When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FUN.2022.13
URN: urn:nbn:de:0030-drops-159839
URL: https://drops.dagstuhl.de/opus/volltexte/2022/15983/
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How Fast Can We Play Tetris Greedily with Rectangular Pieces?

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Abstract

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 from the Multiphase problem [Pătraşcu, 2010] that on a board of width w = Θ(n), if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time O(n^{1/2-ε}) 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 an n^o(1) factor.

BibTeX - Entry

```@InProceedings{dallant_et_al:LIPIcs.FUN.2022.13,
author =	{Dallant, Justin and Iacono, John},
title =	{{How Fast Can We Play Tetris Greedily with Rectangular Pieces?}},
booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
pages =	{13:1--13:19},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-232-7},
ISSN =	{1868-8969},
year =	{2022},
volume =	{226},
editor =	{Fraigniaud, Pierre and Uno, Yushi},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},