TY - JOUR

T1 - Tight Bounds on Online Checkpointing Algorithms

AU - Bar-On, Achiya

AU - Dinur, Itai

AU - Dunkelman, Orr

AU - Hod, Rani

AU - Keller, Nathan

AU - Ronen, Eyal

AU - Shamir, Adi

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/6

Y1 - 2020/6

N2 - The problem of online checkpointing is a classical problem with numerous applications that has 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, Doerr, Neumann, and Sliacan 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 ln 4-o(1)≈ 1.39 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 article, we solve the main problems left open in the above-mentioned paper by proving that ln 4 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 ≤ 10. In the last part of the article, we describe some new applications of this online checkpointing problem.

AB - The problem of online checkpointing is a classical problem with numerous applications that has 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, Doerr, Neumann, and Sliacan 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 ln 4-o(1)≈ 1.39 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 article, we solve the main problems left open in the above-mentioned paper by proving that ln 4 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 ≤ 10. In the last part of the article, we describe some new applications of this online checkpointing problem.

KW - Checkpoints

KW - competitive analysis

KW - online algorithms

UR - http://www.scopus.com/inward/record.url?scp=85088707298&partnerID=8YFLogxK

U2 - 10.1145/3379543

DO - 10.1145/3379543

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AN - SCOPUS:85088707298

SN - 1549-6325

VL - 16

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 3

M1 - 31

ER -