1,984 research outputs found
Fun with Fonts: Algorithmic Typography
Over the past decade, we have designed six typefaces based on mathematical
theorems and open problems, specifically computational geometry. These
typefaces expose the general public in a unique way to intriguing results and
hard problems in hinged dissections, geometric tours, origami design,
computer-aided glass design, physical simulation, and protein folding. In
particular, most of these typefaces include puzzle fonts, where reading the
intended message requires solving a series of puzzles which illustrate the
challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font.
Original version in Proceedings of the 7th International Conference on Fun
with Algorithm
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
Bidimensionality of Geometric Intersection Graphs
Let B be a finite collection of geometric (not necessarily convex) bodies in
the plane. Clearly, this class of geometric objects naturally generalizes the
class of disks, lines, ellipsoids, and even convex polygons. We consider
geometric intersection graphs GB where each body of the collection B is
represented by a vertex, and two vertices of GB are adjacent if the
intersection of the corresponding bodies is non-empty. For such graph classes
and under natural restrictions on their maximum degree or subgraph exclusion,
we prove that the relation between their treewidth and the maximum size of a
grid minor is linear. These combinatorial results vastly extend the
applicability of all the meta-algorithmic results of the bidimensionality
theory to geometrically defined graph classes
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
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
Verification in Staged Tile Self-Assembly
We prove the unique assembly and unique shape verification problems,
benchmark measures of self-assembly model power, are
-hard and contained in (and in
for staged systems with stages). En route,
we prove that unique shape verification problem in the 2HAM is
-complete.Comment: An abstract version will appear in the proceedings of UCNC 201
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