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1. Speeding Up the RUL¯ Dynamic-Controllability-Checking Algorithm for Simple Temporal Networks with Uncertainty

   https://doi.org/10.1609/aaai.v36i9.21213  

   Hunsberger, Luke; Posenato, Roberto (June 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   A Simple Temporal Network with Uncertainty (STNU) includes real-valued variables, called time-points; binary difference constraints on those time-points; and contingent links that represent actions with uncertain durations. STNUs have been used for robot control, web-service composition, and business processes. The most important property of an STNU is called dynamic controllability (DC); and algorithms for checking this property are called DC-checking algorithms. The DC-checking algorithm for STNUs with the best worst-case time-complexity is the RUL¯ algorithm due to Cairo, Hunsberger and Rizzi. Its complexity is O(mn + k²n + kn log n), where n is the number of time-points, m is the number of constraints, and k is the number of contingent links. It is expected that this worst-case complexity cannot be improved upon. However, this paper provides a new algorithm, called RUL2021, that improves its performance in practice by an order of magnitude, as demonstrated by a thorough empirical evaluation. 

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2. Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation

   Aggarwal, Amol and (July 2022, Computational complexity)

   Polynomial approximations for e−x and ex have applications to the design of algorithms for many problems, and our degree bounds show both the power and limitations of these algorithms. We focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in Θ(logn) dimensions with error δ=n−Θ(1). We show that the running time one can achieve depends on the square of the diameter of the point set, B, with a transition at B=Θ(logn) mirroring the corresponding transition in dB;δ(e−x): - When B=o(logn), we give the first algorithm running in time n1+o(1). - When B=κlogn for a small constant κ>0, we give an algorithm running in time n1+O(loglogκ−1/logκ−1). The loglogκ−1/logκ−1 term in the exponent comes from analyzing the behavior of the leading constant in our computation of dB;δ(e−x). - When B=ω(logn), we show that time n2−o(1) is necessary assuming SETH. 

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3. Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

   N. Pashanasangi, C.Seshadhri (August 2021, SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) 2021)

   null (Ed.)

   Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)-temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)-temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing state-of-the-art temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine. 

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4. Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

   Pashanasangi, Noujan; Seshadhri, C. (January 2021, KDD)

   null (Ed.)

   Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)-temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)-temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing state-of-the-art temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine. 

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5. An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

   https://doi.org/10.3390/math11194191  

   Zhong, Huicong; Feng, Xiaobing (October 2023, Mathematics)

   This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(d3Nb)(b≤2) or better, compared to the exponential order O(N(logN)d−1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration. 

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