Implicit B-trees: a new data structure for the dictionary problem

Abstract

An implicit data structure for the dictionary problem maintains n data values in the first n locations of an array in such a way that it efficiently supports the operations insert, delete and search. No information other than that in O(1) memory cells and in the input data is to be retained; and the only operations performed on the data values (other than reads and writes) are comparisons. This paper describes the implicit B-tree, a new data structure supporting these operations in 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 B n+r/B) block transfers. En route a number of space efficient techniques for handling segments of a large array in a memory hierarchy are developed. Being implicit, the proposed data structure occupies exactly ⌈ n/ B⌉ blocks of memory after each update, where n is the number of keys after each update and B is the number of keys contained in a memory block. In main memory, the time complexity of the operations is O( log 2 n/ log log n) , disproving a conjecture of the mid 1980s.