Tight Bounds on Online Checkpointing Algorithms

The problem of online checkpointing is a classical problem with numerous applications which had been studied in various forms for almost 50 years. In the simplest version of this problem, a user has to maintain $k$ memorized checkpoints during a long computation, where the only allowed operation is to move one of the checkpoints from its old time to the current time, and his goal is to keep the checkpoints as evenly spread out as possible at all times. Bringmann et al. studied this problem as a special case of an online/offline optimization problem in which the deviation from uniformity is measured by the natural discrepancy metric of the worst case ratio between real and ideal segment lengths. They showed this discrepancy is smaller than $1.59-o(1)$ for all $k$, and smaller than $\ln4-o(1)\approx1.39$ for the sparse subset of $k$'s which are powers of 2. In addition, they obtained upper bounds on the achievable discrepancy for some small values of $k$. In this paper we solve the main problems left open in the above-mentioned paper by proving that $\ln4$ is a tight upper and lower bound on the asymptotic discrepancy for all large $k$, and by providing tight upper and lower bounds (in the form of provably optimal checkpointing algorithms, some of which are in fact better than those of Bringmann et al.) for all the small values of $k \leq 10$. In the last part of the paper we describe some new applications of this online checkpointing problem.