Optimal Solutions for the Temporal Precedence Problem

Abstract. In this paper we refer to the Temporal Precedence Problem on Pure Pointer Machines . This problem asks for the design of a data structure, maintaining a set of stored elements and supporting the following two operations: insert and precedes . The operation insert (a) introduces a new element a in the structure, while the operation precedes (a,b) returns true iff element a was inserted before element b temporally. In [11] a solution was provided to the problem with worst-case time complexity O (log log n ) per operation and O(n log log n) space, where n is the number of elements inserted. It was also demonstrated that the precedes operation has a lower bound of Ω (log log n ) for the Pure Pointer Machine model of computation. In this paper we present two simple solutions with linear space and worst-case constant insertion time. In addition, we describe two algorithms that can handle the precedes (a,b) operation in O (log log d ) time, where d is the temporal distance between the elements a and b .