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I want to try trying executing the worst-case sequence on Splay tree.

But what is the worst-case sequence on Splay-trees? And are there any way to calculate this sequence easily given the keys which is inserted into the tree?

Any can help me with this?

templatetypedef
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Theis F. Hinz
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2 Answers2

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Unless someone corrects me, I'm going to go with "no one actually knows what the worst-case series of operations is on a splay tree or what the complexity is in that case."

While we do know many results about the efficiency of splay trees, we actually don't know all that much about how to bound the time complexity of a splay tree. There's a conjecture called the dynamic optimality conjecture that says that in the worst case, any sufficiently long series of operations on a splay tree will take no more than a constant amount of time more than the best possible self-adjusting binary search tree on that series of operations. One of the challenges we're having in trying to prove this is that no one actually knows how to determine the cost of the best possible BST on all inputs. Another is that finding upper bounds on the runtimes of various input combinations to splay trees is hard - as of now, no one knows whether it takes time O(n) to treat a splay tree as a deque!

Hope this helps!

templatetypedef
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I don't know if an attempt of an answer after more than five years is of any use to you, but sorry, I made my Master in CS only recently :-) In the wake of that, I played around exactly with your question. Consider the sequence S(3,2) (it should be obvious how S(m,n) works generally if you graph it): S(3,2)=[5,13,6,14,3,15,4,16,1,17,2,18,11,19,12,20,9,21,10,22,7,23,8,24]. Splay is so lousy on this sequence type that the competive ratio r to the "Greedy Future" algorithm (see Demaine) is S[infty,infty]=2. I was never able to get over 2 even though Greedy Future is also not completely optimal and I could shave off a few operations. enter image description here (Legend: black,grey,blue: S(7,4); purple,orange,red: Splay must access these points too. Shown in the Demaine formulation.) But note that your question is somewhat ill defined! If you ask for the absolutely worst sequence, then take e.g. the bit-reversal sequence, ANY tree algorithm needs O(n log n) for that. But if you ask for the competetive ratio r as implied in templatetypdef's answer, then indeed nobody knows (but I would make bets on r=2, see above). Feel free to email me for details, I'm easily googled.

Hauke Reddmann
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