Algorithms / Heap Invariants
Least You Need to Know: Heaps, Heap Invariants, and Fast Extremes
A heap stores items so the **root is always the minimum or maximum** according to a priority rule. It does not fully sort everything, but it lets you update the frontier quickly when you repeatedly need the next best element.
جو کم از کم جاننا ضروری ہے
- A heap is a complete binary tree stored compactly in an array.
- The heap property compares each node only with its children, not with every later element.
- Push and pop take logarithmic time because only one root-to-leaf path may need repair.
- A heap is ideal when you repeatedly need the smallest or largest current item.
- A heap is not the same as a fully sorted array.
اہم علامتیں
min-heap
smallest key at the root
max-heap
largest key at the root
sift up / sift down
restore heap property after insert or remove
مختصر حل شدہ مثال
- Put incoming tasks into a min-heap keyed by priority.
- The root is always the next task with the smallest key.
- After removing the root, move the last element to the top and sift down.
- Only one path changes, so the repair is logarithmic rather than linear.
عام غلطیاں
- Students often think a heap means the entire array is globally sorted.
- Students often forget that only parent-child order is guaranteed.
- Students often choose a heap when they only need one final sort and no repeated frontier updates.
اس قسم کے سوال کو کیسے پہچانیں
- You repeatedly need the smallest or largest current element.
- Items are arriving, expiring, or being updated over time.
- A full sort each step would be wasteful.
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