202 research outputs found
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
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Approximate range searching☆☆A preliminary version of this paper appeared in the Proc. of the 11th Annual ACM Symp. on Computational Geometry, 1995, pp. 172–181.
AbstractThe range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and ε>0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance εw of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in Rd can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(1/ε)d) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O(logn+(1/ε)d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/ε)d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times
Evaluation of Labeling Strategies for Rotating Maps
We consider the following problem of labeling points in a dynamic map that
allows rotation. We are given a set of points in the plane labeled by a set of
mutually disjoint labels, where each label is an axis-aligned rectangle
attached with one corner to its respective point. We require that each label
remains horizontally aligned during the map rotation and our goal is to find a
set of mutually non-overlapping active labels for every rotation angle so that the number of active labels over a full map rotation of
2 is maximized. We discuss and experimentally evaluate several labeling
models that define additional consistency constraints on label activities in
order to reduce flickering effects during monotone map rotation. We introduce
three heuristic algorithms and compare them experimentally to an existing
approximation algorithm and exact solutions obtained from an integer linear
program. Our results show that on the one hand low flickering can be achieved
at the expense of only a small reduction in the objective value, and that on
the other hand the proposed heuristics achieve a high labeling quality
significantly faster than the other methods.Comment: 16 pages, extended version of a SEA 2014 pape
On Optimal Top-K String Retrieval
Let = be a given set of
(string) documents of total length . The top- document retrieval problem
is to index such that when a pattern of length , and a
parameter come as a query, the index returns the most relevant
documents to the pattern . Hon et. al. \cite{HSV09} gave the first linear
space framework to solve this problem in time. This was
improved by Navarro and Nekrich \cite{NN12} to . These results are
powerful enough to support arbitrary relevance functions like frequency,
proximity, PageRank, etc. In many applications like desktop or email search,
the data resides on disk and hence disk-bound indexes are needed. Despite of
continued progress on this problem in terms of theoretical, practical and
compression aspects, any non-trivial bounds in external memory model have so
far been elusive. Internal memory (or RAM) solution to this problem decomposes
the problem into subproblems and thus incurs the additive factor of
. In external memory, these approaches will lead to I/Os instead
of optimal I/O term where is the block-size. We re-interpret the
problem independent of , as interval stabbing with priority over tree-shaped
structure. This leads us to a linear space index in external memory supporting
top- queries (with unsorted outputs) in near optimal I/Os for any constant { and
}. Then we get space index
with optimal I/Os.Comment: 3 figure
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