34 research outputs found
I/O-Efficient Planar Range Skyline and Attrition Priority Queues
In the planar range skyline reporting problem, we store a set P of n 2D
points in a structure such that, given a query rectangle Q = [a_1, a_2] x [b_1,
b_2], the maxima (a.k.a. skyline) of P \cap Q can be reported efficiently. The
query is 3-sided if an edge of Q is grounded, giving rise to two variants:
top-open (b_2 = \infty) and left-open (a_1 = -\infty) queries.
All our results are in external memory under the O(n/B) space budget, for
both the static and dynamic settings:
* For static P, we give structures that answer top-open queries in O(log_B n
+ k/B), O(loglog_B U + k/B), and O(1 + k/B) I/Os when the universe is R^2, a U
x U grid, and a rank space grid [O(n)]^2, respectively (where k is the number
of reported points). The query complexity is optimal in all cases.
* We show that the left-open case is harder, such that any linear-size
structure must incur \Omega((n/B)^e + k/B) I/Os for a query. We show that this
case is as difficult as the general 4-sided queries, for which we give a static
structure with the optimal query cost O((n/B)^e + k/B).
* We give a dynamic structure that supports top-open queries in O(log_2B^e
(n/B) + k/B^1-e) I/Os, and updates in O(log_2B^e (n/B)) I/Os, for any e
satisfying 0 \le e \le 1. This leads to a dynamic structure for 4-sided queries
with optimal query cost O((n/B)^e + k/B), and amortized update cost O(log
(n/B)).
As a contribution of independent interest, we propose an I/O-efficient
version of the fundamental structure priority queue with attrition (PQA). Our
PQA supports FindMin, DeleteMin, and InsertAndAttrite all in O(1) worst case
I/Os, and O(1/B) amortized I/Os per operation.
We also add the new CatenateAndAttrite operation that catenates two PQAs in
O(1) worst case and O(1/B) amortized I/Os. This operation is a non-trivial
extension to the classic PQA of Sundar, even in internal memory.Comment: Appeared at PODS 2013, New York, 19 pages, 10 figures. arXiv admin
note: text overlap with arXiv:1208.4511, arXiv:1207.234
Online Data Structures in External Memory
The original publication is available at www.springerlink.comThe data sets for many of today's computer applications are
too large to t within the computer's internal memory and must instead
be stored on external storage devices such as disks. A major performance
bottleneck can be the input/output communication (or I/O) between
the external and internal memories. In this paper we discuss a variety of
online data structures for external memory, some very old and some very
new, such as hashing (for dictionaries), B-trees (for dictionaries and 1-D
range search), bu er trees (for batched dynamic problems), interval trees
with weight-balanced B-trees (for stabbing queries), priority search trees
(for 3-sided 2-D range search), and R-trees and other spatial structures.
We also discuss several open problems along the way
I/O-Efficient Dynamic Planar Range Skyline Queries
We present the first fully dynamic worst case I/O-efficient data structures
that support planar orthogonal \textit{3-sided range skyline reporting queries}
in \bigO (\log_{2B^\epsilon} n + \frac{t}{B^{1-\epsilon}}) I/Os and updates
in \bigO (\log_{2B^\epsilon} n) I/Os, using \bigO
(\frac{n}{B^{1-\epsilon}}) blocks of space, for input planar points,
reported points, and parameter . We obtain the result
by extending Sundar's priority queues with attrition to support the operations
\textsc{DeleteMin} and \textsc{CatenateAndAttrite} in \bigO (1) worst case
I/Os, and in \bigO(1/B) amortized I/Os given that a constant number of blocks
is already loaded in main memory. Finally, we show that any pointer-based
static data structure that supports \textit{dominated maxima reporting
queries}, namely the difficult special case of 4-sided skyline queries, in
\bigO(\log^{\bigO(1)}n +t) worst case time must occupy space, by adapting a similar lower bounding argument for
planar 4-sided range reporting queries.Comment: Submitted to SODA 201
I/O-efficient 2-d orthogonal range skyline and attrition priority queues
In the planar range skyline reporting problem, we store a set P of n 2D points in a structure such that, given a query rectangle Q = [a_1, a_2] x [b_1, b_2], the maxima (a.k.a. skyline) of P \cap Q can be reported efficiently. The query is 3-sided if an edge of Q is grounded, giving rise to two variants: top-open (b_2 = \infty) and left-open (a_1 = -\infty) queries. All our results are in external memory under the O(n/B) space budget, for both the static and dynamic settings: * For static P, we give structures that answer top-open queries in O(log_B n + k/B), O(loglog_B U + k/B), and O(1 + k/B) I/Os when the universe is R^2, a U x U grid, and a rank space grid [O(n)]^2, respectively (where k is the number of reported points). The query complexity is optimal in all cases. * We show that the left-open case is harder, such that any linear-size structure must incur \Omega((n/B)^e + k/B) I/Os for a query. We show that this case is as difficult as the general 4-sided queries, for which we give a static structure with the optimal query cost O((n/B)^e + k/B). * We give a dynamic structure that supports top-open queries in O(log_2B^e (n/B) + k/B^1-e) I/Os, and updates in O(log_2B^e (n/B)) I/Os, for any e satisfying 0 \le e \le 1. This leads to a dynamic structure for 4-sided queries with optimal query cost O((n/B)^e + k/B), and amortized update cost O(log (n/B)). As a contribution of independent interest, we propose an I/O-efficient version of the fundamental structure priority queue with attrition (PQA). Our PQA supports FindMin, DeleteMin, and InsertAndAttrite all in O(1) worst case I/Os, and O(1/B) amortized I/Os per operation. We also add the new CatenateAndAttrite operation that catenates two PQAs in O(1) worst case and O(1/B) amortized I/Os. This operation is a non-trivial extension to the classic PQA of Sundar, even in internal memory
08081 Abstracts Collection -- Data Structures
From February 17th to 22nd 2008, the Dagstuhl Seminar 08081 ``Data Structures\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl. It brought together 49 researchers from four continents to discuss recent developments concerning data structures in terms of research but also in terms of new technologies that impact how data can be stored, updated,
and retrieved.
During the seminar a fair number of participants presented their current
research. There was discussion of ongoing work, and in addition an open problem
session was held. This paper first describes the seminar topics and goals in general, then gives the minutes of the open problem session, and concludes with
abstracts of the presentations given during the seminar.
Where appropriate and available, links to extended abstracts or full papers are provided