The dynamic optimality conjecture states that splay trees have constant competitive ratio, but this remains unproven.
2.
This improved upon the previous best known competitive ratio, which was O ( \ log n ).
3.
If no additional restrictions on the graph are given, the optimal competitive ratio is only slightly sublinear.
4.
So every conservative algorithm attains the \ dfrac { k } { k-h + 1 }-competitive ratio.
5.
Papadimitriou ( 1995 ) proved that Work Function Algorithm ( WFA ) has competitive ratio 2 " k "-1.
6.
While this is not dynamically optimal, the competitive ratio of \ log \ log n is still very small for reasonable values of n.
7.
However, for interval graphs, a constant competitive ratio is possible, while for bipartite graphs and sparse graphs a logarithmic ratio can be achieved.
8.
The best online competitive ratio for the search on the line is 9 but it can be reduced to 4.6 by using a randomized strategy.
9.
However, despite the efforts of many other researchers, reducing the competitive ratio to " k " or providing an improved lower bound remains open.
10.
The splay tree is conjectured to have a constant competitive ratio compared to the dynamically optimal tree in all cases, though this has not yet been proven.
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