows, such as managing candidates in beam search, finding nearest neighbors (when combined with other structures), or implementing selection algorithms. Its performance ( $O(\log n)$ for push/pop, $O(n)$ for heapify) makes it suitable for handling large datasets where efficiency is paramount.

Python's heapq Module