1,149 research outputs found
Solitaire Clobber
Clobber is a new two-player board game. In this paper, we introduce the
one-player variant Solitaire Clobber where the goal is to remove as many stones
as possible from the board by alternating white and black moves. We show that a
checkerboard configuration on a single row (or single column) can be reduced to
about n/4 stones. For boards with at least two rows and two columns, we show
that a checkerboard configuration can be reduced to a single stone if and only
if the number of stones is not a multiple of three, and otherwise it can be
reduced to two stones. We also show that in general it is NP-complete to decide
whether an arbitrary Clobber configuration can be reduced to a single stone.Comment: 14 pages. v2 fixes small typ
PushPush and Push-1 are NP-hard in 2D
We prove that two pushing-blocks puzzles are intractable in 2D. One of our
constructions improves an earlier result that established intractability in 3D
[OS99] for a puzzle inspired by the game PushPush. The second construction
answers a question we raised in [DDO00] for a variant we call Push-1. Both
puzzles consist of unit square blocks on an integer lattice; all blocks are
movable. An agent may push blocks (but never pull them) in attempting to move
between given start and goal positions. In the PushPush version, the agent can
only push one block at a time, and moreover when a block is pushed it slides
the maximal extent of its free range. In the Push-1 version, the agent can only
push one block one square at a time, the minimal extent---one square. Both
NP-hardness proofs are by reduction from SAT, and rely on a common
construction.Comment: 10 pages, 11 figures. Corrects an error in the conference version:
Proc. of the 12th Canadian Conference on Computational Geometry, August 2000,
pp. 211-21
PushPush is NP-hard in 2D
We prove that a particular pushing-blocks puzzle is intractable in 2D,
improving an earlier result that established intractability in 3D [OS99]. The
puzzle, inspired by the game *PushPush*, consists of unit square blocks on an
integer lattice. An agent may push blocks (but never pull them) in attempting
to move between given start and goal positions. In the PushPush version, the
agent can only push one block at a time, and moreover, each block, when pushed,
slides the maximal extent of its free range. We prove this version is NP-hard
in 2D by reduction from SAT.Comment: 18 pages, 13 figures, 1 table. Improves cs.CG/991101
Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes
We investigate how to make the surface of a convex polyhedron (a polytope) by
folding up a polygon and gluing its perimeter shut, and the reverse process of
cutting open a polytope and unfolding it to a polygon. We explore basic
enumeration questions in both directions: Given a polygon, how many foldings
are there? Given a polytope, how many unfoldings are there to simple polygons?
Throughout we give special attention to convex polygons, and to regular
polygons. We show that every convex polygon folds to an infinite number of
distinct polytopes, but that their number of combinatorially distinct gluings
is polynomial. There are, however, simple polygons with an exponential number
of distinct gluings.
In the reverse direction, we show that there are polytopes with an
exponential number of distinct cuttings that lead to simple unfoldings. We
establish necessary conditions for a polytope to have convex unfoldings,
implying, for example, that among the Platonic solids, only the tetrahedron has
a convex unfolding. We provide an inventory of the polytopes that may unfold to
regular polygons, showing that, for n>6, there is essentially only one class of
such polytopes.Comment: 54 pages, 33 figure
Enumerating Foldings and Unfoldings between Polygons and Polytopes
We pose and answer several questions concerning the number of ways to fold a
polygon to a polytope, and how many polytopes can be obtained from one polygon;
and the analogous questions for unfolding polytopes to polygons. Our answers
are, roughly: exponentially many, or nondenumerably infinite.Comment: 12 pages; 10 figures; 10 references. Revision of version in
Proceedings of the Japan Conference on Discrete and Computational Geometry,
Tokyo, Nov. 2000, pp. 9-12. See also cs.CG/000701
Data Sketches for Disaggregated Subset Sum and Frequent Item Estimation
We introduce and study a new data sketch for processing massive datasets. It
addresses two common problems: 1) computing a sum given arbitrary filter
conditions and 2) identifying the frequent items or heavy hitters in a data
set. For the former, the sketch provides unbiased estimates with state of the
art accuracy. It handles the challenging scenario when the data is
disaggregated so that computing the per unit metric of interest requires an
expensive aggregation. For example, the metric of interest may be total clicks
per user while the raw data is a click stream with multiple rows per user. Thus
the sketch is suitable for use in a wide range of applications including
computing historical click through rates for ad prediction, reporting user
metrics from event streams, and measuring network traffic for IP flows.
We prove and empirically show the sketch has good properties for both the
disaggregated subset sum estimation and frequent item problems. On i.i.d. data,
it not only picks out the frequent items but gives strongly consistent
estimates for the proportion of each frequent item. The resulting sketch
asymptotically draws a probability proportional to size sample that is optimal
for estimating sums over the data. For non i.i.d. data, we show that it
typically does much better than random sampling for the frequent item problem
and never does worse. For subset sum estimation, we show that even for
pathological sequences, the variance is close to that of an optimal sampling
design. Empirically, despite the disadvantage of operating on disaggregated
data, our method matches or bests priority sampling, a state of the art method
for pre-aggregated data and performs orders of magnitude better on skewed data
compared to uniform sampling. We propose extensions to the sketch that allow it
to be used in combining multiple data sets, in distributed systems, and for
time decayed aggregation
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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