In this paper we focus on the intersection of tile assembling systems, edge-matching puzzles, combinatorial games, and knot construction and identity. As a basis, we utilize the game Celtic!, which is a 2-player board game where the goal of the game is to construct knots where one knot uses more of a player’s pieces than the other player over all knots. All pieces must build off an existing knot and a valid knot must be closed. We consider three variations: a 0-player self-assembly variation that deterministically places pieces to form a closed knot of some length, a 1-player puzzle variation where the goal is to form a closed knot of some length, and the original 2-player game with restricted pieces. We show these are P-complete, NP-complete (depending on the pieces), and PSPACE-complete (for a first-player win), respectively. We nearly fully characterize the hardness of the 1-player puzzle based on the pieces. We prove these results through standard hardness reductions and with constraint logic. Finally, we note some combinatorial game theory strategies to show certain configurations are a draw through strategy stealing.

Recently, there has been a significant advancement in designing Self-Supervised Learning (SSL) frameworks for time series data to reduce the dependency on data labels. Among these works, hierarchical contrastive learning-based SSL frameworks, which learn representations by contrasting data embeddings at multiple resolutions, have gained considerable attention. Due to their ability to gather more information, they exhibit better generalization in various downstream tasks. However, when the time series data length is significant long, the computational cost is often significantly higher than that of other SSL frameworks. In this paper, to address this challenge, we propose an efficient way to train hierarchical contrastive learning models. Inspired by the fact that each resolution’s data embedding is highly dependent, we introduce importance-aware resolution selection based training framework to reduce the computational cost. In the experiment, we demonstrate that the proposed method significantly improves training time while preserving the original model’s integrity in extensive time series classification performance evaluations. Our code could be found here: https://github.com/KEEBVIN/IARS