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Sliding-window aggregation is a foundational stream processing primitive that efficiently summarizes recent data. The state-of-the-art algorithms for sliding-window aggregation are highly efficient when stream data items are evicted or inserted one at a time, even when some of the insertions occur out-of-order. However, real-world streams are often not only out-of-order but also burtsy, causing data items to be evicted or inserted in larger bulks. This paper introduces a new algorithm for sliding-window aggregation with bulk eviction and bulk insertion. For the special case of single insert and evict, our algorithm matches the theoretical complexity of the best previous out-of-order algorithms. For the case of bulk evict, our algorithm improves upon the theoretical complexity of the best previous algorithm for that case and also outperforms it in practice. For the case of bulk insert, there are no prior algorithms, and our algorithm improves upon the naive approach of emulating bulk insert with a loop over single inserts, both in theory and in practice. Overall, this paper makes high-performance algorithms for sliding window aggregation more broadly applicable by efficiently handling the ubiquitous cases of out-of-order data and bursts., Comment: Extended version for VLDB 2023 paper

Sliding-window aggregation is a widely-used approach for extracting insights from the most recent portion of a data stream. The aggregations of interest can usually be expressed as binary operators that are associative but not necessarily commutative nor invertible. Non-invertible operators, however, are difficult to support efficiently. In a 2017 conference paper, we introduced DABA, the first algorithm for sliding-window aggregation with worst-case constant time. Before DABA, if a window had size $n$, the best published algorithms would require $O(\log n)$ aggregation steps per window operation---and while for strictly in-order streams, this bound could be improved to $O(1)$ aggregation steps on average, it was not known how to achieve an $O(1)$ bound for the worst-case, which is critical for latency-sensitive applications. This article is an extended version of our 2017 paper. Besides describing DABA in more detail, this article introduces a new variant, DABA Lite, which achieves the same time bounds in less memory. Whereas DABA requires space for storing $2n$ partial aggregates, DABA Lite only requires space for $n+2$ partial aggregates. Our experiments on synthetic and real data support the theoretical findings.

We present the architecture of a cloud native version of IBM Streams, with Kubernetes as our target platform. Streams is a general purpose streaming system with its own platform for managing applications and the compute clusters that execute those applications. Cloud native Streams replaces that platform with Kubernetes. By using Kubernetes as its platform, Streams is able to offload job management, life cycle tracking, address translation, fault tolerance and scheduling. This offloading is possible because we define custom resources that natively integrate into Kubernetes, allowing Streams to use Kubernetes' eventing system as its own. We use four design patterns to implement our system: controllers, conductors, coordinators and causal chains. Composing controllers, conductors and coordinators allows us to build deterministic state machines out of an asynchronous distributed system. The resulting implementation eliminates 75% of the original platform code. Our experimental results show that the performance of Kubernetes is an adequate replacement in most cases, but it has problems with oversubscription, networking latency, garbage collection and pod recovery., Comment: 18 pages, 11 figures, submitted to OSDI 2020

Sliding-window aggregation summarizes the most recent information in a data stream. Users specify how that summary is computed, usually as an associative binary operator because this is the most general known form for which it is possible to avoid naively scanning every window. For strictly in-order arrivals, there are algorithms with $O(1)$ time per window change assuming associative operators. Meanwhile, it is common in practice for streams to have data arriving slightly out of order, for instance, due to clock drifts or communication delays. Unfortunately, for out-of-order streams, one has to resort to latency-prone buffering or pay $O(\log n)$ time per insert or evict, where $n$ is the window size. This paper presents the design, analysis, and implementation of FiBA, a novel sliding-window aggregation algorithm with an amortized upper bound of $O(\log d)$ time per insert or evict, where $d$ is the distance of the inserted or evicted value to the closer end of the window. This means $O(1)$ time for in-order arrivals and nearly $O(1)$ time for slightly out-of-order arrivals, with a smooth transition towards $O(\log n)$ as $d$ approaches $n$. We also prove a matching lower bound on running time, showing optimality. Our algorithm is as general as the prior state-of-the-art: it requires associativity, but not invertibility nor commutativity. At the heart of the algorithm is a careful combination of finger-searching techniques, lazy rebalancing, and position-aware partial aggregates. We further show how to answer range queries that aggregate subwindows for window sharing. Finally, our experimental evaluation shows that FiBA performs well in practice and supports the theoretical findings.

We show that for any bounded function $f:[a,b]\rightarrow{\mathbb R}$ and $\epsilon>0$ there is a partition $P$ of $[a,b]$ with respect to which the Riemann sum of $f$ using right endpoints is within $\epsilon$ of the upper Darboux sum of $f$. This leads to an elementary proof of the theorem of Gillespie \cite{G} showing that Cauchy's and Riemann's definitions of integrability coincide., Comment: 4 pages

We discuss combinatorial conditions for the existence of various types of reductions between equivalence relations, and in particular identify necessary and sufficient conditions for the existence of injective reductions., Comment: 7 pages, 2 figures

Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel function $f:X\to Y$ that is both a reduction from $E$ to $E'$ and a reduction from $F$ to $F'$. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs $E\subseteq F$ of smooth countable Borel equivalence relations up to simultaneous Borel bireducibility and biembeddability, and a significant portion of such pairs up to simultaneous Borel isomorphism. We generalize Mauldin's notion of Borel parametrization in order to identify large natural subclasses of pairs of smooth countable equivalence relations and of singleton smooth (not necessarily countable) equivalence relations for which the natural combinatorial isomorphism invariants are complete, and we present counterexamples outside these subclasses. Finally, we relate isomorphism of smooth equivalence relations and of pairs of smooth countable equivalence relations to Borel equivalence of Borel functions as discussed in Komisarski, Michalewski, and Milewski., Comment: 35 pages

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. This extends previous results obtained by Gao-Jackson for abelian groups and by Jackson-Kechris-Louveau for finitely generated nilpotent-by-finite groups. Our proof is based on a mixture of coarse geometric properties of locally nilpotent groups together with an adaptation of the Gao-Jackson machinery.

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting number $\mathfrak s$ coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics.

We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures., Comment: 7 pages