Searching the Integers

Consider an ordered universe U , and a set T ⊂ U with |T | = n. The goal is to preprocess T , such that the following query can be answered efficiently: given x ∈ U , report the predecessor of x, i.e. max{y ∈ T | y < x}. We can also consider the dynamic problem, where elements are inserted and deleted into T . Let tq be the query time, and tu the update time. This is a fundamental search problem, with an impressive number of applications. Later, we will discuss IP lookup (forwarding packets on the Internet), orthogonal range queries and persistent data structures as examples. The problem was considered in many computational models. In fact, most models below were initially defined to study the predecessor problem.