Online Checkpointing with Improved Worst-Case Guarantees

Abstract

Abstract. In the online checkpointing problem, the task is to continuously maintain a set of k checkpoints that allow to rewind an ongoing computation faster than by a full restart. The only operation allowed is to remove an old checkpoint and to store the current state instead. Our aim are checkpoint placement strategies that minimize rewinding cost, i.e., such that at all times T when requested to rewind to some time t ≤ T the number of computation steps that need to be redone to get to t from a checkpoint before t is as small as possible. In particular, we want that the closest checkpoint earlier than t is not further away from t than p k times the ideal distance T /(k + 1), where p k is a small constant. Improving over earlier work showing 1 + 1/k ≤ p k ≤ 2, we show that p k can be chosen less than 2 uniformly for all k. More precisely, we show the uniform bound p k ≤ 1.7 for all k, and present algorithms with asymptotic performance p k ≤ 1.59 + o(1) valid for all k and p k ≤ ln(4) + o(1) ≤ 1.39 + o(1) valid for k being a power of two. For small values of k, we show how to use a linear programming approach to compute good checkpointing algorithms. This gives performances of less than 1.53 for k ≤ 10. One the more theoretical side, we show the first lower bound that is asymptotically more than one, namely p k ≥ 1.30 − o(1). We also show that optimal algorithms (yielding the infimum performance) exist for all k.

Tight Bounds on Online Checkpointing Algorithms

“…Clearly, q k ≥ 1 for all k ≥ 2, and cannot be too close to 1 since any snapshot in which all subintervals have roughly the same length will be transformed by the next update operation to a snapshot in which one of the subintervals will be the union of two previous subintervals, and thus will be about twice as long as the other subintervals. 4 On the other hand, there is a very simple subinterval doubling algorithm from [1, Section 3.1] which is 2-efficient: Assuming WLOG that k is even, the algorithm starts with the snapshot (1, 2, 3, . .…”

“…Analyzing this problem is surprisingly difficult, and so far there had been no tight bounds on the best possible efficiencies q k of online checkpointing algorithms in this model. The main results in [4] are two online checkpointing algorithms whose asymptotic efficiencies are ln 4 + o(1) ≈ 1.39 for the sparse subset of k's which are powers of 2, and 1.59 for general k. In addition, they proved in their model the first nontrivial asymptotic lower bound of 2 − ln 2 − o(1) ≈ 1.30. However, since the upper and lower bounds did not match, it was not clear whether the checkpointing algorithms they proposed were asymptotically optimal.…”

“…In this paper we solve the main open problems related to the mathematical formulation of the problem which was defined and studied in [4]. In particular, we develop a new checkpointing algorithm with an asymptotic efficiency of ln 4 for all values of k, and prove its optimality by providing a matching asymptotic lower bound.…”

“…The first paper dealing with this problem seems to be "Rollback and Recovery Strategies for Computer Programs" [5], published in 1972, while the first paper which tried to solve it optimally was "On the Optimum Checkpoint Interval" [7], published in 1979. Over the years, dozens of academic research papers were published in this area, most notably [8] in 1984, [3] in 1994, and [1,4] in 2013. However, many of these papers either dealt with concrete applications of the problem in other areas (especially in distributed computing where the notion of a timeline is different), or used other optimization criteria (which make their optimal solutions incomparable with ours).…”

“…However, many of these papers either dealt with concrete applications of the problem in other areas (especially in distributed computing where the notion of a timeline is different), or used other optimization criteria (which make their optimal solutions incomparable with ours). The mathematical problem we are dealing with in this paper was mentioned in [1] and studied in [4] which was published at ICALP'13, and we closely follow their model and notation.…”