12 research outputs found
Complexity of Interlocking Polyominoes
Polyominoes are a subset of polygons which can be constructed from
integer-length squares fused at their edges. A system of polygons P is
interlocked if no subset of the polygons in P can be removed arbitrarily far
away from the rest. It is already known that polyominoes with four or fewer
squares cannot interlock. It is also known that determining the interlockedness
of polyominoes with an arbitrary number of squares is PSPACE hard. Here, we
prove that a system of polyominoes with five or fewer squares cannot interlock,
and that determining interlockedness of a system of polyominoes including
hexominoes (polyominoes with six squares) or larger polyominoes is PSPACE hard.Comment: 18 pages, 15 figure
Any Monotone Function Is Realized by Interlocked Polygons
Suppose there is a collection of n simple polygons in the plane, none of which overlap each other. The polygons are interlocked if no subset can be separated arbitrarily far from the rest. It is natural to ask the characterization of the subsets that makes the set of interlocked polygons free (not interlocked). This abstracts the essence of a kind of sliding block puzzle. We show that any monotone Boolean function ƒ on n variables can be described by m = O(n) interlocked polygons. We also show that the decision problem that asks if given polygons are interlocked is PSPACE-complete
Picture-Hanging Puzzles
We show how to hang a picture by wrapping rope around n nails, making a
polynomial number of twists, such that the picture falls whenever any k out of
the n nails get removed, and the picture remains hanging when fewer than k
nails get removed. This construction makes for some fun mathematical magic
performances. More generally, we characterize the possible Boolean functions
characterizing when the picture falls in terms of which nails get removed as
all monotone Boolean functions. This construction requires an exponential
number of twists in the worst case, but exponential complexity is almost always
necessary for general functions.Comment: 18 pages, 8 figures, 11 puzzles. Journal version of FUN 2012 pape
Picture-hanging puzzles
We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.National Science Foundation (U.S.) (NSF grant CCF-1018388
Logic circuits from zero forcing
We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of “back forcing” as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we show that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity. Moreover, in the light of applications of zero forcing in quantum mechanics, the link with Boolean functions may suggest a new directions in quantum control theory and in the study of engineered quantum spin systems. It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Any Monotone Boolean Function Can Be Realized by Interlocked Polygons
We show how to construct interlocked collections of simple
polygons in the plane that fall apart upon removing
certain combinations of pieces. Precisely, interior-disjoint
simple planar polygons are interlocked if no
subset can be separated arbitrarily far from the rest,
moving each polygon as a rigid object as in a sliding-block
puzzle. Removing a subset S of these polygons
might keep them interlocked or free the polygons, allowing
them to separate. Clearly freeing removal sets
satisfy monotonicity: if S S [prime] and removing S frees
the polygons, then so does S [prime]. In this paper, we show
that any monotone Boolean function f on n variables
can be described by m > n interlocked polygons: n of
the m polygons represent the n variables, and removing
a subset of these n polygons frees the remaining
polygons if and only if f is 1 when the corresponding
variables are 1
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version