An approximate membership query data structure (AMQ)-such as a Bloom, quotient, or cuckoo filter-maintains a compact, probabilistic representation of a set S of keys from a universe U. It supports lookups and inserts. Some AMQs also support deletes. A query for x ∈ S returns PRESENT. A query for x ∈ S returns PRESENT with a tunable false-positive probability ε, and otherwise returns ABSENT.AMQs are widely used to speed up dictionaries that are stored remotely (e.g., on disk or across a network). The AMQ is stored locally (e.g., in memory). The remote dictionary is only accessed when the AMQ returns PRESENT. Thus, the primary performance metric of an AMQ is how often it returns ABSENT for negative queries.Existing AMQs offer weak guarantees on the number of false positives in a sequence of queries. The false-positive probability ε holds only for a single query. It is easy for an adversary to drive an AMQ's false-positive rate towards 1 by simply repeating false positives.This paper shows what it takes to get strong guarantees on the number of false positives. We say that an AMQ is adaptive if it guarantees a false-positive probability of ε for every query, regardless of answers to previous queries.We establish upper and lower bounds for adaptive AMQs. Our lower bound shows that it is impossible to build a small adaptive AMQ, even when the AMQ is immediately told whenever a query is a false positive. On the other hand, we show that it is possible to maintain an AMQ that uses the same amount of local space as a non-adaptive AMQ (up to lower order terms), performs all queries and updates in constant time, and guarantees that each negative query to the dictionary accesses remote storage with probability ε, independent of the results of past queries. Thus, we show that adaptivity can be achieved effectively for free.
We introduce the cache-adaptive model, which generalizes the external-memory model to apply to environments in which the amount of memory available to an algorithm can fluctuate. The cache-adaptive model applies to operating systems, databases, and other systems where the allocation of memory to processes changes over time. We prove that if an optimal cache-oblivious algorithm has a particular recursive structure, then it is also an optimal cache-adaptive algorithm. Cache-oblivious algorithms having this form include Floyd-Warshall all pairs shortest paths, naïve recursive matrix multiplication, matrix transpose, and Gaussian elimination. While the cache-oblivious sorting algorithm Lazy Funnel Sort does not have this recursive structure, we prove that it is nonetheless optimally cache-adaptive. We also establish that if a cache-oblivious algorithm is optimal on "square" (well-behaved) memory profiles then, given resource augmentation it is optimal on all memory profiles. We give paging algorithms for the case where the cache size changes dynamically. We prove that LRU with 4memory and 4-speed augmentation is competitive with optimal. Moreover, Belady's algorithm remains optimal even when the cache size changes. Cache-obliviousness is distinct from cache-adaptivity. We exhibit a cache-oblivious algorithm that is not cacheadaptive and a cache-adaptive algorithm for a problem having no optimal cache-oblivious solution.
Interactive proofs (IP) model a world where a verifier delegates computation to an untrustworthy prover, verifying the prover's claims before accepting them. IP protocols have applications in areas such as verifiable computation outsourcing, computation delegation, cloud computing, etc. In these applications, the verifier may pay the prover based on the quality of his work. Rational interactive proofs (RIP), introduced by Azar and Micali (2012), are an interactive-proof system with payments, in which the prover is rational rather than untrustworthy-he may lie, but only to increase his payment. Rational proofs leverage the prover's rationality to obtain simple and efficient protocols. Azar and Micali show that RIP=IP(=PSPACE), i.e., the set of provable languages stay the same with a single rational prover (compared to classic IP). They leave the question of whether multiple provers are more powerful than a single prover for rational and classical proofs as an open problem.In this paper we introduce multi-prover rational interactive proofs (MRIP). Here, a verifier cross-checks the provers' answers with each other and pays them according to the messages exchanged. The provers are cooperative and maximize their total expected payment if and only if the verifier learns the correct answer to the problem. We further refine the model of MRIP to incorporate utility gaps, which is the loss in payment suffered by provers who mislead the verifier to the wrong answer.We define the class of MRIP protocols with constant, noticeable and negligible utility gaps-the payment loss due to a wrong answer is O(1), 1/n O(1) and 1/2 n O(1) respectively, where n is the length of the input. We give tight characterization for all three MRIP classes. On the way, we resolve Azar and Micali's open problem-under standard complexity-theoretic assumptions, MRIP is not only more powerful than RIP, but also more powerful than MIP (classic multi-prover IP); and this is true even the utility gap is required to be constant. We further show that the full power of each MRIP class can be achieved using only two provers and three rounds of communication.
Integrated Stockpile Evaluation (ISE) is a program to test nuclear weapons periodically. Tests are performed by machines that may require occasional calibration. These calibrations are expensive, so finding a schedule that minimizes calibrations allows more testing to be done for a given amount of money.This paper introduces a theoretical framework for ISE. Machines run jobs with release times and deadlines. Calibrating a machine requires unit cost. The machine remains calibrated for T time steps, after which it must be recalibrated before it can resume running jobs. The objective is to complete all jobs while minimizing the number of calibrations.The paper gives several algorithms to solve the ISE problem for the case where jobs have unit processing times. For one available machine, there is an optimal polynomial-time algorithm. For multiple machines, there is a 2-approximation algorithm, which finds an optimal solution when all jobs have distinct deadlines.
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