387 research outputs found
Data Structure Lower Bounds for Document Indexing Problems
We study data structure problems related to document indexing and pattern
matching queries and our main contribution is to show that the pointer machine
model of computation can be extremely useful in proving high and unconditional
lower bounds that cannot be obtained in any other known model of computation
with the current techniques. Often our lower bounds match the known space-query
time trade-off curve and in fact for all the problems considered, there is a
very good and reasonable match between the our lower bounds and the known upper
bounds, at least for some choice of input parameters. The problems that we
consider are set intersection queries (both the reporting variant and the
semi-group counting variant), indexing a set of documents for two-pattern
queries, or forbidden- pattern queries, or queries with wild-cards, and
indexing an input set of gapped-patterns (or two-patterns) to find those
matching a document given at the query time.Comment: Full version of the conference version that appeared at ICALP 2016,
25 page
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 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
Simplex range reporting on a pointer machine
AbstractWe give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P ∩ q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(nδ + r), where r is the number of points to be reported and δ is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Ω(nd(1 − δ)− ε), for any fixed ε > 0. This lower bound is tight within a factor of nε
Adapt Or Die: Polynomial Lower Bounds For Non-Adaptive Dynamic Data Structures
In this paper, we study the role non-adaptivity plays in maintaining dynamic data structures. Roughly speaking, a data structure is non-adaptive if the memory locations it reads and/or writes when processing a query or update depend only on the query or update and not on the contents of previously read cells. We study such non-adaptive data structures in the cell probe model. The cell probe model is one of the least restrictive lower bound models and in particular, cell probe lower bounds apply to data structures developed in the popular word-RAM model. Unfortunately, this generality comes at a high cost: the highest lower bound proved for any data structure problem is only polylogarithmic (if allowed adaptivity). Our main result is to demonstrate that one can in fact obtain polynomial cell probe lower bounds for non-adaptive data structures. To shed more light on the seemingly inherent polylogarithmic lower bound barrier, we study several different notions of non-adaptivity and identify key properties that must be dealt with if we are to prove polynomial lower bounds without restrictions on the data structures. Finally, our results also unveil an interesting connection between data structures and depth-2 circuits. This allows us to translate conjectured hard data structure problems into good candidates for high circuit lower bounds; in particular, in the area of linear circuits for linear operators. Building on lower bound proofs for data structures in slightly more restrictive models, we also present a number of properties of linear operators which we believe are worth investigating in the realm of circuit lower bounds
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