284 research outputs found
On Range Searching with Semialgebraic Sets II
Let be a set of points in . We present a linear-size data
structure for answering range queries on with constant-complexity
semialgebraic sets as ranges, in time close to . It essentially
matches the performance of similar structures for simplex range searching, and,
for , significantly improves earlier solutions by the first two authors
obtained in~1994. This almost settles a long-standing open problem in range
searching.
The data structure is based on the polynomial-partitioning technique of Guth
and Katz [arXiv:1011.4105], which shows that for a parameter , , there exists a -variate polynomial of degree such that
each connected component of contains at most points
of , where is the zero set of . We present an efficient randomized
algorithm for computing such a polynomial partition, which is of independent
interest and is likely to have additional applications
On the complexity of range searching among curves
Modern tracking technology has made the collection of large numbers of
densely sampled trajectories of moving objects widely available. We consider a
fundamental problem encountered when analysing such data: Given polygonal
curves in , preprocess into a data structure that answers
queries with a query curve and radius for the curves of that
have \Frechet distance at most to .
We initiate a comprehensive analysis of the space/query-time trade-off for
this data structuring problem. Our lower bounds imply that any data structure
in the pointer model model that achieves query time, where is
the output size, has to use roughly space in
the worst case, even if queries are mere points (for the discrete \Frechet
distance) or line segments (for the continuous \Frechet distance). More
importantly, we show that more complex queries and input curves lead to
additional logarithmic factors in the lower bound. Roughly speaking, the number
of logarithmic factors added is linear in the number of edges added to the
query and input curve complexity. This means that the space/query time
trade-off worsens by an exponential factor of input and query complexity. This
behaviour addresses an open question in the range searching literature: whether
it is possible to avoid the additional logarithmic factors in the space and
query time of a multilevel partition tree. We answer this question negatively.
On the positive side, we show we can build data structures for the \Frechet
distance by using semialgebraic range searching. Our solution for the discrete
\Frechet distance is in line with the lower bound, as the number of levels in
the data structure is , where denotes the maximal number of vertices
of a curve. For the continuous \Frechet distance, the number of levels
increases to
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Lower Bounds for Semialgebraic Range Searching and Stabbing Problems
In the semialgebraic range searching problem, we are to preprocess points
in s.t. for any query range from a family of constant complexity
semialgebraic sets, all the points intersecting the range can be reported or
counted efficiently. When the ranges are composed of simplices, the problem can
be solved using space and with query time with and this trade-off is almost tight. Consequently, there exists
low space structures that use space with query
time and fast query structures that use space with
query time. However, for the general semialgebraic ranges, only low space
solutions are known, but the best solutions match the same trade-off curve as
the simplex queries. It has been conjectured that the same could be done for
the fast query case but this open problem has stayed unresolved.
Here, we disprove this conjecture. We give the first nontrivial lower bounds
for semilagebraic range searching and related problems. We show that any data
structure for reporting the points between two concentric circles with
query time must use space, meaning, for
, space must be used. We also study
the problem of reporting the points between two polynomials of form
where are given at the
query time. We show . So
for , we must use space. For
the dual semialgebraic stabbing problems, we show that in linear space, any
data structure that solves 2D ring stabbing must use query
time. This almost matches the linearization upper bound. For general
semialgebraic slab stabbing problems, again, we show an almost tight lower
bounds.Comment: Submitted to SoCG'21; this version: readjust the table and other
minor change
Complexity in Automation of SOS Proofs: An Illustrative Example
We present a case study in proving invariance
for a chaotic dynamical system, the logistic map, based on
Positivstellensatz refutations, with the aim of studying the
problems associated with developing a completely automated
proof system. We derive the refutation using two different forms
of the Positivstellensatz and compare the results to illustrate the
challenges in defining and classifying the ‘complexity’ of such
a proof. The results show the flexibility of the SOS framework
in converting a dynamics problem into a semialgebraic one as
well as in choosing the form of the proof. Yet it is this very
flexibility that complicates the process of automating the proof
system and classifying proof ‘complexity.
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