Why Heaps Are Often Implemented as Arrays
🧠 What Is a Heap?
A heap is a binary tree with two strict rules:
-
✅ Shape Property: It's a complete binary tree
(Every level is fully filled left-to-right, except possibly the last) -
✅ Heap Property: Each parent obeys a priority rule:
- In a min-heap, every parent is ≤ its children
- In a max-heap, every parent is ≥ its children
This makes heaps partially ordered — you’re guaranteed the min or max at the top, but no guarantee among siblings, cousins, or other nodes.
🔽 Min-Heap Example
Array Representation:
Index: 0 1 2 3 4 5
Value: [1, 3, 5, 7, 9, 6]
- ✅ Every parent is ≤ its children
- ❌ The array is not sorted
- ❌ No guarantee that 3 < 5, or 6 < 7, etc.
That’s partial ordering in action.
🔧 What Are Heaps Good For?
Heaps are the engine behind priority queues, and shine when you want quick access to the best item — without fully sorting the data.
Real-world use cases:
| Use Case | Why Heap? |
|---|---|
| Streaming min/max tracking | Always get current min/max in O(1) |
| Top K elements | Keep best K values without sorting everything |
| Dijkstra’s / A* | Always expand next shortest path |
| Task scheduling / simulations | Always process the next-most-urgent item |
| Median of a stream | Combine min/max heaps for balance |
🧱 Heaps Are Perfectly Shaped for Arrays
Heaps have a complete tree shape — which means you can pack them tightly into an array with no gaps.
You don’t need to store pointers or nodes — just use index math to navigate the tree:
| Relationship | Formula |
|---|---|
| Parent(i) | floor((i - 1) / 2) |
| Left(i) | 2i + 1 |
| Right(i) | 2i + 2 |
This makes array-based heaps:
- Efficient
- Easy to implement
- Space-compact
🚫 Why Not Use Tree Nodes?
A regular binary tree can look like this:
This shape can’t be stored tightly in an array — you’d waste a ton of space:
Index: 0 1 2 ... 1000+
Value: [1, _, 2, _, _, _, _, _, _, _, 3, ...]
Worse: You’d have to use pointers or objects to manage the structure.
🤯 Bonus: Heap Arrays Mirror BFS Traversal
If you perform a level-order (BFS) traversal of a heap, you'll get its array representation.
Array: [10, 20, 30, 40, 50, 60, 70]
No need to convert between structures — it's already lined up.
🐍 Python Example (Min-Heap)
Python’s heapq module uses a list as the heap:
import heapq
h = [7, 5, 11, 3, 24]
heapq.heapify(h) # now h is a valid min-heap
heapq.heappush(h, 2)
print(heapq.heappop(h)) # 2 (smallest element)
Note: heapify() modifies the list in place and returns nothing.
Python only supports min-heaps. For max-heaps, invert the values (-x trick).
TL;DR
- ✅ Heaps are complete binary trees with partial ordering
- ✅ They guarantee fast access to min or max
- ✅ Their tight shape makes them perfect for arrays
- ❌ Using tree nodes would add complexity and waste
- ✅ Python’s
heapquses this exact idea — functional, fast, but minimal