16,655 research outputs found
Further Results on Colored Range Searching
We present a number of new results about range searching for colored (or
"categorical") data:
1. For a set of colored points in three dimensions, we describe
randomized data structures with space that can
report the distinct colors in any query orthogonal range (axis-aligned box) in
expected time, where is the number of
distinct colors in the range, assuming that coordinates are in
. Previous data structures require query time. Our result also implies improvements in higher constant
dimensions.
2. Our data structures can be adapted to halfspace ranges in three dimensions
(or circular ranges in two dimensions), achieving expected query
time. Previous data structures require query time.
3. For a set of colored points in two dimensions, we describe a data
structure with space that can answer colored
"type-2" range counting queries: report the number of occurrences of every
distinct color in a query orthogonal range. The query time is , where is the number of distinct colors in
the range. Naively performing uncolored range counting queries would
require time.
Our data structures are designed using a variety of techniques, including
colored variants of randomized incremental construction (which may be of
independent interest), colored variants of shallow cuttings, and bit-packing
tricks.Comment: full version of a SoCG'20 pape
Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication
This paper develops a new technique for proving amortized, randomized
cell-probe lower bounds on dynamic data structure problems. We introduce a new
randomized nondeterministic four-party communication model that enables
"accelerated", error-preserving simulations of dynamic data structures.
We use this technique to prove an cell-probe
lower bound for the dynamic 2D weighted orthogonal range counting problem
(2D-ORC) with updates and queries, that holds even
for data structures with success probability. This
result not only proves the highest amortized lower bound to date, but is also
tight in the strongest possible sense, as a matching upper bound can be
obtained by a deterministic data structure with worst-case operational time.
This is the first demonstration of a "sharp threshold" phenomenon for dynamic
data structures.
Our broader motivation is that cell-probe lower bounds for exponentially
small success facilitate reductions from dynamic to static data structures. As
a proof-of-concept, we show that a slightly strengthened version of our lower
bound would imply an lower bound for the
static 3D-ORC problem with space. Such result would give a
near quadratic improvement over the highest known static cell-probe lower
bound, and break the long standing barrier for static data
structures
Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles
In this paper we study two geometric data structure problems in the special
case when input objects or queries are fat rectangles. We show that in this
case a significant improvement compared to the general case can be achieved.
We describe data structures that answer two- and three-dimensional orthogonal
range reporting queries in the case when the query range is a \emph{fat}
rectangle. Our two-dimensional data structure uses words and supports
queries in time, where is the number of points in the
data structure, is the size of the universe and is the number of points
in the query range. Our three-dimensional data structure needs
words of space and answers queries in time. We also consider the rectangle stabbing problem on a set of
three-dimensional fat rectangles. Our data structure uses space and
answers stabbing queries in time.Comment: extended version of a WADS'19 pape
Towards Tight Lower Bounds for Range Reporting on the RAM
In the orthogonal range reporting problem, we are to preprocess a set of
points with integer coordinates on a grid. The goal is to support
reporting all points inside an axis-aligned query rectangle. This is one of
the most fundamental data structure problems in databases and computational
geometry. Despite the importance of the problem its complexity remains
unresolved in the word-RAM. On the upper bound side, three best tradeoffs
exists: (1.) Query time with words
of space for any constant . (2.) Query time with words of space. (3.) Query time
with optimal words of space. However, the
only known query time lower bound is , even for linear
space data structures.
All three current best upper bound tradeoffs are derived by reducing range
reporting to a ball-inheritance problem. Ball-inheritance is a problem that
essentially encapsulates all previous attempts at solving range reporting in
the word-RAM. In this paper we make progress towards closing the gap between
the upper and lower bounds for range reporting by proving cell probe lower
bounds for ball-inheritance. Our lower bounds are tight for a large range of
parameters, excluding any further progress for range reporting using the
ball-inheritance reduction
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