The reordering buffer problem on the line revisited

The reordering bu↵er problem (or also sorting bu↵er problem) was introduced by Räcke, Sohler, and Westermann in 2002 [14] and has been extensively studied since then. In this problem, a metric space is given1 and a sequence of items arrive online. Each item is associated with a point in the metric space. We allow multiple items to be associated with the same point. An online algorithm can store up to k items in a bu↵er, but once the bu↵er is full, the algorithm has to process at least one of the items stored in the bu↵er. To process an item from the bu↵er, the algorithm moves a single server in the metric space to the point corresponding to that item. The goal is to minimize the total distance that the server has to travel to process the entire input sequence. The problem is reasonably well understood for some metric spaces. For uniform metric spaces for example, a deterministic O( p log k)-competitive algorithm is known, which is close to the lower bound of ⌦( p log k/ log log k) [1]. Similarly, [4] gives a O(log log k)-competitive randomized online algorithm, which is asymptotically tight [1]. For other metric spaces however, the picture is less clear. We will refrain from listing all known results in detail, but there have been a number of papers investigating this online problem for di↵erent metrics spaces and settings [2, 3, 7, 8, 9, 10, 11, 12, 13]. However, in this column, we will focus on line metric spaces. The last notable result for this metric was obtained eleven years ago by Gamzu and Segev [11]. Their main result is a deterministic O(log n)-competitive online algorithm for a line metric space with n evenly spaced points. In the reminder, we will sketch a slightly simplified and improved version of this result.