Online Service with Delay

In this paper, we introduce the online service with delay problem. In this problem, there are $n$ points in a metric space that issue service requests over time, and a server that serves these requests. The goal is to minimize the sum of distance traveled by the server and the total delay in serving the requests. This problem models the fundamental tradeoff between batching requests to improve locality and reducing delay to improve response time, that has many applications in operations management, operating systems, logistics, supply chain management, and scheduling. Our main result is to show a poly-logarithmic competitive ratio for the online service with delay problem. This result is obtained by an algorithm that we call the preemptive service algorithm. The salient feature of this algorithm is a process called preemptive service, which uses a novel combination of (recursive) time forwarding and spatial exploration on a metric space. We hope this technique will be useful for related problems such as reordering buffer management, online TSP, vehicle routing, etc. We also generalize our results to $k > 1$ servers.Comment: 30 pages, 11 figures, Appeared in 49th ACM Symposium on Theory of Computing (STOC), 201

Reordering Buffer Management with a Logarithmic Guarantee in General Metric Spaces

In the reordering buffer management problem a sequence of requests arrive online in a finite metric space, and have to be processed by a single server. This server is equipped with a request buffer of size k and can decide at each point in time, which request from its buffer to serve next. Servicing of a request is simply done by moving the server to the location of the request. The goal is to process all requests while minimizing the total distance that the server is traveling inside the metric space. In this paper we present a deterministic algorithm for the reordering buffer management problem that achieves a competitive ratio of O(log Delta + min {log n,log k}) in a finite metric space of n points and aspect ratio Delta. This is the first algorithm that works for general metric spaces and has just a logarithmic dependency on the relevant parameters. The guarantee is memory-robust, i.e., the competitive ratio decreases only slightly when the buffer-size of the optimum is increased to h=(1+epsilon)k. For memory robust guarantees our bounds are close to optimal

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 3 n) in expectation against an oblivious adversary. This algorithm also yields a competitive ratio of O(α −1 log 3 n) if we are allowed to use a buffer of size αk for any 1 ≤ α ≤ log n. 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.