Tight Bounds for Parallel Paging and Green Paging
In the parallel paging problem, there are p processors that share a cache of size k. The goal is to partition the cache among the processors over time in order to minimize their average completion time. For this long-standing open problem, we give tight upper and lower bounds of ⇥(log p) on the competitive ratio with O(1) resource augmentation. A key idea in both our algorithms and lower bounds is to relate the problem of parallel paging to the seemingly unrelated problem of green paging. In green paging, there is an energy-optimized processor that can temporarily turn off one or more of its cache banks (thereby reducing power consumption), so that the cache size varies between a maximum size k and a minimum size k/p. The goal is to minimize the total energy consumed by the computation, which is proportional to the integral of the cache size over time. We show that any efficient solution to green paging can be converted into an efficient solution to parallel paging, and that any lower bound for green paging can be converted into a lower bound for parallel paging, in both cases in a black-box fashion. We then show that, with O(1) resource augmentation, the more »
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Publication Date:
NSF-PAR ID:
10298509
Journal Name:
Proceedings of the annual ACMSIAM symposium on discrete algorithms
ISSN:
2160-1445
1. In the parallel paging problem, there are $\pP$ processors that share a cache of size $k$. The goal is to partition the cache among the \procs over time in order to minimize their average completion time. For this long-standing open problem, we give tight upper and lower bounds of $\Theta(\log \p)$ on the competitive ratio with $O(1)$ resource augmentation. A key idea in both our algorithms and lower bounds is to relate the problem of parallel paging to the seemingly unrelated problem of green paging. In green paging, there is an energy-optimized processor that can temporarily turn off one or more of its cache banks (thereby reducing power consumption), so that the cache size varies between a maximum size $k$ and a minimum size $k/\p$. The goal is to minimize the total energy consumed by the computation, which is proportional to the integral of the cache size over time. We show that any efficient solution to green paging can be converted into an efficient solution to parallel paging, and that any lower bound for green paging can be converted into a lower bound for parallel paging, in both cases in a black-box fashion. We then show that, with $O(1)$ resourcemore »
2. Abstract We study a fundamental online job admission problem where jobs with deadlines arrive online over time at their release dates, and the task is to determine a preemptive single-server schedule which maximizes the number of jobs that complete on time. To circumvent known impossibility results, we make a standard slackness assumption by which the feasible time window for scheduling a job is at least $$1+\varepsilon$$ 1 + ε times its processing time, for some $$\varepsilon >0$$ ε > 0 . We quantify the impact that different provider commitment requirements have on the performance of online algorithms. Our main contribution is one universal algorithmic framework for online job admission both with and without commitments. Without commitment, our algorithm with a competitive ratio of  $$\mathcal {O}(1/\varepsilon )$$ O ( 1 / ε ) is the best possible (deterministic) for this problem. For commitment models, we give the first non-trivial performance bounds. If the commitment decisions must be made before a job’s slack becomes less than a $$\delta$$ δ -fraction of its size, we prove a competitive ratio of $$\mathcal {O}(\varepsilon /((\varepsilon -\delta )\delta ^2))$$ O ( ε / ( ( ε - δ ) δ 2 ) ) ,more »
4. We study an online hypergraph matching problem with delays, motivated by ridesharing applications. In this model, users enter a marketplace sequentially, and are willing to wait up to $d$ timesteps to be matched, after which they will leave the system in favor of an outside option. A platform can match groups of up to $k$ users together, indicating that they will share a ride. Each group of users yields a match value depending on how compatible they are with one another. As an example, in ridesharing, $k$ is the capacity of the service vehicles, and $d$ is the amount of time a user is willing to wait for a driver to be matched to them. We present results for both the utility maximization and cost minimization variants of the problem. In the utility maximization setting, the optimal competitive ratio is $\frac{1}{d}$ whenever $k \geq 3$, and is achievable in polynomial-time for any fixed $k$. In the cost minimization variation, when $k = 2$, the optimal competitive ratio for deterministic algorithms is $\frac{3}{2}$ and is achieved by a polynomial-time thresholding algorithm. When $k>2$, we show that a polynomial-time randomized batching algorithm is $(2 - \frac{1}{d}) \log k$-competitive, and it is NP-hardmore »