Predecessor queries in dynamic integer sets

Abstract. We consider the problem of maintaining a set of n integers in the range 0::2 w 1 under the operations of insertion, deletion, pre-decessor 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 mini-mum 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.