Online Sorting Buffers on Line

We consider the online scheduling problem for sorting buffers on a line metric. This problem is motivated by an application to disc scheduling. The input to this problem is a sequence of requests. Each request is a block of data to be written on a specified track of the disc. The disc is modeled as a number of tracks arranged on a line. To write a block on a particular track, the scheduler has to bring the disc head to that track. The cost of moving the disc head from a track to another is the distance between those tracks. A sorting buffer that can store at most k requests at a time is available to the scheduler. This buffer can be used to rearrange the input sequence. The objective is to minimize the total cost of head movement while serving the requests. On a disc with n uniformly-spaced tracks, we give a randomized online algorithm with a competitive ratio of O(log n) in expectation against an oblivious adversary. This algorithm also yields a competitive ratio of O(α log n) if we are allowed to use a buffer of size αk for any 1 ≤ α ≤ logn. This is the first non-trivial approximation for the sorting buffers problem on a line metric. Our technique is based on probabilistically embedding the line metric into hierarchically well-separated trees. We show that any deterministic strategy which makes scheduling decisions based only on the contents of the buffer has a competitive ratio of Ω(k). Category: Algorithms and Data Structures.