356 research outputs found
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
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
An O(n^3)-Time Algorithm for Tree Edit Distance
The {\em edit distance} between two ordered trees with vertex labels is the
minimum cost of transforming one tree into the other by a sequence of
elementary operations consisting of deleting and relabeling existing nodes, as
well as inserting new nodes. In this paper, we present a worst-case
-time algorithm for this problem, improving the previous best
-time algorithm~\cite{Klein}. Our result requires a novel
adaptive strategy for deciding how a dynamic program divides into subproblems
(which is interesting in its own right), together with a deeper understanding
of the previous algorithms for the problem. We also prove the optimality of our
algorithm among the family of \emph{decomposition strategy} algorithms--which
also includes the previous fastest algorithms--by tightening the known lower
bound of ~\cite{Touzet} to , matching our
algorithm's running time. Furthermore, we obtain matching upper and lower
bounds of when the two trees have
different sizes and~, where .Comment: 10 pages, 5 figures, 5 .tex files where TED.tex is the main on
Locked and Unlocked Polygonal Chains in 3D
In this paper, we study movements of simple polygonal chains in 3D. We say
that an open, simple polygonal chain can be straightened if it can be
continuously reconfigured to a straight sequence of segments in such a manner
that both the length of each link and the simplicity of the chain are
maintained throughout the movement. The analogous concept for closed chains is
convexification: reconfiguration to a planar convex polygon. Chains that cannot
be straightened or convexified are called locked. While there are open chains
in 3D that are locked, we show that if an open chain has a simple orthogonal
projection onto some plane, it can be straightened. For closed chains, we show
that there are unknotted but locked closed chains, and we provide an algorithm
for convexifying a planar simple polygon in 3D with a polynomial number of
moves.Comment: To appear in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan.
199
Reconstructing David Huffman's Origami Tessellations
David A. Huffman (1925–1999) is best known in computer science for his work in information theory, particularly Huffman codes, and best known in origami as a pioneer of curved-crease folding. But during his early paper folding in the 1970s, he also designed and folded over a 100 different straight-crease origami tessellations. Unlike most origami tessellations designed in the past 20 years, Huffman's straight-crease tessellations are mostly three-dimensional, rigidly foldable, and have no locking mechanism. In collaboration with Huffman's family, our goal is to document all of his designs by reverse-engineering his models into the corresponding crease patterns, or in some cases, matching his models with his sketches of crease patterns. Here, we describe several of Huffman's origami tessellations that are most interesting historically, mathematically, and artistically.National Science Foundation (U.S.) (Origami Design for Integration of Self-assembling Systems for Engineering Innovation Grant EFRI-1240383)National Science Foundation (U.S.) (Expedition Grant CCF-1138967
Contraction Bidimensionality: the Accurate Picture
We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory -- the meta algorithmic framework to design efficient parameterized and approximation algorithms for contraction closed parameters
On the Structure of Equilibria in Basic Network Formation
We study network connection games where the nodes of a network perform edge
swaps in order to improve their communication costs. For the model proposed by
Alon et al. (2010), in which the selfish cost of a node is the sum of all
shortest path distances to the other nodes, we use the probabilistic method to
provide a new, structural characterization of equilibrium graphs. We show how
to use this characterization in order to prove upper bounds on the diameter of
equilibrium graphs in terms of the size of the largest -vicinity (defined as
the the set of vertices within distance from a vertex), for any
and in terms of the number of edges, thus settling positively a conjecture of
Alon et al. in the cases of graphs of large -vicinity size (including graphs
of large maximum degree) and of graphs which are dense enough.
Next, we present a new swap-based network creation game, in which selfish
costs depend on the immediate neighborhood of each node; in particular, the
profit of a node is defined as the sum of the degrees of its neighbors. We
prove that, in contrast to the previous model, this network creation game
admits an exact potential, and also that any equilibrium graph contains an
induced star. The existence of the potential function is exploited in order to
show that an equilibrium can be reached in expected polynomial time even in the
case where nodes can only acquire limited knowledge concerning non-neighboring
nodes.Comment: 11 pages, 4 figure
Integrated music and math projects in secondary education
The introduction of projects involving music and mathematics in
Secondary Education should allow the integration of these disciplines by nonspecialists.
In the present work we describe an experience carried out with future
mathematics teachers with solid scientific-technical training, but little musical
training. This pilot contributes with concrete orientations and results para the
creation and development of STEAM activities, which can be found in [5].This research work has been carried out within the framework of the project EDU2017-84979-R, of the Spanish State Program of R&D and Innovation Oriented to the Challenges of the Society
Beyond Worst-Case Analysis for Joins with Minesweeper
We describe a new algorithm, Minesweeper, that is able to satisfy stronger
runtime guarantees than previous join algorithms (colloquially, `beyond
worst-case guarantees') for data in indexed search trees. Our first
contribution is developing a framework to measure this stronger notion of
complexity, which we call {\it certificate complexity}, that extends notions of
Barbay et al. and Demaine et al.; a certificate is a set of propositional
formulae that certifies that the output is correct. This notion captures a
natural class of join algorithms. In addition, the certificate allows us to
define a strictly stronger notion of runtime complexity than traditional
worst-case guarantees. Our second contribution is to develop a dichotomy
theorem for the certificate-based notion of complexity. Roughly, we show that
Minesweeper evaluates -acyclic queries in time linear in the certificate
plus the output size, while for any -cyclic query there is some instance
that takes superlinear time in the certificate (and for which the output is no
larger than the certificate size). We also extend our certificate-complexity
analysis to queries with bounded treewidth and the triangle query.Comment: [This is the full version of our PODS'2014 paper.
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