24,822 research outputs found
Assembling and Rearranging Digital Objects in Physical Space with Tongs, a Gluegun, and a Lightsaber
We present an interface for the arrangement of objects in three-dimensional space. Physical motions of the user are mapped to interface commands through tangible props. Tongs move objects freely, a gluegun binds objects together, and a lightsaber breaks these bonds. The experimental interface is implemented on the Responsive Workbench, a semi-immersive 3D computer. We conducted a small user study comparing our approach with the 2D interface of Maya. The results suggest that our system is much faster than Maya for object assembly. Users qualitatively found the system to be far more intuitive than the monitor-based alternative
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
When almost all sets are difference dominated
We investigate the relationship between the sizes of the sum and difference
sets attached to a subset of {0,1,...,N}, chosen randomly according to a
binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the
random subset is almost surely difference dominated, as N --> oo, for any
choice of p(N) tending to zero, thus confirming a conjecture of Martin and
O'Bryant. The proofs use recent strong concentration results.
Furthermore, we exhibit a threshold phenomenon regarding the ratio of the
size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all
sums and differences in the random subset are almost surely distinct, and in
particular the difference set is almost surely about twice as large as the
sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost
surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost
surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to
infinity (i.e., as the threshold is crossed), the same ratio almost surely
decreases continuously from two to one according to an explicitly given
function of c.
We also extend our results to the comparison of the generalized difference
sets attached to an arbitrary pair of binary linear forms. For certain pairs of
forms f and g, we show that there in fact exists a sharp threshold at c_{f,g}
N^{-1/2}, for some computable constant c_{f,g}, such that one form almost
surely dominates below the threshold, and the other almost surely above it.
The heart of our approach involves using different tools to obtain strong
concentration of the sizes of the sum and difference sets about their mean
values, for various ranges of the parameter p.Comment: Version 2.1. 24 pages. Fixed a few typos, updated reference
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