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Amortized Analysis Examples

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Queue Enqueue with Two Stacks

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Amortized complexity is O(1) for Enqueue operation, due to the two-stack approach, wherein the cost of reversing the stack is spread out over multiple Enqueue operations, making the average cost per operation constant.

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Binary Heap Insertion

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Amortized complexity of insertion in a binary heap is O(1), because while the worst-case time complexity is O(log n) for bubble up operation, the insertion itself is O(1) and the bubble up does not always traverse the entire height of the tree.

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Splay Tree Access

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Amortized complexity for splaying in a splay tree is O(log n). Splaying brings the accessed node to the root of the tree, and while single operation might need O(n) in the worst case, the splay tree properties ensure that frequent operations adjust the tree in such a way that keeps the average access time logarithmic.

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Array Doubling

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Amortized complexity is O(1) because although the worst-case scenario of inserting an element when the array is full is O(n), the cost is spread out over the n insertions that filled the array, resulting in a constant average cost per insertion.

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Incrementing a Binary Counter

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Amortized complexity is O(1) for incrementing a binary counter because the frequent O(1) operations dominate the occasional O(n) operation when all bits flip from 1 to 0.

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