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 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 − (cid:15) ) 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.