Online bottleneck matching on a line

We study the online bottleneck matching problem on a line, which is to match a sequence of m requests arriving one-by-one in an online fashion to a given set of m servers, such that each server is matched exactly once and the maximum distance between any request and its server is minimized. When the distances between any two adjacent servers are the same, we present an optimal online algorithm with a competitive ratio of m+ 1. When m = 3, we present an optimal online algorithm whose competitive ratio is determined by the relative distance between adjacent servers and no more than 3 + root 2, which matches the previous best lower bound proposed thirty years ago.