Implicit B-trees: New results for the dictionary problem

We reopen the issue of finding an implicit data structure for the dictionary problem. In particular, we examine the problem of maintaining n data values in the first n locations of an array in such a way that we can efficiently perform the operations insert, delete and search. No information other than n and the data is to be retained; and the only operations which we may perform on the data values (other than reads and writes) are comparisons. Our structure supports these operations in O(log² n/log log n) time, marking the first improvement on the problem since the mid 1980´s. En route we develop a number of space efficient techniques for handling segments of a large array in a memory hierarchy. We achieve a cost of O(log_B n) block transfers like in regular B-trees, under the realistic assumption that a block stores B = Ω(log n) keys, so that reporting r consecutive keys in sorted order has a cost of O(log_Bn+r/B) block transfers. Being implicit, our B-tree occupies exactly [n/B] blocks after each update.