Heap: The Data Structure Almost Every Developer Ignores (Until They Need to Solve a Hard Problem)

Most developers spend years writing code without ever implementing a Heap.

Yet Heaps are everywhere: routing systems, search engines, databases, operating systems, graph algorithms, task processing queues, video games, artificial intelligence, and virtually every platform that needs to quickly decide which element should be processed next.

The interesting part is that many developers are familiar with arrays, linked lists, stacks, queues, trees, and hash tables, but have never truly studied Heaps.

Until the day a technical interview or a LeetCode problem practically screams for one.

At that point, trying to solve the problem using only arrays usually leads to solutions that are either too slow or unnecessarily complex.

In this article, we'll take an in-depth look at how this data structure works, its different variants, its time complexities, and how to use it in PHP, always with practical examples and real-world applications.

What Is a Heap?

A Heap is a data structure based on a complete binary tree whose primary purpose is to keep the element with the highest (or lowest) priority readily accessible at all times.

Unlike a Binary Search Tree (BST), where each node maintains an ordering relationship with its entire subtree, a Heap only guarantees a local relationship between parent and child nodes. This property is enough to ensure that the most important element is always at the root, making operations such as retrieving or removing it extremely efficient.

In other words, a Heap does not keep all elements sorted. Instead, it keeps the highest-priority element in the correct position.

This simple idea makes the Heap one of the most widely used data structures in Computer Science.

It appears in classic algorithms such as:

• Dijkstra's shortest path algorithm
• Prim's minimum spanning tree algorithm
• A* search
• Heap Sort
• Merge K Sorted Lists
• Kth Largest Element
• Median Finder

Real-world systems also rely on Heaps to implement:

• print queues;
• operating system process schedulers;
• asynchronous task queues;
• ranking systems;
• search engines;
• recommendation algorithms;
• event simulators.

Whenever your application needs to answer the question:

"Which element should be processed next?"

there is a good chance that a Heap is the most appropriate data structure.

Heap Does Not Mean Heap Memory

When researching this topic, you'll often come across references to Heap Memory. Despite the similar name, it has nothing to do with the Heap data structure.

These are two completely different concepts.

Heap (Data Structure)

A Heap is a data structure used to organize elements according to their priority.

Its goal is to efficiently support operations such as:

• inserting elements;
• removing the largest or smallest element;
• quickly retrieving the highest-priority element.

It can be implemented using arrays, trees, or other underlying structures.

Heap Memory

Heap Memory, on the other hand, is part of a programming language's memory management system.

It is where dynamically allocated objects and data structures are typically stored.

For example, in PHP:

class User
{
    public string $name = 'Thiago';
}

$user = new User();

The User object is allocated in the language's heap memory, while the $user variable only stores a reference to that object.

Likewise, in Python:

numbers = [10, 20, 30]

The list is allocated in the interpreter's heap memory.

Notice that this does not mean the list is organized as a Heap data structure. It simply resides in the region of memory reserved for dynamically allocated objects.

In summary:

Although they share the same name, these two concepts serve completely different purposes.

The Heap Property

Every Heap follows a very simple rule known as the Heap Property.

This property defines only the relationship between a node and its immediate children.

Depending on the type of Heap, there are two possibilities.

In a Max Heap

The value of a parent is always greater than or equal to the values of its children.

For example:

          100
         /   \
       80     60
      / \    / \
    40 50  20 10

Notice that:

• 100 is greater than 80 and 60;
• 80 is greater than 40 and 50;
• 60 is greater than 20 and 10.

Nothing else is guaranteed.

For example, can 50 be greater than 60? No.

But can 40 be greater than 20? Yes.

That does not violate any Heap rule.

The Heap Property only concerns parent-child relationships.

In a Min Heap

The reasoning is exactly the opposite.

Each parent is always less than or equal to its children.

          10
         /  \
       20    30
      / \    / \
    40 50  60 80

Again, there is no guarantee about the ordering of elements that belong to different branches.

This is an important characteristic.

A Heap is not a sorted collection.

If we iterate over its elements in storage order, we'll most likely see what appears to be a random sequence.

The only element whose position is always guaranteed is the root.

It is precisely this property that allows us to retrieve the largest (or smallest) element in constant time (O(1)), without paying the cost of keeping the entire collection sorted.

Min Heap

In a Min Heap, the smallest element in the structure is always kept at the root.

This means retrieving the minimum value is an extremely fast operation.

For example:

          5
        /   \
      12     18
     / \     / \
   30 25   40 50

The smallest element is always 5.

If we insert a new element with the value 2, it is initially placed in the next available position in the tree. It is then compared with its parent and, if necessary, repeatedly swapped upward until the Heap Property is restored.

This process is known as Heapify Up (or Sift Up).

Thanks to this mechanism, insertion has a time complexity of O(log n).

Min Heaps are widely used in problems that repeatedly need to retrieve the smallest available element, such as:

• Dijkstra's algorithm;
• Prim's algorithm;
• Merge K Sorted Lists;
• task schedulers;
• event simulators;
• systems that process items with the lowest numerical priority first.

In Python, for example, the standard heapq module implements a Min Heap, making this type of data structure extremely common in the language.

Max Heap

A Max Heap works in the opposite way.

The largest element is always kept at the root.

For example:

          90
        /    \
      70      60
     / \      / \
   40 30    20 10

Whenever we need the largest element in the collection, we simply look at the root.

This operation runs in O(1) time.

When the largest element is removed, the Heap automatically reorganizes itself through a process known as Heapify Down (or Sift Down), restoring the Heap Property in O(log n) time.

Max Heaps are commonly used in scenarios such as:

• leaderboards;
• game scoreboards;
• recommendation systems;
• priority management;
• selecting the largest values from a collection;
• algorithms that repeatedly need the next largest element.

In PHP, the SplMaxHeap class provides a ready-to-use implementation of this data structure, while SplPriorityQueue supports arbitrary priorities, making it an even more flexible option for many use cases.

A Heap Is a Tree... But Not Exactly

When we say that a Heap is a tree, that is true only from a conceptual perspective.

In practice, almost every implementation stores its elements in an array, without creating node objects or maintaining pointers to parents and children.

This is possible because a Heap has a very important characteristic: it is always a complete binary tree.

A complete binary tree is one in which:

• every level is completely filled, except possibly the last one;
• the last level is filled from left to right.

For example:

          10
        /    \
      20      30
     /  \    /
   40   50  60

This organization eliminates "gaps" in the tree, allowing all nodes to be stored sequentially in an array.

This characteristic is precisely what makes Heaps so memory-efficient.

While a traditional binary tree usually requires each node to store references to its parent, left child, and right child, a Heap only requires an array.

As a result, it:

• uses less memory;
• has better cache locality;
• performs fewer object allocations;
• generally delivers better performance in intensive workloads.

This is one of the reasons why virtually every programming language and standard library implements Heaps using arrays.

How a Heap Is Stored in an Array

Consider the following Max Heap:

          90
        /    \
      70      60
     /  \    /  \
   40   30 20   10

Internally, it can be stored as:

[90, 70, 60, 40, 30, 20, 10]

Notice that there is no explicit information indicating which node is the parent or the child of another.

Those relationships are calculated entirely from the element's position in the array.

If an element is located at index i, then:

• parent:

(i - 1) / 2

• left child:

2 * i + 1

• right child:

2 * i + 2

For example, consider index 1:

Index: 0   1   2   3   4   5   6
Value: 90 70 60 40 30 20 10

The element at index 1 has the value 70.

Its children are:

left child  = 2 × 1 + 1 = 3
right child = 2 × 1 + 2 = 4

Therefore:

70
├── 40
└── 30

Likewise, the parent of the element at index 4 is:

(4 - 1) / 2 = 1

That means its parent is the element stored at index 1.

These formulas make navigating a Heap extremely efficient while completely eliminating the need for pointers.

Insertion (Heapify Up)

Inserting an element into a Heap is a straightforward process.

First, the new element is placed in the next available position in the array.

This preserves the complete binary tree property.

Suppose we have the following Max Heap:

          90
        /    \
      70      60
     /  \
   40   30

Represented as:

[90, 70, 60, 40, 30]

Now let's insert the value 80.

Initially, it is appended to the end of the array:

[90, 70, 60, 40, 30, 80]

The tree now looks like this:

          90
        /    \
      70      60
     /  \    /
   40   30 80

At this point, the Heap Property has been violated.

The value 80 is larger than its parent (60).

So they are swapped:

          90
        /    \
      70      80
     /  \    /
   40   30 60

Now the new parent is 90.

Since 80 is smaller than 90, the process stops.

This upward movement toward the root is called Heapify Up (also known as Sift Up or Bubble Up).

In the worst case, the element moves from the last leaf all the way to the root.

Because the height of a Heap is log₂(n), insertion has a time complexity of:

O(log n)

Removal (Heapify Down)

Removing an element from a Heap usually means removing its root, since that is where the element with the highest (or lowest) priority is stored.

Consider the following Max Heap again:

          90
        /    \
      70      80
     /  \    /
   40   30 60

When removing 90, we cannot simply delete the root, as that would leave a gap in the tree.

Instead, we move the last element to the root.

The tree becomes:

          60
        /    \
      70      80
     /  \
   40   30

At this point, the Heap Property has been violated.

The value 60 is smaller than both of its children.

It is therefore compared with each of them.

Since 80 is the larger child, the two elements are swapped:

          80
        /    \
      70      60
     /  \
   40   30

At this point, the Heap once again satisfies its property.

This process of moving an element downward until it reaches its correct position is called Heapify Down (or Sift Down).

Just like insertion, the element travels at most the height of the tree.

As a result, removing the root also has a time complexity of:

O(log n)

Time Complexity

One of the greatest strengths of a Heap is that it provides very efficient operations for managing prioritized elements.

At first glance, one result often surprises people: why does building an entire Heap take O(n) instead of O(n log n)?

The answer lies in the algorithm used.

A naive approach would insert the elements one by one into an initially empty Heap. Since each insertion costs O(log n), the total cost would indeed be O(n log n).

However, there is a more efficient algorithm known as Floyd's Heap Construction, or simply Build Heap, which starts with an array that is already populated and reorganizes its elements from the bottom up.

Instead of performing a Heapify Up operation for every inserted element, it performs Heapify Down only on the internal nodes, starting from the last parent and working its way up to the root.

Since most nodes are located near the bottom of the tree—where they require only a few swaps—the total amount of work grows linearly with the number of elements.

This is a classic result in algorithm analysis and one of the most elegant properties of the Heap data structure.

In practice, this means that when all your data is already available in an array, it is significantly more efficient to build the Heap in one step than to insert each element individually. The performance difference becomes increasingly noticeable as the dataset grows.

Heap vs. Sorted Array

At first glance, a Heap and a sorted array may seem to solve the same problem: both allow you to quickly retrieve the largest or smallest element. However, their characteristics are quite different.

In a sorted array, the elements remain ordered at all times.

[10, 20, 30, 40, 50, 60, 70]

This makes retrieving the smallest or largest element trivial, depending on the sorting order.

The problem arises when we need to insert or remove elements.

Imagine inserting the value 35 into the array above.

[10, 20, 30, 35, 40, 50, 60, 70]

Every element after the insertion point must be shifted one position to the right.

Likewise, removing an element requires shifting multiple values to fill the resulting gap.

As a consequence, although lookups are extremely fast, updates typically cost O(n).

With a Heap, only the parent-child relationship is maintained.

The internal representation may look "unsorted":

[70, 50, 60, 20, 40, 30, 10]

Even so, the largest element is still immediately available at the root.

Because only a few elements need to move during insertions and removals, these operations run in O(log n) time.

The comparison can be summarized as follows:

Therefore, if your application frequently needs to iterate over every element in sorted order, a sorted array is the better choice.

On the other hand, if you repeatedly need to access only the highest- or lowest-priority element while continuously inserting and removing values, a Heap is usually much more efficient.

Heap vs. Binary Search Tree

Another common comparison is between a Heap and a Binary Search Tree (BST).

Although both are binary trees, they were designed to solve very different problems.

A BST organizes its elements so that:

• every value in the left subtree is smaller than the current node;
• every value in the right subtree is larger than the current node.

For example:

          50
        /    \
      30      70
     /  \    /  \
   20   40 60   80

This organization enables efficient searching.

A Heap, however, has no such property.

Consider the following Max Heap:

          80
        /    \
      70      60
     /  \    /  \
   20   40 30   50

Notice that the value 50 is in the right subtree of 60, even though it is smaller.

This is perfectly valid because a Heap was never designed for searching.

Its only guarantee is that each parent satisfies the Heap Property with respect to its immediate children.

As a result, finding an arbitrary value in a Heap generally requires scanning most—or even all—of its elements.

The comparison becomes clearer in the following table:

* In balanced trees such as AVL Trees and Red-Black Trees. A regular BST can degrade to O(n) in the worst case.

In summary:

• choose a Heap when your primary goal is to repeatedly retrieve the largest or smallest element as efficiently as possible;
• choose a BST when your application performs many searches for specific values or frequently traverses the elements in sorted order.

Priority Queue

A Priority Queue is an abstract data type whose behavior differs from that of a traditional queue.

In a regular queue (FIFO – First In, First Out), the first element inserted is always the first one removed.

A → B → C → D

The processing order depends solely on when each element entered the queue.

In a priority queue, however, each element has an associated priority.

The element with the highest priority is processed first, regardless of its insertion order.

For example:

The execution order will be:

1. Process payment
2. Generate report
3. Backup
4. Send email

Notice that "Process payment" was the third task inserted, yet it is executed first because it has the highest priority.

This is precisely where a Heap comes into play.

The most common implementation of a Priority Queue uses a Heap internally.

This allows us to:

• insert new elements in O(log n);
• remove the highest-priority element in O(log n);
• peek at the next element in O(1).

In practice, you'll often use a Heap without even realizing it, as many libraries expose a Priority Queue implementation built on top of this data structure.

Heap in PHP

PHP provides built-in Heap implementations through the SPL (Standard PHP Library).

In most real-world applications, there is no need to implement a Heap from scratch.

The main classes are:

• SplMinHeap
• SplMaxHeap
• SplPriorityQueue

Although they have similar behavior, each one was designed to solve a different kind of problem.

SplMinHeap

Always keeps the smallest element at the top.

$heap = new SplMinHeap();

$heap->insert(30);
$heap->insert(10);
$heap->insert(50);
$heap->insert(20);

echo $heap->extract(); // 10
echo $heap->extract(); // 20

Elements are removed in ascending order.

SplMaxHeap

Works in the opposite way.

$heap = new SplMaxHeap();

$heap->insert(30);
$heap->insert(10);
$heap->insert(50);
$heap->insert(20);

echo $heap->extract(); // 50
echo $heap->extract(); // 30

Now the elements are extracted from largest to smallest.

SplPriorityQueue

This is the most flexible implementation.

Instead of storing only values, it allows you to associate an independent priority with each element.

$queue = new SplPriorityQueue();

$queue->insert('Send email', 1);
$queue->insert('Generate report', 3);
$queue->insert('Process payment', 10);

echo $queue->extract(); // Process payment

Notice that the returned value is not the priority itself, but rather the element with the highest priority.

This class is especially useful when the data has an associated weight that determines its processing order.

Practical PHP Examples

Once you understand how a Heap works, it becomes much easier to recognize situations where it greatly simplifies the solution.

Finding the Top 10 Largest Numbers

Imagine you need to find the ten largest values in a file containing millions of numbers.

A naive solution would load everything into memory and sort the array:

sort($numbers);

$top10 = array_slice($numbers, -10);

Besides sorting the entire dataset (O(n log n)), this approach requires keeping every number in memory.

With a Min Heap limited to ten elements, you only need to keep track of the ten largest values found so far.

Each new number is compared with the smallest element currently stored in the Heap. If it is larger, it replaces that element.

As a result, memory consumption remains constant regardless of the size of the input file.

Real-Time Leaderboards

Imagine an online game that continuously displays the top 100 players.

Whenever a player's score changes, you simply update the Heap responsible for the leaderboard.

There is no need to sort the entire list of players again.

Task Scheduling

Processing queues commonly rely on priorities.

For example:

$queue = new SplPriorityQueue();

$queue->insert('Send invoice', 2);
$queue->insert('Generate invoice', 5);
$queue->insert('Refresh cache', 1);

while (!$queue->isEmpty()) {
    echo $queue->extract() . PHP_EOL;
}

Output:

Generate invoice
Send invoice
Refresh cache

The most important tasks are always processed first.

Event Processing

Simulation systems, game engines, and trading platforms often need to execute events in chronological order.

Each event receives its scheduled execution time as its priority.

The Heap guarantees that the next event is always immediately available while keeping insertions and removals efficient, even when millions of events are waiting to be processed.

In all of these scenarios, the primary advantage of a Heap is that it avoids repeatedly sorting the entire collection. Instead, it maintains only the information necessary to ensure that the next highest- or lowest-priority element is always available with minimal computational cost.

The Equivalent in Python

Unlike PHP, which provides ready-to-use classes such as SplMinHeap and SplMaxHeap, Python's standard library offers only the heapq module, which implements a Min Heap.

The Heap is built on top of a regular list.

import heapq

heap = []

heapq.heappush(heap, 30)
heapq.heappush(heap, 10)
heapq.heappush(heap, 50)
heapq.heappush(heap, 20)

print(heap)
# [10, 20, 50, 30]

print(heapq.heappop(heap))
# 10

print(heapq.heappop(heap))
# 20

Although the list appears to be partially sorted, it is not fully sorted.

The only guarantee is that the smallest element is always located at the first position (heap[0]).

Simulating a Max Heap

A common question among Python developers is:

How can I create a Max Heap if heapq only implements a Min Heap?

The traditional solution is to store negative values.

import heapq

heap = []

heapq.heappush(heap, -30)
heapq.heappush(heap, -10)
heapq.heappush(heap, -50)
heapq.heappush(heap, -20)

print(-heapq.heappop(heap))
# 50

print(-heapq.heappop(heap))
# 30

Since the smallest negative number corresponds to the largest positive number, the Min Heap behaves exactly like a Max Heap.

This technique is widely used in competitive programming, technical interviews, and published solutions to LeetCode problems.

Classic Problems Solved with Heaps

Heaps appear so frequently in technical interviews that learning to recognize when one should be used is often more valuable than knowing how to implement one from scratch.

Whenever a problem involves repeatedly retrieving the largest or smallest element from a dynamic collection, a Heap is worth considering.

Some classic examples include:

Kth Largest Element

Find the kth largest element in a collection.

Instead of sorting the entire array, simply maintain a Min Heap of size k.

Complexity:

• Time: O(n log k)
• Space: O(k)

Top K Frequent Elements

Given a collection of values, find the k most frequent ones.

After counting occurrences using a map (HashMap or dict), a Heap keeps track of only the most relevant elements.

Merge K Sorted Lists

A well-known LeetCode problem.

Instead of continuously comparing all lists, a Heap stores the smallest element from each one.

Whenever the smallest element is removed, the next element from the same list is inserted into the Heap.

This strategy significantly reduces the overall complexity of the solution.

Median Finder

Another classic interview problem.

The most efficient solution uses two Heaps:

• a Max Heap to store the lower half of the values;
• a Min Heap to store the upper half.

This allows the median to be retrieved in constant time after each insertion.

Dijkstra's Algorithm

Perhaps the most famous Heap application.

During execution, the algorithm repeatedly needs to determine which vertex currently has the shortest known distance.

A Min Heap makes this operation highly efficient.

Prim's Algorithm

Used to compute a minimum spanning tree.

Just like Dijkstra's algorithm, it relies on a Heap to quickly select the next edge with the lowest cost.

A* Search

The A* algorithm uses a priority queue to repeatedly select the most promising state during the search.

It is widely used in game development, robotics, and navigation systems.

Task Scheduling

Application servers, operating systems, and asynchronous processing platforms frequently rely on Heaps to determine which task should be executed next.

In practice, whenever a problem includes concepts such as:

• priority;
• minimum;
• maximum;
• next;
• Top K;
• ranking;
• scheduling;
• scheduler;
• processing queue;

there is a good chance that a Heap is an excellent solution.

Heap Sort

Heap Sort is a sorting algorithm built directly on top of the Heap data structure.

It works in two stages:

1. build a Max Heap from the input data;
2. repeatedly remove the largest element, placing it at the end of the array.

For example:

[8, 3, 6, 1, 9, 5]

After building the Heap:

[9, 8, 6, 1, 3, 5]

The largest element is then moved to the end of the array, the Heap is rebuilt for the remaining elements, and the process continues until the entire array is sorted.

The main characteristics of Heap Sort are:

One of Heap Sort's biggest advantages is that its running time remains O(n log n) regardless of the input, unlike Quick Sort, which can degrade to O(n²) in certain scenarios.

However, in practice, Heap Sort is often slower than Quick Sort because it performs more non-sequential memory accesses, resulting in poorer CPU cache utilization.

Since Heap Sort is a rich topic in its own right, we'll dedicate an entire article to it in the future.

Common Mistakes

Developers who are learning about Heaps often make the same mistakes.

Assuming the Heap Is Fully Sorted

This is probably the most common misconception.

A Heap is not a sorted array.

[100, 70, 90, 20, 60, 40, 80]

This array represents a perfectly valid Heap, even though it is clearly neither sorted in ascending nor descending order.

Calling sort() Before Solving the Problem

Many problems can indeed be solved by sorting the entire array.

While that approach works, it often produces a solution that is slower than necessary.

If all you need is the largest element, the smallest element, or the k largest elements, there is usually no reason to sort the entire collection.

Ignoring Time Complexity

Inserting an element into a Heap costs O(log n).

Sorting the entire collection repeatedly costs O(n log n).

The difference may seem insignificant for small datasets, but it becomes substantial as the amount of data grows.

Forgetting to Restore the Heap

After removing the root or inserting a new element, the Heap Property must be restored.

Otherwise, the structure is no longer a valid Heap.

Choosing the Wrong Heap

Surprisingly often, developers implement a Max Heap when the algorithm actually requires a Min Heap—or vice versa.

Before writing any code, always ask yourself:

Do I need to repeatedly retrieve the smallest element or the largest one?

The answer almost always determines which type of Heap you should use.

When NOT to Use a Heap

Although a Heap is an extremely useful data structure, it is not the ideal solution for every problem.

Avoid using one in the following situations:

You Need to Search for Specific Elements

Finding an arbitrary value in a Heap requires traversing most—or even all—of the structure.

In this scenario, a balanced tree or a hash table is usually a much better choice.

All Elements Must Remain Sorted

If your application frequently needs to iterate over data in ascending or descending order, data structures such as balanced trees—or even a sorted array—may be more appropriate.

A Heap only guarantees that the highest- (or lowest-) priority element is in the correct position.

The Dataset Rarely Changes

If your data is loaded once and remains unchanged, simply sorting it may be much simpler than maintaining a Heap.

The Dataset Is Small

For collections containing only a few dozen elements, the performance difference is usually negligible.

In these cases, a simpler solution may provide better readability without any noticeable impact on performance.

As with almost every data structure, the best choice depends entirely on the problem you are trying to solve.

Conclusion

At first glance, a Heap may seem like a complex data structure, especially because it is often introduced as a binary tree.

However, its core idea is surprisingly simple: ensure that the element with the highest or lowest priority is always available at the root.

This single property is enough to provide insertions and removals in O(log n) and retrieval of the highest-priority element in O(1), making the Heap one of the most efficient data structures for priority-based operations.

Throughout this article, we've seen that a Heap:

• does not keep all elements sorted;
• is typically stored in an array rather than as a tree of linked nodes;
• serves as the foundation for Priority Queues;
• is used by classic algorithms such as Dijkstra, Prim, A*, and Heap Sort;
• has built-in implementations in both PHP (SplMinHeap, SplMaxHeap, and SplPriorityQueue) and Python (heapq).

More important than memorizing how to implement a Heap is learning to recognize the situations where it is the right tool. In technical interviews, programming competitions, and real-world software, quickly identifying that a problem revolves around priorities is often the key to finding an efficient solution.

Finally, even if your programming language already provides a Heap implementation, it is well worth implementing one yourself at least once. Doing so helps you understand how operations such as Heapify Up and Heapify Down work, why they run in O(log n) time, and—most importantly—develops the intuition needed to choose the right data structure when solving new problems.

Once you truly understand how a Heap works, you'll begin to see many problems from a different perspective. Solutions that once seemed to require full sorting or sophisticated algorithms can often be reduced to just a few operations on a priority queue—and that shift in thinking is one of the greatest benefits of studying data structures.