اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAlgorithms for the Problems of Length-Constrained Heaviest Segments
124
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 $k$ 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.
قيم البحث
اقرأ أيضاً
Given a set $P$ of $n$ points and a set $S$ of $m$ weighted disks in the plane, the disk coverage problem asks for a subset of disks of minimum total weight that cover all points of $P$. The problem is NP-hard. In this paper, we consider a line-const
rained version in which all disks are centered on a line $L$ (while points of $P$ can be anywhere in the plane). We present an $O((m+n)log(m+n)+kappalog m)$ time algorithm for the problem, where $kappa$ is the number of pairs of disks that intersect. Alternatively, we can also solve the problem in $O(nmlog(m+n))$ time. For the unit-disk case where all disks have the same radius, the running time can be reduced to $O((n+m)log(m+n))$. In addition, we solve in $O((m+n)log(m+n))$ time the $L_{infty}$ and $L_1$ cases of the problem, in which the disks are squares and diamonds, respectively. As a by-product, the 1D version of the problem where all points of $P$ are on $L$ and the disks are line segments on $L$ is also solved in $O((m+n)log(m+n))$ time. We also show that the problem has an $Omega((m+n)log (m+n))$ time lower bound even for the 1D case. We further demonstrate that our techniques can also be used to solve other geometric coverage problems. For example, given in the plane a set $P$ of $n$ points and a set $S$ of $n$ weighted half-planes, we solve in $O(n^4log n)$ time the problem of finding a subset of half-planes to cover $P$ so that their total weight is minimized. This improves the previous best algorithm of $O(n^5)$ time by almost a linear factor. If all half-planes are lower ones, then our algorithm runs in $O(n^2log n)$ time, which improves the previous best algorithm of $O(n^4)$ time by almost a quadratic factor.
In this paper, we first consider the subpath convex hull query problem: Given a simple path $pi$ of $n$ vertices, preprocess it so that the convex hull of any query subpath of $pi$ can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyin
k [SODA 90] proposed a data structure of $O(n)$ space and $O(log nloglog n)$ query time; reducing the query time to $O(log n)$ increases the space to $O(nloglog n)$. We present an improved result that uses $O(n)$ space while achieving $O(log n)$ query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in $O(log n)$ time each. Our new result leads to improvements for several other problems. In particular, with the help of the above result, we present new algorithms for the ray-shooting problem among segments. Given a set of $n$ (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of $O(nlog n)$ space that can answer each query in $(sqrt{n}log n)$ time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to $O(n)$. All these are classical problems that have been studied extensively. Previously data structures of $widetilde{O}(sqrt{n})$ query time (the notation $widetilde{O}$ suppresses a polylogarithmic factor) were known in early 1990s; nearly no progress has been made for over two decades. For all problems, our results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better.
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear naturally
and frequently in the fields like Machine learning, Data mining, and so on, and (ii) we are interested in extending the algorithms and techniques for single point set (monochromatic) problems to bichromatic case. For all the problems considered in this paper, we design low polynomial time exact algorithms. These algorithms are based on novel techniques which might be of independent interest.
We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the
fewest sensors (the hitting points). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. We prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.
We consider the $k$-center problem in which the centers are constrained to lie on two lines. Given a set of $n$ weighted points in the plane, we want to locate up to $k$ centers on two parallel lines. We present an $O(nlog^2 n)$ time algorithm, which
minimizes the weighted distance from any point to a center. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithms run in $O(nlog^2 n)$ time also in this case.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
