DS-chapter4(Splay Tree,B tree)

# DS-chapter4(Splay Tree,B tree) - 4.5 Splay Trees Target Any...

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4.5 Splay Trees Target : Any M consecutive tree operations starting from an empty tree take at most O( M log N ) time. Does it mean that every operation takes O(log N ) time? No. It means that the amortized time is O(log N ). So a single operation might still take O( N ) time? Then what’s the point? The bound is weaker. But the effect is the same: There are no bad input sequences. But if one node takes O( N ) time to access, we can keep accessing it for M times, can’t we? Sure we can – that only means that whenever a node is accessed, it must be moved . Idea : After a node is accessed, it is pushed to the root by a series of AVL tree rotations.

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k 5 F k 4 E k 3 D k 2 A k 1 C B Splay Trees k 5 F k 4 E k 3 D k 2 B A k 1 C k 5 F k 4 E k 2 B A k 1 k 3 D C k 5 F k 4 E k 2 B A k 1 k 3 D C k 4 E k 5 F k 2 B A k 1 k 3 D C Does NOT work!
Splay Trees An even worse case: 1 2 2 1 3 3 2 1 Insert : 1, 2, 3, … N 3 2 1 N Find : 1 3 2 1 N Find : 2 3 1 2 N …… Find : N 3 2 1 N T ( N ) = O ( N 2 )

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§5 Splay Trees Try again -- For any nonroot node X
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DS-chapter4(Splay Tree,B tree) - 4.5 Splay Trees Target Any...

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