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

Notions of explainability and evaluation approaches for explainable artificial intelligence

Explainable Artificial Intelligence (XAI) has experienced a significant growth over the last few years. This is due to the widespread application of machine learning, particularly deep learning, that has led to the development of highly accurate models that lack explainability and interpretability. A plethora of methods to tackle this problem have been proposed, developed and tested, coupled with several studies attempting to define the concept of explainability and its evaluation. This systematic review contributes to the body of knowledge by clustering all the scientific studies via a hierarchical system that classifies theories and notions related to the concept of explainability and the evaluation approaches for XAI methods. The structure of this hierarchy builds on top of an exhaustive analysis of existing taxonomies and peer-reviewed scientific material. Findings suggest that scholars have identified numerous notions and requirements that an explanation should meet in order to be easily understandable by end-users and to provide actionable information that can inform decision making. They have also suggested various approaches to assess to what degree machine-generated explanations meet these demands. Overall, these approaches can be clustered into human-centred evaluations and evaluations with more objective metrics. However, despite the vast body of knowledge developed around the concept of explainability, there is not a general consensus among scholars on how an explanation should be defined, and how its validity and reliability assessed. Eventually, this review concludes by critically discussing these gaps and limitations, and it defines future research directions with explainability as the starting component of any artificial intelligent system

Data-Driven Supervised Learning for Life Science Data

Life science data are often encoded in a non-standard way by means of alpha-numeric sequences, graph representations, numerical vectors of variable length, or other formats. Domain-specific or data-driven similarity measures like alignment functions have been employed with great success. The vast majority of more complex data analysis algorithms require fixed-length vectorial input data, asking for substantial preprocessing of life science data. Data-driven measures are widely ignored in favor of simple encodings. These preprocessing steps are not always easy to perform nor particularly effective, with a potential loss of information and interpretability. We present some strategies and concepts of how to employ data-driven similarity measures in the life science context and other complex biological systems. In particular, we show how to use data-driven similarity measures effectively in standard learning algorithms