Is there a algorithm to perform insertions into a heap with atmost one swap (O(log n) comparisons are allowed)
Insertions into a max heap with atmost one swap
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No.
Consider this heap:
Suppose you add 200. Obviously it will have to become the new root.
So where does 100 go? It can't become a child of 3, and that's what it would have to do if you only have one swap.