Implicit Dictionaries with O(1) Modifications per Update and Fast Search

The implicit dictionary problem is that of maintaining a dynamic ordered set, S, under the operations search, insert and delete, so that the elements of S are stored in the first |S| locations of an array. No operations are permitted on the data other than comparisons (≤) and interchanges. The only auxiliary memory permitted is a constant number of O(log |S|) bit integers. The organization will, then, rely heavily on the permutations of the relative order of the values in which the data is stored. While such a structure can be maintained in O(log |S|) time, the most interesting lower bound on the topic is that of Borodin, Fich, Meyer auf der Heide, Upfal and Wigderson [3]. They proved a tradeoff between search and update time in implicit dictionaries: if the update cost (comparisons and exchanges) is O(1), then the search cost must be Ω(|S| ε), for some constant ε ≥ 0. The authors left open the question of whether such a tradeoff would hold if only the modifications performed during an update were considered. They conjectured that any implicit dictionary performing only O(1) exchanges per update should very quickly become "disorganized", and so require Ω(|S| ε) comparisons per search. We answer this long-standing open question by disproving the conjecture.