Tight Bounds on Online Checkpointing Algorithms

Achiya Bar-On, Itai Dinur, Orr Dunkelman, Rani Hod, Nathan Keller, Eyal Ronen, Adi Shamir

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Article number31
JournalACM Transactions on Algorithms
Volume16
Issue number3
DOIs
StatePublished - Jun 2020

Bibliographical note

Funding Information:
The work was partially supported by the the European Research Council under the ERC starting grant agreement no. 757731 (LightCrypt), the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office, and by the Israeli Science Foundation through grant no. 573/16. The work was also supported in part by the Israel Ministry of Science and Technology. Eyal Ronen is a member of CPIIS, Tel Aviv University. Authors’ addresses: A. Bar-On and N. Keller, Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel; emails: abo1000@gmail.com, nkeller@math.biu.ac.il; I. Dinur, Computer Science Department, Ben-Gurion University, Beer-Sheba, Israel; email: dinuri@cs.bgu.ac.il; O. Dunkelman, Computer Science Department, University of Haifa, Haifa, Israel; email: orrd@cs.haifa.ac.il; R. Hod, School of Computer Science, Tel Aviv University, Tel Aviv, Israel; email: ranihod@tau.ac.il; E. Ronen, School of Computer Science, Tel Aviv University, Tel Aviv, Israel, Department of Electrical Engineering ESAT, KU Leuven, Leuven, Belgium; email: eyal.ronen@cs.tau.ac.il; A. Shamir, Computer Science Department, The Weizmann Institute, Rehovot, Israel; email: adi.shamir@weizmann.ac.il. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. 1549-6325/2020/05-ART31 $15.00 https://doi.org/10.1145/3379543

Publisher Copyright:
© 2020 ACM.

Keywords

  • Checkpoints
  • competitive analysis
  • online algorithms

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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