95 research outputs found
Algorithms for the Problems of Length-Constrained Heaviest Segments
We present algorithms for length-constrained maximum sum segment and maximum
density segment problems, in particular, and the problem of finding
length-constrained heaviest segments, in general, for a sequence of real
numbers. Given a sequence of n real numbers and two real parameters L and U (L
<= U), the maximum sum segment problem is to find a consecutive subsequence,
called a segment, of length at least L and at most U such that the sum of the
numbers in the subsequence is maximum. The maximum density segment problem is
to find a segment of length at least L and at most U such that the density of
the numbers in the subsequence is the maximum. For the first problem with
non-uniform width there is an algorithm with time and space complexities in
O(n). We present an algorithm with time complexity in O(n) and space complexity
in O(U). For the second problem with non-uniform width there is a combinatorial
solution with time complexity in O(n) and space complexity in O(U). We present
a simple geometric algorithm with the same time and space complexities.
We extend our algorithms to respectively solve the length-constrained k
maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the
length-constrained maximum density segments problem in O(n min{k, U-L})
time and O(U+k) space. We present extensions of our algorithms to find all the
length-constrained segments having user specified sum and density in O(n+m) and
O(nlog (U-L)+m) times respectively, where m is the number of output.
Previously, there was no known algorithm with non-trivial result for these
problems. We indicate the extensions of our algorithms to higher dimensions.
All the algorithms can be extended in a straight forward way to solve the
problems with non-uniform width and non-uniform weight.Comment: 21 pages, 12 figure
Tight Hardness Results for Maximum Weight Rectangles
Given weighted points (positive or negative) in dimensions, what is
the axis-aligned box which maximizes the total weight of the points it
contains?
The best known algorithm for this problem is based on a reduction to a
related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in
time . It was conjectured [Barbay et al., CCCG'13] that this runtime is
tight up to subpolynomial factors. We answer this conjecture affirmatively by
providing a matching conditional lower bound. We also provide conditional lower
bounds for the special case when points are arranged in a grid (a well studied
problem known as Maximum Subarray problem) as well as for other related
problems.
All our lower bounds are based on assumptions that the best known algorithms
for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique
problem in edge-weighted graphs are essentially optimal
OPTYMALIZACJA PROBLEMU NAJWIĘKSZEJ PODTABLICY DLA SPECYFICZNYCH DANYCH
The maximum subarray problem (MSP) is to the find maximum contiguous sum in an array. This paper describes a method of Kadanes algorithm (the state of the art) optimization for specific data (continuous sequences of zeros or negative real numbers). When the data are unfavourable, the modification of the algorithm causes a non significant performance loss (1% > decrease in performance). The modification does not improve time complexity but reduces the number of elementary operations. Various experimental data sets have been used to evaluate possible time efficiency improvement. For the most favourable data sets an increase in efficiency of 25% can be achieved.Problem najwiekszej podtablicy to inaczej znalezienie podciągu, którego suma na największą wartość. Artykuł opisuje optymalizację algorytmu Kadane dla specyficznych danych (z powtarzającymi się ciągami zer lub liczb negatywnych). W przypadku niekorzystnych danych wejściowych zaproponowa modyfikacja nieznacznie spowalnia działanie algorytmu (mniej niż 1% szybkości działania). Ulepszenie algorytmu nie zmienia rzędu asymptotycznego tempa wzrostu, lecz zmniejsza ilość elementarnych operacji. Eksperymenty wykazały, że dla sprzyjających danych możemy zmniejszyć efektywny czas działania algorytmu o 25%
Algorithms in the Ultra-Wide Word Model
The effective use of parallel computing resources to speed up algorithms in
current multi-core parallel architectures remains a difficult challenge, with
ease of programming playing a key role in the eventual success of various
parallel architectures. In this paper we consider an alternative view of
parallelism in the form of an ultra-wide word processor. We introduce the
Ultra-Wide Word architecture and model, an extension of the word-RAM model that
allows for constant time operations on thousands of bits in parallel. Word
parallelism as exploited by the word-RAM model does not suffer from the more
difficult aspects of parallel programming, namely synchronization and
concurrency. For the standard word-RAM algorithms, the speedups obtained are
moderate, as they are limited by the word size. We argue that a large class of
word-RAM algorithms can be implemented in the Ultra-Wide Word model, obtaining
speedups comparable to multi-threaded computations while keeping the simplicity
of programming of the sequential RAM model. We show that this is the case by
describing implementations of Ultra-Wide Word algorithms for dynamic
programming and string searching. In addition, we show that the Ultra-Wide Word
model can be used to implement a nonstandard memory architecture, which enables
the sidestepping of lower bounds of important data structure problems such as
priority queues and dynamic prefix sums. While similar ideas about operating on
large words have been mentioned before in the context of multimedia processors
[Thorup 2003], it is only recently that an architecture like the one we propose
has become feasible and that details can be worked out.Comment: 28 pages, 5 figures; minor change
Parallel Algorithm for Finding All Minimal Maximum Subsequences via Random-walk Theory
A maximum contiguous subsequence of a real-valued sequence is a contiguous subsequence with the maximum cumulative sum. A minimal maximum contiguous subsequence is a minimal contiguous subsequence among all maximum ones of the sequence. Time- and space-efficient algorithms for finding the single or multiple contiguous subsequences of a real-valued sequence with large cumulative sums, in addition to its combinatorial appeal, have major applications such as in bioinformatics, pattern matching, and data mining. We have designed and implemented a domain-decomposed parallel algorithm on cluster systems with Message Passing Interface that finds all minimal maximum subsequences of a random sample sequence from a normal distribution with negative mean. We find the structural decomposition of the sequence with overlapping common subsequences for adjacent processors that allow hosting processors to compute the minimal maximum subsequences of the sequence independently. Our study employs the theory of random walk to derive an approximate probabilistic length bound for common subsequences in an appropriate probabilistic setting, which is incorporated in the algorithm to facilitate the concurrent computation of all minimal maximum subsequences in hosting processors. We also present an empirical study of the speedup and efficiency achieved by the parallel algorithm with synthetic random data.Computer Scienc
Combinatorial algorithms in the approximate computing paradigm
Data-intensive computing has led to the emergence of data centers with massive processor counts and main memory sizes. However, the demand for shared resources has surpassed their capacity, resulting in increased costs and limited access. Commodity hardware, although accessible, has limited computational resources. This poses a challenge when performing computationally intensive tasks with large amounts of data on systems with restricted memory. To address these issues, Approximate Computing offers a solution by allowing selective solution approximation, leading to improved resource efficiency.
This dissertation focuses on the trade-off between output quality and computational resource usage in sorting and searching problems. It introduces the concept of Approximate Sorting, which aims to reduce resource usage while maintaining an accepted level of sorting quality. Quality metrics are defined to assess the ”sortedness” of approximately sorted arrays. The dissertation also proposes a general framework for incorporating approximate computing into sorting algorithms, presenting an algorithm for approximate sorting with guaranteed upper bounds. The algorithms operate under a constraint on the number of comparisons performed.
The dissertation continues to explore searching algorithms, specifically binary search algorithms on approximately sorted arrays. It addresses cases where metrics are given for the input array and cases where metrics are not available. Efficient and optimal algorithms are developed for multidimensional range searches and catalog searches on approximately sorted input. The dissertation further proposes algorithms that analyze patterns in input order to optimize sorting. These algorithms identify underlying patterns and sequences, facilitating faster sorting approaches.
Additionally, the dissertation discusses the growing popularity of approximate computing in the field of High-Performance Computing (HPC). It presents a novel approach to comparison-based sorting by incorporating parallel approximate computing. The dissertation also proposes algorithms for various queries on approximately sorted arrays, such as determining the rank or position of an element. The time complexity of these querying algorithms is proportional to the input metric.
The dissertation concludes by emphasizing the wide range of applications for sorting and searching algorithms. In the context of packet classification in router buffers, approximate sorting offers advantages by reducing the time-consuming sorting step. By capping the number of comparisons, approximate sorting becomes a practical solution for efficiently handling the large volume of incoming packets.
This dissertation contributes to the field of approximate computing by addressing resource limitations and cost issues in data-intensive computing. It provides insights into approximate sorting and searching algorithms, and their application in various domains, offering a valuable contribution to the advancement of efficient, scalable, and accessible data processing
New Sublinear Algorithms and Lower Bounds for LIS Estimation
Estimating the length of the longest increasing subsequence (LIS) in an array
is a problem of fundamental importance. Despite the significance of the LIS
estimation problem and the amount of attention it has received, there are
important aspects of the problem that are not yet fully understood. There are
no better lower bounds for LIS estimation than the obvious bounds implied by
testing monotonicity (for adaptive or nonadaptive algorithms). In this paper,
we give the first nontrivial lower bound on the complexity of LIS estimation,
and also provide novel algorithms that complement our lower bound.
Specifically, for every constant , every nonadaptive
algorithm that outputs an estimate of the length of the LIS in an array of
length to within an additive error of has to make
queries. Next, we design nonadaptive LIS
estimation algorithms whose complexity decreases as the the number of distinct
values, , in the array decreases. We first present a simple algorithm that
makes queries and approximates the LIS length with an
additive error bounded by . We then use it to construct a
nonadaptive algorithm with query complexity that, for an array with LIS length at least , outputs a multiplicative -approximation to the LIS length.
Finally, we describe a nonadaptive erasure-resilient tester for sortedness,
with query complexity . Our result implies that nonadaptive tolerant
testing is strictly harder than nonadaptive erasure-resilient testing for the
natural property of monotonicity.Comment: 32 pages, 3 figure
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