30,168 research outputs found
Optimal Encodings for Range Min-Max and Top-k
In this paper we consider various encoding problems for range queries on arrays. In these problems, the goal is that the encoding occupies the information theoretic minimum space required to answer a particular set of range queries. Given an array a range top- query on an arbitrary range asks us to return the ordered set of indices such that is the -th largest element in . We present optimal encodings for range top- queries, as well as for a new problem which we call range min-max, in which the goal is to return the indices of both the minimum and maximum element in a range
LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations
LRM-Trees are an elegant way to partition a sequence of values into sorted
consecutive blocks, and to express the relative position of the first element
of each block within a previous block. They were used to encode ordinal trees
and to index integer arrays in order to support range minimum queries on them.
We describe how they yield many other convenient results in a variety of areas,
from data structures to algorithms: some compressed succinct indices for range
minimum queries; a new adaptive sorting algorithm; and a compressed succinct
data structure for permutations supporting direct and indirect application in
time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur
Combined Data Structure for Previous- and Next-Smaller-Values
Let be a static array storing elements from a totally ordered set. We
present a data structure of optimal size at most
bits that allows us to answer the following queries on in constant time,
without accessing : (1) previous smaller value queries, where given an index
, we wish to find the first index to the left of where is strictly
smaller than at , and (2) next smaller value queries, which search to the
right of . As an additional bonus, our data structure also allows to answer
a third kind of query: given indices , find the position of the minimum in
. Our data structure has direct consequences for the space-efficient
storage of suffix trees.Comment: to appear in Theoretical Computer Scienc
Towards a compact representation of temporal rasters
Big research efforts have been devoted to efficiently manage spatio-temporal
data. However, most works focused on vectorial data, and much less, on raster
data. This work presents a new representation for raster data that evolve along
time named Temporal k^2 raster. It faces the two main issues that arise when
dealing with spatio-temporal data: the space consumption and the query response
times. It extends a compact data structure for raster data in order to manage
time and thus, it is possible to query it directly in compressed form, instead
of the classical approach that requires a complete decompression before any
manipulation. In addition, in the same compressed space, the new data structure
includes two indexes: a spatial index and an index on the values of the cells,
thus becoming a self-index for raster data.Comment: This research has received funding from the European Union's Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie
Actions H2020-MSCA-RISE-2015 BIRDS GA No. 690941. Published in SPIRE 201
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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