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1. The White-Box Adversarial Data Stream Model

   https://doi.org/10.1145/3517804.3526228  

   Ajtai, M. ; Braverman, V. ; Jayram, T.S. ; Silwal, S. ; Sun, A. ; Woodruff, D.P. ; Zhou, S. ( January 2022 , Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS 2022))

   There has been a flurry of recent literature studying streaming algorithms for which the input stream is chosen adaptively by a black-box adversary who observes the output of the streaming algorithm at each time step. However, these algorithms fail when the adversary has access to the internal state of the algorithm, rather than just the output of the algorithm. We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the L1-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our L1-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in amore »stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for randomized algorithms robust to a white-box adversary. In particular, our results show that for all p ≥ 0, there exists a constant Cp > 1 such that any Cp-approximation algorithm for Fp moment estimation in insertion-only streams with a white-box adversary requires Ω(n) space for a universe of size n. Similarly, there is a constant C > 1 such that any C-approximation algorithm in an insertion-only stream for matrix rank requires Ω(n) space with a white-box adversary. These results do not contradict our upper bounds since they assume the adversary has unbounded computational power. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. Finally, we prove a lower bound of Ω(log n) bits for the fundamental problem of deterministic approximate counting in a stream of 0’s and 1’s, which holds even if we know how many total stream updates we have seen so far at each point in the stream. Such a lower bound for approximate counting with additional information was previously unknown, and in our context, it shows a separation between multiplayer deterministic maximum communication and the white-box space complexity of a streaming algorithm« less
2. Spiking Neural Networks Through the Lens of Streaming Algorithms

   Hitron, Y. ; Musco, C. ; Parter, M. ( January 2020 , 34th International Symposium on Distributed Computing (DISC))

   We initiate the study of biologically-inspired spiking neural networks from the perspective of streaming algorithms. Like computers, human brains face memory limitations, which pose a significant obstacle when processing large scale and dynamically changing data. In computer science, these challenges are captured by the well-known streaming model, which can be traced back to Munro and Paterson `78 and has had significant impact in theory and beyond. In the classical streaming setting, one must compute a function f of a stream of updates 𝒮 = {u₁,…,u_m}, given restricted single-pass access to the stream. The primary complexity measure is the space used by the algorithm. In contrast to the large body of work on streaming algorithms, relatively little is known about the computational aspects of data processing in spiking neural networks. In this work, we seek to connect these two models, leveraging techniques developed for streaming algorithms to better understand neural computation. Our primary goal is to design networks for various computational tasks using as few auxiliary (non-input or output) neurons as possible. The number of auxiliary neurons can be thought of as the "space" required by the network. Previous algorithmic work in spiking neural networks has many similarities with streaming algorithms.more »However, the connection between these two space-limited models has not been formally addressed. We take the first steps towards understanding this connection. On the upper bound side, we design neural algorithms based on known streaming algorithms for fundamental tasks, including distinct elements, approximate median, and heavy hitters. The number of neurons in our solutions almost match the space bounds of the corresponding streaming algorithms. As a general algorithmic primitive, we show how to implement the important streaming technique of linear sketching efficiently in spiking neural networks. On the lower bound side, we give a generic reduction, showing that any space-efficient spiking neural network can be simulated by a space-efficient streaming algorithm. This reduction lets us translate streaming-space lower bounds into nearly matching neural-space lower bounds, establishing a close connection between the two models.« less
3. New hardness results for planar graph problems in p and an algorithm for sparsest cut

   https://doi.org/10.1145/3357713.3384310  

   Abboud, Amir ; Cohen-Addad, Vincent ; Klein, Philip N. ( June 2020 , Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing)

   The Sparsest Cut is a fundamental optimization problem that have been extensively studied. For planar inputs the problem is in P and can be solved in Õ(n 3 ) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90’s and can only achieve O(log n)-approximation in Õ(n) time or 3.5-approximation in Õ(n 2 ) time [Rao, STOC92]. Our main result is an Ω(n 2−ε ) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approxima- tions are inevitable in the near-linear time regime. To complement the lower bound, we provide a 3.3-approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. We also show that our lower bound is not far from optimal by observing an exact algorithm with running time Õ(n 5/2 ) improving upon the Õ(n 3 ) algorithm of Park and Phillips [STOC93]. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Building on our construction we prove near-quadratic lower bounds under SETHmore »for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGET model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.« less
4. Batched Predecessor and Sorting with Size-Priced Information in External Memory

   Bender, Michael ; Goswami, Mayank ; Medjedovic, Dzejla ; Montes, Pablo ; Tsichlas, Kostas ( January 2021 , Latin American Theoretical Informatics Symposium)

   In the unit-cost comparison model, a black box takes an input two items and outputs the result of the comparison. Problems like sorting and searching have been studied in this model, and it has been general- ized to include the concept of priced information, where different pairs of items (say database records) have different comparison costs. These comparison costs can be arbitrary (in which case no algorithm can be close to optimal (Charikar et al. STOC 2000)), structured (for exam- ple, the comparison cost may depend on the length of the databases (Gupta et al. FOCS 2001)), or stochastic (Angelov et al. LATIN 2008). Motivated by the database setting where the cost depends on the sizes of the items, we consider the problems of sorting and batched predecessor where two non-uniform sets of items A and B are given as input. (1) In the RAM setting, we consider the scenario where both sets have n keys each. The cost to compare two items in A is a, to compare an item of A to an item of B is b, and to compare two items in B is c. We give upper and lower bounds for the case a ≤more »b ≤ c, the case that serves as a warmup for the generalization to the external-memory model. Notice that the case b = 1,a = c = ∞ is the famous “nuts and bolts” problem. ) In the Disk-Access Model (DAM), where transferring elements between disk and internal memory is the main bottleneck, we con- sider the scenario where elements in B are larger than elements in A. The larger items take more I/Os to be brought into memory, consume more space in internal memory, and are required in their entirety for comparisons. A key observation is that the complexity of sorting depends heavily on the interleaving of the small and large items in the final sorted order. If all large elements come after all small elements in the final sorted order, sorting each type separately and concatenating is optimal. However, if the set of predecessors of B in A has size k ≪ n, one must solve an associated batched predecessor problem in order to achieve optimality. We first give output-sensitive lower and upper bounds on the batched predecessor problem, and use these to derive bounds on the complexity of sorting in the two models. Our bounds are tight in most cases, and require novel generalizations of the classical lower bound techniques in external memory to accommodate the non-uniformity of keys.« less
5. Timely Reporting of Heavy Hitters using External Memory

   Singh, S ; Pandey, P ; Bender, M. ; Berry, J. ; Farach-Colton, M ; Johnson, R ; Kroeger, T. ; Phillips, C ( January 2021 , SIGMOD record)

   Given an input stream of size N , a -heavy hiter is an item that occurs at least N times in S. The problem of finding heavy-hitters is extensively studied in the database literature. We study a real-time heavy-hitters variant in which an element must be reported shortly after we see its T = N - th occurrence (and hence becomes a heavy hitter). We call this the Timely Event Detection (TED) Problem. The TED problem models the needs of many real-world monitoring systems, which demand accurate (i.e., no false negatives) and timely reporting of all events from large, high-speed streams, and with a low reporting threshold (high sensitivity). Like the classic heavy-hitters problem, solving the TED problem without false-positives requires large space ((N ) words). Thus in-RAM heavy-hitters algorithms typically sacrfice accuracy (i.e., allow false positives), sensitivity, or timeliness (i.e., use multiple passes). We show how to adapt heavy-hitters algorithms to exter- nal memory to solve the TED problem on large high-speed streams while guaranteeing accuracy, sensitivity, and timeli- ness. Our data structures are limited only by I/O-bandwidth (not latency) and support a tunable trade-off between report- ing delay and I/O overhead. With a small bounded reporting delay, ourmore »algorithms incur only a logarithmic I/O overhead. We implement and validate our data structures empirically using the Firehose streaming benchmark. Multi-threaded ver- sions of our structures can scale to process 11M observations per second before becoming CPU bound. In comparison, a naive adaptation of the standard heavy-hitters algorithm to external memory would be limited by the storage device’s random I/O throughput, i.e., approx 100K observations per second.« less