Sorted List vs Min-Heap: What’s Better for Top-K in Python?

When solving problems like “find the top K largest elements”, you have two common strategies:

1. Use a sorted list to maintain the top K
2. Use a min-heap (via Python’s heapq) to do the same

At first glance, it may seem both approaches are similar — especially since we’re just storing K elements — but their performance and behavior differ significantly.

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🔍 Scenario: Maintain Top 3 Elements

Say you’re scanning a list or stream of numbers like this:

[5, 6, 7]

Now you want to insert a new number: 4

Let’s compare what happens in both approaches.

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✅ Method 1: Sorted List Approach

Current list:

[5, 6, 7]

Insert 4 (maintain sort):

1. Use binary search to find the insert position → index 0 ✅ O(log K)
2. Insert 4 at index 0 ❌ requires shifting all elements right → O(K)

🔧 Result:

[4, 5, 6, 7]

If you’re only allowed to store 3 elements (Top K), you’d remove the smallest (now at the start) → back to [5, 6, 7]

⏱ Complexity:

• Insert position: O(log K)
• Actual insert (shift): O(K)
• Total: O(K)

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✅ Method 2: Min-Heap Approach

We store the smallest of the top K at the root.

Heap (min-heap) contents (internally):

[5, 6, 7]

Insert 4:

• Compare 4 with heap[0] (which is 5)
• Since 4 < 5, we don’t insert → discard ✅ O(1)

Or if 4 were bigger:

• Replace the root with 4 → heapq.heappushpop() → ✅ O(log K)

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🔧 Visual Example:

Heap Structure (before insert):

      5
     / \
    6   7

Attempt to insert 4:

• 4 < 5 → too small → discard

Attempt to insert 8:

• Replace 5 → bubble 8 down → maintain heap order

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🔁 Comparison Summary

Feature	Sorted List	Min-Heap (heapq)
Insert position search	✅ O(log K)	🚫 Not needed
Insertion (actual shift)	❌ O(K)	✅ O(log K)
Overall insert cost	❌ O(K)	✅ O(log K)
Space complexity	✅ O(K)	✅ O(K)
Maintains sorted order	✅ Yes	❌ No (only top element known)
Best for…	Small K, final output	Large streams, fast updates

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✅ Final Thoughts (Answering Your Original Doubt)

You might ask :

“If we already have a sorted list of K items, shouldn’t insert be fast?”

The answer is:

• Searching where to insert is fast (O(log K) via binary search)
• But inserting itself is costly (O(K)), because all items after the insert point must shift right in memory (Python uses arrays, not linked lists)

On the other hand, min-heaps:

• Allow fast insertions (O(log K))
• Only track the top K, not full sort
• Are ideal when you don’t need full ordering — just the top results efficiently

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🧪 Optional Code (Min-Heap Top K Tracker)

import heapq

class TopKHeap:
    def __init__(self, k):
        self.k = k
        self.heap = []

    def add(self, num):
        if len(self.heap) < self.k:
            heapq.heappush(self.heap, num)
        elif num > self.heap[0]:
            heapq.heappushpop(self.heap, num)

    def get_top_k(self):
        return sorted(self.heap, reverse=True)

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✅ In Summary

Use a min-heap (heapq) when:

• You’re processing a large stream of data
• You need fast, efficient top-K tracking
• You don’t need full sorting, just the top few items

Use a sorted list when:

• K is small
• You need the list to stay sorted
• You don’t have frequent updates