35,432 research outputs found
Encodings of Range Maximum-Sum Segment Queries and Applications
Given an array A containing arbitrary (positive and negative) numbers, we consider the problem of supporting range maximum-sum segment queries on A: i.e., given an arbitrary range [i,j], return the subrange [i' ,j' ] \subseteq [i,j] such that the sum of the numbers in A[i'..j'] is maximized. Chen and Chao [Disc. App. Math. 2007] presented a data structure for this problem that occupies {\Theta}(n) words, can be constructed in {\Theta}(n) time, and supports queries in {\Theta}(1) time. Our first result is that if only the indices [i',j'] are desired (rather than the maximum sum achieved in that subrange), then it is possible to reduce the space to {\Theta}(n) bits, regardless the numbers stored in A, while retaining the same construction and query time. We also improve the best known space lower bound for any data structure that supports range maximum-sum segment queries from n bits to 1.89113n - {\Theta}(lg n) bits, for sufficiently large values of n. Finally, we provide a new application of this data structure which simplifies a previously known linear time algorithm for finding k-covers: i.e., given an array A of n numbers and a number k, find k disjoint subranges [i_1 ,j_1 ],...,[i_k ,j_k ], such that the total sum of all the numbers in the subranges is maximized
Encodings of Range Maximum-Sum Segment Queries and Applications
Given an array A containing arbitrary (positive and negative) numbers, we
consider the problem of supporting range maximum-sum segment queries on A:
i.e., given an arbitrary range [i,j], return the subrange [i' ,j' ] \subseteq
[i,j] such that the sum of the numbers in A[i'..j'] is maximized. Chen and Chao
[Disc. App. Math. 2007] presented a data structure for this problem that
occupies {\Theta}(n) words, can be constructed in {\Theta}(n) time, and
supports queries in {\Theta}(1) time. Our first result is that if only the
indices [i',j'] are desired (rather than the maximum sum achieved in that
subrange), then it is possible to reduce the space to {\Theta}(n) bits,
regardless the numbers stored in A, while retaining the same construction and
query time. We also improve the best known space lower bound for any data
structure that supports range maximum-sum segment queries from n bits to
1.89113n - {\Theta}(lg n) bits, for sufficiently large values of n. Finally, we
provide a new application of this data structure which simplifies a previously
known linear time algorithm for finding k-covers: i.e., given an array A of n
numbers and a number k, find k disjoint subranges [i_1 ,j_1 ],...,[i_k ,j_k ],
such that the total sum of all the numbers in the subranges is maximized.Comment: 19 pages + 2 page appendix, 4 figures. A shortened version of this
paper will appear in CPM 201
Pattern Matching for sets of segments
In this paper we present algorithms for a number of problems in geometric
pattern matching where the input consist of a collections of segments in the
plane. Our work consists of two main parts. In the first, we address problems
and measures that relate to collections of orthogonal line segments in the
plane. Such collections arise naturally from problems in mapping buildings and
robot exploration.
We propose a new measure of segment similarity called a \emph{coverage
measure}, and present efficient algorithms for maximising this measure between
sets of axis-parallel segments under translations. Our algorithms run in time
O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for
the case when all segments are horizontal. In addition, we show that when
restricted to translations that are only vertical, the Hausdorff distance
between two sets of horizontal segments can be computed in time roughly
O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over
the general algorithm of Chew et al. that takes time . In the
second part of this paper we address the problem of matching polygonal chains.
We study the well known \Frd, and present the first algorithm for computing the
\Frd under general translations. Our methods also yield algorithms for
computing a generalization of the \Fr distance, and we also present a simple
approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200
Ranking Large Temporal Data
Ranking temporal data has not been studied until recently, even though
ranking is an important operator (being promoted as a firstclass citizen) in
database systems. However, only the instant top-k queries on temporal data were
studied in, where objects with the k highest scores at a query time instance t
are to be retrieved. The instant top-k definition clearly comes with
limitations (sensitive to outliers, difficult to choose a meaningful query time
t). A more flexible and general ranking operation is to rank objects based on
the aggregation of their scores in a query interval, which we dub the aggregate
top-k query on temporal data. For example, return the top-10 weather stations
having the highest average temperature from 10/01/2010 to 10/07/2010; find the
top-20 stocks having the largest total transaction volumes from 02/05/2011 to
02/07/2011. This work presents a comprehensive study to this problem by
designing both exact and approximate methods (with approximation quality
guarantees). We also provide theoretical analysis on the construction cost, the
index size, the update and the query costs of each approach. Extensive
experiments on large real datasets clearly demonstrate the efficiency, the
effectiveness, and the scalability of our methods compared to the baseline
methods.Comment: VLDB201
Static Data Structure for Discrete Advance Bandwidth Reservations on the Internet
In this paper we present a discrete data structure for reservations of
limited resources. A reservation is defined as a tuple consisting of the time
interval of when the resource should be reserved, , and the amount of the
resource that is reserved, , formally .
The data structure is similar to a segment tree. The maximum spanning
interval of the data structure is fixed and defined in advance. The granularity
and thereby the size of the intervals of the leaves is also defined in advance.
The data structure is built only once. Neither nodes nor leaves are ever
inserted, deleted or moved. Hence, the running time of the operations does not
depend on the number of reservations previously made. The running time does not
depend on the size of the interval of the reservation either. Let be the
number of leaves in the data structure. In the worst case, the number of
touched (i.e. traversed) nodes is in any operation , hence the
running time of any operation is also
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