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1. Calibration schemes with O(N log N) scaling for large-N radio interferometers built on a regular grid

   https://doi.org/10.1093/mnras/staa3229  

   Gorthi, Deepthi B; Parsons, Aaron R; Dillon, Joshua S (November 2020, Monthly Notices of the Royal Astronomical Society)

   null (Ed.)

   ABSTRACT Future generations of radio interferometers targeting the 21 cm signal at cosmological distances with N ≫ 1000 antennas could face a significant computational challenge in building correlators with the traditional architecture, whose computational resource requirement scales as $$\mathcal {O}(N^2)$$ with array size. The fundamental output of such correlators is the cross-correlation products of all antenna pairs in the array. The FFT-correlator architecture reduces the computational resources scaling to $$\mathcal {O}(N\log {N})$$ by computing cross-correlation products through a spatial Fourier transform. However, the output of the FFT-correlator is meaningful only when the input antenna voltages are gain- and phase-calibrated. Traditionally, interferometric calibration has used the $$\mathcal {O}(N^2)$$ cross-correlations produced by a standard correlator. This paper proposes two real-time calibration schemes that could work in parallel with an FFT-correlator as a self-contained $$\mathcal {O}(N\log {N})$$ correlator system that can be scaled to large-N redundant arrays. We compare the performance and scalability of these two calibration schemes and find that they result in antenna gains whose variance decreases as 1/log N with increase in the size of the array. 

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2. Vertex Sparsification for Edge Connectivity

   https://doi.org/10.1137/1.9781611976465.74  

   Chalermsook, Parinya; Das, Syamantak Das; Kook, Yunbum; Laekhanukit, Bundit; Liu, Yang P.; Peng, Richard Peng; Sellke, Mark; Vaz, Daniel (January 2021, Proceedings of the 2021 {ACM-SIAM} Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021)

   null (Ed.)

   Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $$(1+\epsilon)$$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we initiate the study of a thresholded version of the problem: for a given parameter $$c$$, find a smaller graph, which we call \emph{connectivity-$$c$$ mimicking network}, which preserves connectivity among $$k$$ terminals exactly up to the value of $$c$$. We show that connectivity-$$c$$ mimicking networks of size $O(kc^4)$ exist and can be found in time $$m(c\log n)^{O(c)}$$. We also give a separate algorithm that constructs such graphs of size $$k \cdot O(c)^{2c}$$ in time $$mc^{O(c)}\log^{O(1)}n$$. These results lead to the first offline data structures for answering fully dynamic $$c$$-edge-connectivity queries for $$c \ge 4$$ in polylogarithmic time per query as well as more efficient algorithms for survivable network design on bounded treewidth graphs. 

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3. Summing $$\mu (n)$$: a faster elementary algorithm

   https://doi.org/10.1007/s40993-022-00408-8  

   Helfgott, Harald Andrés; Thompson, Lola (March 2023, Research in Number Theory)

   Abstract We present a new elementary algorithm that takes $$ \textrm{time} \ \ O_\epsilon \left( x^{\frac{3}{5}} (\log x)^{\frac{8}{5}+\epsilon } \right) \ \ \textrm{and} \ \textrm{space} \ \ O\left( x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right) $$ time O ϵ x 3 5 ( log x ) 8 5 + ϵ and space O x 3 10 ( log x ) 13 10 (measured bitwise) for computing $$M(x) = \sum _{n \le x} \mu (n),$$ M ( x ) = ∑ n ≤ x μ ( n ) , where $$\mu (n)$$ μ ( n ) is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $$O(x^{1/5} (\log x)^{5/3})$$ O ( x 1 / 5 ( log x ) 5 / 3 ) by the use of (Helfgott in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of $$x^{3/5} (\log x)^2 \log \log x$$ x 3 / 5 ( log x ) 2 log log x . 

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4. A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

   https://doi.org/10.1109/FOCS46700.2020.00111  

   Chuzhoy, Julia; Gao, Yu; Li, Jason; Nanongkai, Danupon; Peng, Richard; Saranurak, Thatchaphol (November 2020, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We consider the classical Minimum Balanced Cut problem: given a graph $$G$$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $$n$$-vertex $$m$$-edge graph $$G$$ and any parameter $$1\leq r\leq O(\log n)$$, computes a $$(\log m)^{r^2}$$-approximation for Minimum Balanced Cut on $$G$$, in time $$O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$$. In particular, we obtain a $$(\log m)^{1/\epsilon}$$-approximation in time $$m^{1+O(1/\sqrt{\epsilon})}$$ for any constant $$\epsilon$$, and a $$(\log m)^{f(m)}$$-approximation in time $$m^{1+o(1)}$$, for any slowly growing function $$m$$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $$G$$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $$n$$-vertex graph is $$n^{o(1)}$$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $$n$$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs. 

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5. Adaptive Out-Orientations with Applications

   https://doi.org/10.1137/1.9781611977912.110  

   Chekuri, Chandra; Christiansen, Aleksander Bjørn; Holm, Jacob; van der Hoog, Ivor; Quanrud, Kent; Rotenberg, Eva; Schwiegelshohn, Chris. (January 2024, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out-degree is proportional to the arboricity $$\alpha$$ of the graph, in, either, an amortised update time of $$O(\log^2 n \log \alpha)$$, or a worst-case update time of $$O(\log^3 n \log \alpha)$$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either $$O(\log n \log \alpha)$$, amortised, or $$O(\log ^2 n \log \alpha)$$, worst-case, for the problem of maintaining an edge-orientation with at most $$O(\alpha + \log n)$$ out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $$n$$ and $$\alpha$$. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a $$(1+\varepsilon)$$ approximation of the maximum subgraph density, $$\rho$$, of the dynamic graph. Our algorithms have update times of $$O(\varepsilon^{-6}\log^3 n \log \rho)$$ worst-case, and $$O(\varepsilon^{-4}\log^2 n \log \rho)$$ amortised, respectively. We may output a subgraph $$H$$ of the input graph where its density is a $$(1+\varepsilon)$$ approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the $$O(\varepsilon^{-6}\log ^4 n)$$ algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an $$O(\varepsilon^{-6}\log^3 n \log \alpha)$$ worst-case update time algorithm for maintaining a $$(1~+~\varepsilon)\textnormal{OPT} + 2$$ approximation of the optimal out-orientation of a graph with adaptive arboricity $$\alpha$$, improving the $$O(\varepsilon^{-6}\alpha^2 \log^3 n)$$ algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $$O(\alpha)$$ forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, $$\Delta+1$$ colouring, and matrix vector multiplication. All update times are worst-case $$O(\alpha+\log^2n \log \alpha)$$, where $$\alpha$$ is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time $$O(\alpha^2 + \log^2 n)$$, and by Neiman and Solomon from STOC 2013 runs in time $$O(\sqrt{m})$$. We give improved running times whenever the arboricity $$\alpha \in \omega( \log n\sqrt{\log\log n})$$. 

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