We present the first known pivot Gray code for spanning trees of complete graphs, listing all spanning trees such that consecutive trees differ by pivoting a single edge around a vertex. This pivot Gray code thus addresses an open problem posed by Knuth in The Art of Computer Programming, Volume 4 (Exercise 101, Section 7.2.1.6, [Knuth 2011]), rated at a difficulty level of 46 out of 50, and imposes stricter conditions than existing revolving-door or edge-exchange Gray codes for spanning trees of complete graphs. Our recursive algorithm generates each spanning tree in constant amortized time using \(O(n^2)\) space. In addition, we provide a novel proof of Cayley’s formula, \(n^{n-2}\), for the number of spanning trees in a complete graph, derived from our recursive approach. We extend the algorithm to generate edge-exchange Gray codes for general graphs with \(n\) vertices, achieving \(O(n^2)\) time per tree using \(O(n^2)\) space. For specific graph classes, the algorithm can be optimized to generate edge-exchange Gray codes for spanning trees in constant amortized time per tree for complete bipartite graphs, \(O(n)\)-amortized time per tree for fan graphs, and \(O(n)\)-amortized time per tree for wheel graphs, all using \(O(n^2)\) space.

Although many complex models were proposed to analyze time series data, some studies have demonstrated remarkable performance with simpler structures. A recent study proposed a non-parametric framework for 3D point cloud classification, which has the potential to be adapted for time series forecasting and enable interpretability. Inspired by the previous works, we present TSNN, a non-parametric and interpretable framework for traffic time series forecasting. TSNN consists of multiple layers that decouple the time series by matching the entries in a memory bank, where the memory bank is constructed using a similar matching process within the training set. It leverages the periodicity in traffic data to enhance forecasting accuracy while maintaining a simple model architecture. The proposed model operates without trainable parameters, preserving its inherent interpretability. In the experiments, TSNN achieves competitive performance compared to the typical deep learning models in four real-world traffic flow datasets. We also visualize the decoupling process to show the effectiveness of the components. Finally, we demonstrate the interpretability of the model and illustrate the contribution of each time step within the memory bank. Our code is available athttps://github.com/pzzzzzm/TSNN_release.

Dual challenges of computational efficiency and representation effectiveness exist in processing point clouds. Inspired by the TDA (Topological Data Analysis), we propose to convert the point cloud to a transposed fully connected and weighted (t-FCW) graph in order to significantly decrease the computational complexity in the following processing steps. We design a TDA pipeline called Point-FCW with a series of vectorization techniques for the 3D object point cloud feature extraction, which is plugged into the non-parametric classification head. Our experimental results demonstrate that Point-FCW achieves 75.28% accuracy on the ModelNet40 dataset with 512 points, providing a tiny, consistent, and effective representation for TDA. Furthermore, when integrated with the state-of-the-art non-parametric network Point-NN, the mixture model performs better, with an improvement of 4.47% in the OBJ-BG split of the ScanObjectNN dataset. Similarly, when integrating Point-FCW into the parametric network, PointMLP yields a performance improvement of 3.54% in the PB-T50-RS split of the ScanObjectNN dataset. The proposed Point-FCW can serve as a complementary enhancement feature when integrated into the Point-NN and PointMLP models. Moreover, the t-FCW graph representation can be efficiently converted at a rate of 3739 samples/second. Our code is available inhttps://github.com/hawkinglai/Point-FCW.