On Energy-Efficient Computations With Advice

Abstract

Abstract. Online problems have always played an important role in computer science. Here, not the whole input is known at the beginning, but it is only revealed gradually. These problems frequently occur in practice, and therefore the performance of algorithms for such problems is of great theoretical and practical interest. One such online problem is energy management in electronic devices, e. g., smartphones. As such a device is usually not being used permanently, it is reasonable to change to a lowerenergy state (like hibernation) after a certain idle time. Resuming from hibernation, however, also needs a certain amount of energy; therefore, hibernation should only happen if the idle period is long. Advice complexity is a recent approach for measuring the information content of an online problem, i. e., the amount of knowledge about the future parts of the input that is necessary to compute a high-quality solution. The approach allows for a more fine-grained analysis of the hardness of online problems than the classical competitive analysis. We analyze the advice complexity of this problem. For systems with two states, we construct an online algorithm with advice that is 1.8-competitive with one advice bit and 1.6-competitive with five advice bits, whereas every deterministic algorithm without advice is known to be no better than 2-competitive. Moreover, the algorithm's competitive ratio converges fast towards e/(e − 1) with an increasing number of advice bits. For competitive ratios in the range [1, e/(e − 1)], we present two complementary algorithms: one behaves optimally on a certain prefix, and the other falls asleep on the longest phases. Conversely, we show that every algorithm with a competitive ratio less than 1 + 1/(4w + 2), where w is the wake-up energy, needs to read a linear number of advice bits.

Randomization can be as helpful as a glimpse of the future in online computation

“…Thus, we get a lower bound of e/(e − 1) for algorithms with advice complexity o(n) for 2-SSM. The previous best lower bound for algorithms with sublinear advice was 7/6 and was obtained in [21]. We note that [67] also gives a randomized algorithm with a competitive ratio of e/(e − 1).…”

“…Assumption (20) says that the expected cost of a deterministic algorithm (without advice) should be at least t in each round, no matter what has happened in previous rounds. Regarding assumption (21), note that if M is an upper bound on the cost incurred by any algorithm in a single round, then ( 21) is satisfied by taking s = M . However, in some cases, it is possible to obtain a better estimate and thereby a better advice complexity lower bound.…”

“…The 2-sleep states management problem (2-SSM, see [64,21]) can, obviously, be modeled as a generalized task system, and so it is Σ-repeatable. As observed in [64], the 2-SSM problem is essentially a repeated ski-rental problem.…”

“…We note that [67] also gives a randomized algorithm with a competitive ratio of e/(e − 1). Furthermore, in [21], it is shown that it is possible for a deterministic algorithm to achieve a competitive ratio arbitrarily close to e/(e − 1) using O(1) bits of advice. Since n bits of advice trivially suffice to be optimal for 2-SSM, the asymptotic advice complexity of 2-SSM is now fully determined.…”

Online Simple Knapsack with Reservation Costs

“…This notion of delaying a decision for a fixed percentage of the cost is the model that we want to study in this work. Note that, while the ski-rental problem in its above-mentioned form only models a single buy-or-rent decision (buying or renting a single commodity), iterated versions of it have been discussed in the literature, with important applications, e. g., to energy-efficient computing [14,1,3]. In the following, we investigate the power of delaying decisions for another more involved problem, namely the online knapsack problem.…”