Predecessor Queries in Dynamic Integer Sets

We consider the problem of maintaining a set of n integers in the range 0::2 w ? 1 under the operations of insertion, deletion, predecessor queries, minimum queries and maximum queries on a unit cost RAM with word size w bits. Let f (n) be an arbitrary nondecreasing smooth function satisfying log log n f (n) p log n. A data structure is presented supporting insertions and deletions in worst case O(f (n)) time, predecessor queries in worst case O((log n)=f (n)) time and minimum and maximum queries in worst case constant time. The required space is O(n2 w) for an arbitrary constant > 0. The RAM operations used are addition, arbitrary left and right bit shifts and bit-wise boolean operations. The data structure is the rst supporting predecessor queries in worst case O(log n= log log n) time while having worst case O(log log n) update time.