For several well-studied triangle detection problems in the literature, the trivial enumeration algorithms are known to be optimal (up to the exponent) assuming popular fine-grained conjectures. For example, All-Edges Sparse Triangle and Sparse Monochromatic Triangle where each node has degree n^δ for some δ < 1, and the Exact Triangle where edges have arbitrary weights, all have this property under the 3SUM Conjecture. However, as there are slightly nontrivial algorithms for 3SUM, it is natural to wonder if the trivial algorithm for these tricky triangle tasks might also be improved. Applying a variety of techniques from randomized algorithms, circuit complexity, and communication complexity, we present the first improvements over the trivial algorithms for each of these problems in the Word RAM model. Moreover, our algorithms can be implemented with only polysize AC0 operations on words. Extending our techniques, we also show how to solve the notorious 4-cycle detection problem on n-node graphs in o(n²) time, in a Word-RAM model with word size w > ω(log² n). Along the way, we show how to sort n items over a universe of size 2^u using only AC0 word operations in O(n u log n)/w time.

Accurate forecasting of water quality is vital for safeguarding public health, aquatic ecosystems and ensuring economic stability. This study introduces a novel hybrid approach using two-stage decomposition combined with Bidirectional Long Short-Term Memory and Attention mechanism (BiLSTM-Attention) for short-term forecasting of Dissolved Oxygen (DO). First, Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) decomposes the original data into a set of Intrinsic Mode Functions (IMFs) and residual component. Variational Mode Decomposition(VMD) further decomposes the IMF with the highest frequency into a set of modes. Then, BiLSTM-Attention is employed to generate the final forecast. The existing studies predominantly use MSE or MAE as the loss function for water quality forecasting. We design a customized dynamically weighted Huber loss function (DWHL) to optimize the training of our proposed forecasting model. The model is tested on data from ten locations along the river Ganga, consid-ering the forecasting horizons of one, two, and three hours, and compare it with ten models, both with and without decomposition. The results of the Diebold Mariano test suggests that there is a statistically significant difference in forecast accuracy of the proposed and compared models. The outcomes demonstrate that the proposed CEEMDAN-VMD-DWHL-BiLSTM-Attention approach effectively identifies complex and non-linear patterns in the data, resulting in significantly improved forecasts.