Abstract. We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called an order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely, (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms, and preliminary results for randomized algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. In contrast to the classic competitive ratio measure, where the optimal performance is obtained by deterministic single-threshold algorithms, our results for order-competitive ratio capture the intuition that adaptive algorithms may be more powerful than single-threshold ones, and randomized algorithms outperform deterministic ones.

There is a rising interest for studying the online benchmark as an alternative of the classical offline benchmark in online stochastic settings. Ezra, Feldman, Gravin, and Tang (SODA 2023) introduced the notion of order-competitive ratio, defined as the worst-case ratio between the performance of the best order-unaware algorithm and the best order-aware algorithm, to quantify the loss incurred by the lack of knowledge of the arrival order. They showed in the online single selection setting (a.k.a. the prophet problem), the optimal order-competitive ratio achieved by deterministic algorithms is 1/ϕ ≈ 0.618, and left with an open question whether randomized algorithms can do better.

Cloud systems are now a prevalent platform to host large-scale big-data analytics applications such as machine learning and relational database. However, data privacy remains as a critical concern for public cloud systems. Existing trusted hardware could provide an isolated execution domain on an untrusted platform, but also suffers from access-pattern-based side channels at various levels including memory, disks, and networking. Oblivious algorithms can address these vulnerabilities by hiding the program data access patterns. Unfortunately, current oblivious algorithms for data analytics are limited to single-machine execution, only support simple operations, and/or suffer from significant performance overheads due to the use of expensive global sort and excessive data padding. In this work, we propose SODA, a set of efficient and oblivious algorithms for distributed data analytics operators, including filter, aggregate, and binary equi-join. To improve performance, SODA completely avoids the expensive oblivious global sort primitive, and minimizes the data padding overheads. SODA makes use of low-cost (pseudo-)random communication instead of expensive global sort to ensure uniform data traffic in oblivious filter and aggregate. It also adopts a novel two-level bin-packing approach in oblivious join to alleviate both input redistribution and join product skewness, thus minimizing necessary data padding. Compared to the state-of-the-art system, SODA not only extends the functionality but also improves the performance. It achieves 1.1× to 14.6× speedups on complex multi-operator data analytics workloads.