352 research outputs found
Threes!, Fives, 1024!, and 2048 are Hard
We analyze the computational complexity of the popular computer games
Threes!, 1024!, 2048 and many of their variants. For most known versions
expanded to an m x n board, we show that it is NP-hard to decide whether a
given starting position can be played to reach a specific (constant) tile
value.Comment: 14 pages, 9 figure
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several
partitioning problems for line and hyperplane arrangements. We prove a
ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l
such that in both line sets, for both halfplanes delimited by l, there are
n^{1/2} lines which pairwise intersect in that halfplane, and this bound is
tight; a centerpoint theorem: for any set of n lines there is a point such that
for any halfplane containing that point there are (n/3)^{1/2} of the lines
which pairwise intersect in that halfplane. We generalize those results in
higher dimension and obtain a center transversal theorem, a same-type lemma,
and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This
is done by formulating a generalization of the center transversal theorem which
applies to set functions that are much more general than measures. Back to
Graph Drawing (and in the plane), we completely solve the open problem that
motivated our search: there is no set of n labelled lines that are universal
for all n-vertex labelled planar graphs. As a side note, we prove that every
set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar
graphs
Weighted dynamic finger in binary search trees
It is shown that the online binary search tree data structure GreedyASS
performs asymptotically as well on a sufficiently long sequence of searches as
any static binary search tree where each search begins from the previous search
(rather than the root). This bound is known to be equivalent to assigning each
item in the search tree a positive weight and bounding the search
cost of an item in the search sequence by
amortized. This result is the strongest finger-type bound to be proven for
binary search trees. By setting the weights to be equal, one observes that our
bound implies the dynamic finger bound. Compared to the previous proof of the
dynamic finger bound for Splay trees, our result is significantly shorter,
stronger, simpler, and has reasonable constants.Comment: An earlier version of this work appeared in the Proceedings of the
Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithm
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
The Clique Problem in Ray Intersection Graphs
Ray intersection graphs are intersection graphs of rays, or halflines, in the
plane. We show that any planar graph has an even subdivision whose complement
is a ray intersection graph. The construction can be done in polynomial time
and implies that finding a maximum clique in a segment intersection graph is
NP-hard. This solves a 21-year old open problem posed by Kratochv\'il and
Ne\v{s}et\v{r}il.Comment: 12 pages, 7 figure
Measuring Damages in Survival Actions for Tortious Death
Survival statutes have been adopted to avoid the effect of common law rules preventing claims for the tortious death of a human being. These statutes give the personal representative such causes of action on behalf of the decedent\u27s estate as the decedent would have had were he still alive. The question the statutes do not answer, however, is the effect of the death of a party on the measure of damages. The Washington Supreme Court\u27s decision in Warner v. McCaughan illustrates the problem. Warner arose out of the death of a twenty-one year old college student. Alleging that the death was caused by improper diagnosis and care and by administration of unsafe drugs, her parents, individually, and her father, as administrator of her estate, brought suit for damages against the doctor, hospital, and pharmaceutical company on the grounds of negligence and breach of warranty. The parents\u27 individual claims were dismissed because the parents were not dependents of the decedent, but the estate\u27s claim was entertained. One of the items of damage claimed by the estate was disability in consequence of a medical condition caused by the defendants\u27 tortious acts and resulting in the decedent\u27s death. This claim presented the major issue of the case: whether the prohibition in the general survival statute against recovery for pain and suffering7 prohibited recovery for the decedent\u27s disability. The court rejected the defendants\u27 argument that the statutory prohibition meant that all claims personal to the decedent abated with her death8 and held that the statute allows the broad common-law claim for personal injury, except for pain and suffering. The principal question remaining, which the Warner court did not fully answer, is how these damages for physical injury are to be measured in a tortious death case. The purpose of this article is to discuss the factors relevant to the damages issue and to suggest appropriate standards for measuring them
The Four Bars Problem
A four-bar linkage is a mechanism consisting of four rigid bars which are
joined by their endpoints in a polygonal chain and which can rotate freely at
the joints (or vertices). We assume that the linkage lies in the 2-dimensional
plane so that one of the bars is held horizontally fixed. In this paper we
consider the problem of reconfiguring a four-bar linkage using an operation
called a \emph{pop}. Given a polygonal cycle, a pop reflects a vertex across
the line defined by its two adjacent vertices along the polygonal chain. Our
main result shows that for certain conditions on the lengths of the bars of the
four-bar linkage, the neighborhood of any configuration that can be reached by
smooth motion can also be reached by pops. The proof relies on the fact that
pops are described by a map on the circle with an irrational number of
rotation.Comment: 18 page
Induced Ramsey-type results and binary predicates for point sets
Let and be positive integers and let be a finite point set in
general position in the plane. We say that is -Ramsey if there is a
finite point set such that for every -coloring of
there is a subset of such that and have the same order type
and is monochromatic in . Ne\v{s}et\v{r}il and Valtr proved
that for every , all point sets are -Ramsey. They also
proved that for every and , there are point sets that are
not -Ramsey.
As our main result, we introduce a new family of -Ramsey point sets,
extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result
to show that for every there is a point set such that no function
that maps ordered pairs of distinct points from to a set of size
can satisfy the following "local consistency" property: if attains
the same values on two ordered triples of points from , then these triples
have the same orientation. Intuitively, this implies that there cannot be such
a function that is defined locally and determines the orientation of point
triples.Comment: 22 pages, 3 figures, final version, minor correction
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