Heaps and BSTs are both trees, but their different rules lead to different uses.
- BST Property:
left_child < parent < right_child. This strict, total ordering allows for efficient searching, insertion, and deletion, all in \(O(\log n)\) time on average for balanced trees. - Heap Property:
parent > children(for a max-heap). This partial ordering is less strict. Its only goal is to keep the maximum element at the root for fast access. - You cannot search efficiently in a heap. Finding an arbitrary element requires checking nearly all nodes, an \(O(n)\) operation, as the heap property provides no guidance on which branch to follow.
Can a BST Implement a Priority Queue?
Yes, a balanced BST can implement a priority queue. Operations like insert and extractMax (finding and removing the rightmost node) are both \(O(\log n)\). So, which is better for a Priority Queue: a Heap or a Balanced BST?
While both are theoretically efficient, heaps are almost always preferred for implementing priority queues. Heaps have lower overhead, better cache locality (when using an array), and faster `build` and `peek` operations. The table below breaks down the comparison.
| Operation / Feature | Heap (Array-based) | Balanced BST | Winner for PQ |
|---|---|---|---|
| Insert | \(O(\log n)\) | \(O(\log n)\) | Heap |
| Get Max (Peek) | \(O(1)\) | \(O(\log n)\) | Heap |
| Extract Max | \(O(\log n)\) | \(O(\log n)\) | Heap |
| Build from Array | \(O(n)\) | \(O(n \log n)\) | Heap |
| Find Arbitrary Element | \(O(n)\) | \(O(\log n)\) | BST |
| Ordered Traversal | \(O(n \log n)\) (Heap Sort) | \(O(n)\) (In-order) | BST |
| Memory / Cache Locality | Excellent (Array) | Poor (Pointers) | Heap |