Dynamic ordered sets with exponential search trees
Arne Andersson, Mikkel Thorup · Journal of the ACM · 2007
We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O (√log n /log log n ) for searching and updating a dynamic set X of n integer keys in linear space. Searching X for an integer y means finding the maximum key in X which is smaller than or equal to y . This problem is equivalent to the standard text book problem of maintaining an ordered set. The best previous deterministic linear space bound was O (log n /log log n ) due to Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space. We also get the following worst-case linear space trade-offs between the number n , the word length W , and the maximal key U < 2 W : O (min log log n + log n /log W , log log n ⋅ log log U /log log log U ). These trade-offs are, however, not likely to be optimal. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n .