1,051 research outputs found
Tetris is Hard, Even to Approximate
In the popular computer game of Tetris, the player is given a sequence of
tetromino pieces and must pack them into a rectangular gameboard initially
occupied by a given configuration of filled squares; any completely filled row
of the gameboard is cleared and all pieces above it drop by one row. We prove
that in the offline version of Tetris, it is NP-complete to maximize the number
of cleared rows, maximize the number of tetrises (quadruples of rows
simultaneously filled and cleared), minimize the maximum height of an occupied
square, or maximize the number of pieces placed before the game ends. We
furthermore show the extreme inapproximability of the first and last of these
objectives to within a factor of p^(1-epsilon), when given a sequence of p
pieces, and the inapproximability of the third objective to within a factor of
(2 - epsilon), for any epsilon>0. Our results hold under several variations on
the rules of Tetris, including different models of rotation, limitations on
player agility, and restricted piece sets.Comment: 56 pages, 11 figure
Multi-dimensional Boltzmann Sampling of Languages
This paper addresses the uniform random generation of words from a
context-free language (over an alphabet of size ), while constraining every
letter to a targeted frequency of occurrence. Our approach consists in a
multidimensional extension of Boltzmann samplers \cite{Duchon2004}. We show
that, under mostly \emph{strong-connectivity} hypotheses, our samplers return a
word of size in and exact frequency in
expected time. Moreover, if we accept tolerance
intervals of width in for the number of occurrences of each
letters, our samplers perform an approximate-size generation of words in
expected time. We illustrate these techniques on the
generation of Tetris tessellations with uniform statistics in the different
types of tetraminoes.Comment: 12p
Domains growth and packing properties in driven granular media subject to gravity
We study the dynamical properties of recently introduced frustrated lattice
gas models (IFLG and Tetris) for granular media under gentle shaking. We
consider both the case where grains have inter-grain surface interactions and
the case where they have not, corresponding, for instance, to the presence or
absence of moisture in the packs. To characterise the grains packing structure,
we discuss the properties of density distribution. In particular, we consider
the phenomenon of grains domains formation under compaction. New results
amenable of experimental check are discussed along with some important
differences between the dynamics of the present models.Comment: 7 pages, 6 postscript files for figure
Bust-a-Move/Puzzle Bobble is NP-Complete
We prove that the classic 1994 Taito video game, known as Puzzle Bobble or
Bust-a-Move, is NP-complete. Our proof applies to the perfect-information
version where the bubble sequence is known in advance, and it uses just three
bubble colors.Comment: 9 pages, 9 figures. Corrected mistakes in gadget
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