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1. DOUBLY-ADAPTIVE ARTIFICIAL COMPRESSION METHODS FOR INCOMPRESSIBLE FLOW

   https://doi.org/DOI: https://doi.org/10.1515/jnma-2019-0015  

   LAYTON, W. ; MCLAUGHLIN, M. ( January 2019 , Journal of numerical mathematics)

   This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter ε are independently adapted. The resulting algorithms are supported by analysis and numerical tests. The first and second-order methods are embedded. As a result, the computational, cognitive and space complexities of the adaptive ε,k algorithms are negligibly greater than that of the simplest, first-order, constant ε, constant k artificial compression method. 

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2. Massively Parallel Algorithms for Small Subgraph Counting

   Biswas, Amartya Shankha ; Eden, Talya ; Liu, Quanquan C. ; Rubinfeld, Ronitt Slobodan ( January 2022 , Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022))

   Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph G = (V, E) with n vertices, m edges and T triangles, our first result is an algorithm that outputs a (1+ε)-approximation to T, with asymptotically optimal round and total space complexity provided any S ≥ max{(√ m, n²/m)} space per machine and assuming T = Ω(√{m/n}). Our result gives a quadratic improvement on the bound on T over previous works. We also provide a simple extension of our result to counting any subgraph of k size for constant k ≥ 1. Our second result is an O_δ(log log n)-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the arboricity α of the input graph. We extend this result to exactly counting k-cliques for any constant k. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most 5 can be implemented in the MPC model in Õ_δ(√{log n}) rounds, O(n^δ) space per machine and O(mα³) total space. In addition to our theoretical results, we simulate our triangle counting algorithms in real-world graphs obtained from the Stanford Network Analysis Project (SNAP) database. Our results show that both our approximate and exact counting algorithms exhibit improvements in terms of round complexity and approximation ratio, respectively, compared to two previous widely used algorithms for these problems. 

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3. Parallel Nearest Neighbors in Low Dimensions with Batch Updates

   Blelloch, Guy E. ; Dobson, Magdalen ( January 2022 , 2022 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX))

   We present a set of parallel algorithms for computing exact k-nearest neighbors in low dimensions. Many k-nearest neighbor algorithms use either a kd-tree or the Morton ordering of the point set; our algorithms combine these approaches using a data structure we call the zd-tree. We show that this combination is both theoretically efficient under common assumptions, and fast in practice. For point sets of size n with bounded expansion constant and bounded ratio, the zd-tree can be built in O(n) work with O(n^ε) span for constant ε < 1, and searching for the k-nearest neighbors of a point takes expected O(k log k) time. We benchmark our k-nearest neighbor algorithms against existing parallel k-nearest neighbor algorithms, showing that our implementations are generally faster than the state of the art as well as achieving 75x speedup on 144 hyperthreads. Furthermore, the zd-tree supports parallel batch-dynamic insertions and deletions; to our knowledge, it is the first k-nearest neighbor data structure to support such updates. On point sets with bounded expansion constant and bounded ratio, a batch-dynamic update of size k requires O(k log n/k) work with O(k^ε + polylog(n)) span. 

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4. Set Cover in Sub-linear Time

   Indyk, Piotr ; Mahabadi, Sepideh ; Rubinfeld, Ronitt ; Vakilian, Ali ; Yodpinyanee, Anak ( January 2018 , Annual ACM-SIAM Symposium on Discrete Algorithms)

   We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)^1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is , even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)-approximation algorithm with Õ(mn/kε^2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of query complexity. Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size αk, with the key property that a small number of “almost uniformly distributed” modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems. 

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5. Leibniz International Proceedings in Informatics (LIPIcs):14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

   https://doi.org/10.4230/LIPICS.ITCS.2023.58  

   Goldenberg, Elazar ; Kociumaka, Tomasz ; Krauthgamer, Robert ; Saha, Barna ( January 2023 , SourceOECD ITCS)

   Tauman Kalai, Yael (Ed.)

   The edit distance between strings classically assigns unit cost to every character insertion, deletion, and substitution, whereas the Hamming distance only allows substitutions. In many real-life scenarios, insertions and deletions (abbreviated indels) appear frequently but significantly less so than substitutions. To model this, we consider substitutions being cheaper than indels, with cost 1/a for a parameter a ≥ 1. This basic variant, denoted ED_a, bridges classical edit distance (a = 1) with Hamming distance (a → ∞), leading to interesting algorithmic challenges: Does the time complexity of computing ED_a interpolate between that of Hamming distance (linear time) and edit distance (quadratic time)? What about approximating ED_a? We first present a simple deterministic exact algorithm for ED_a and further prove that it is near-optimal assuming the Orthogonal Vectors Conjecture. Our main result is a randomized algorithm computing a (1+ε)-approximation of ED_a(X,Y), given strings X,Y of total length n and a bound k ≥ ED_a(X,Y). For simplicity, let us focus on k ≥ 1 and a constant ε > 0; then, our algorithm takes Õ(n/a + ak³) time. Unless a = Õ(1), in which case ED_a resembles the standard edit distance, and for the most interesting regime of small enough k, this running time is sublinear in n. We also consider a very natural version that asks to find a (k_I, k_S)-alignment, i.e., an alignment with at most k_I indels and k_S substitutions. In this setting, we give an exact algorithm and, more importantly, an Õ((nk_I)/k_S + k_S k_I³)-time (1,1+ε)-bicriteria approximation algorithm. The latter solution is based on the techniques we develop for ED_a for a = Θ(k_S/k_I), and its running time is again sublinear in n whenever k_I ≪ k_S and the overall distance is small enough. These bounds are in stark contrast to unit-cost edit distance, where state-of-the-art algorithms are far from achieving (1+ε)-approximation in sublinear time, even for a favorable choice of k. 

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