How Fast Can We Play Tetris Greedily With Rectangular Pieces?

Consider a variant of Tetris played on a board of width $w$ and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of $O(n)$ rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction to the Multiphase problem [P\u{a}tra\c{s}cu, 2010] that on a board of width $w=\Theta(n)$, if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time $O(n^{1/2-\epsilon})$ simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in $O(n^{1/2}\log^{3/2}n)$ time on boards of width $n^{O(1)}$, matching the lower bound up to a $n^{o(1)}$ factor.Comment: Correction of typos and other minor correction

Threes!, Fives, 1024!, and 2048 are Hard

We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m x n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.Comment: 14 pages, 9 figure

Level up learning: a national survey on teaching with digital games

Digital games have the potential to transform K-12 education as we know it. But what has been the real experience among teachers who use games in the classroom? In 2013, the Games and Learning Publishing Council conducted a national survey among nearly 700 K-8 teachers. The report reveals key findings from the survey, and looks at how often and why teachers use games in the classroom, as well as issues they encounter in their efforts to implement digital games into their practice

Applicability and Challenges of Deep Reinforcement Learning for Satellite Frequency Plan Design

The study and benchmarking of Deep Reinforcement Learning (DRL) models has become a trend in many industries, including aerospace engineering and communications. Recent studies in these fields propose these kinds of models to address certain complex real-time decision-making problems in which classic approaches do not meet time requirements or fail to obtain optimal solutions. While the good performance of DRL models has been proved for specific use cases or scenarios, most studies do not discuss the compromises and generalizability of such models during real operations. In this paper we explore the tradeoffs of different elements of DRL models and how they might impact the final performance. To that end, we choose the Frequency Plan Design (FPD) problem in the context of multibeam satellite constellations as our use case and propose a DRL model to address it. We identify 6 different core elements that have a major effect in its performance: the policy, the policy optimizer, the state, action, and reward representations, and the training environment. We analyze different alternatives for each of these elements and characterize their effect. We also use multiple environments to account for different scenarios in which we vary the dimensionality or make the environment nonstationary. Our findings show that DRL is a potential method to address the FPD problem in real operations, especially because of its speed in decision-making. However, no single DRL model is able to outperform the rest in all scenarios, and the best approach for each of the 6 core elements depends on the features of the operation environment. While we agree on the potential of DRL to solve future complex problems in the aerospace industry, we also reflect on the importance of designing appropriate models and training procedures, understanding the applicability of such models, and reporting the main performance tradeoffs