Optimal algorithms for rectangle problems on a mesh-connected computer

In this paper, Mesh-Connected Computer (MCC) algorithms for computing several properties of a set of, possibly intersecting rectangles are presented. Given a set of n iso-oriented rectangles, we describe MCC algorithms for determining the following properties: (i) the area of the logic “OR” of these rectangles (i.e., the area of the region covered by at least one rectangle); (ii) the area of the union of pairwise “AND” of the rectangles (i.e., the area of the region covered by two or more rectangles); (iii) the largest number of rectangles that overlap (this solves the fixed-size rectangle placement problem, i.e., given a set of planar points and a rectangle, find a placement of the rectangle in the plane so that the number of points covered by the rectangle is maximal); (iv) the minimum separation between any pair of a set of nonoverlapping rectangles. All these algorithms can be implemented on a 2√n × 2√n MCC in O(√n) time which is optimal. The algorithms compare favorably with the known sequential algorithms that have O(n log n) time complexity.     

Approximate motion planning and the complexity of the boundary of the union of simple geometric figures

We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have a running time of Ω(n ²), wheren is the number of obstacle corners. We introduce thetightness of a motion-planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with a running time ofO((a/b · 1/?_crit + 1)n(logn)²), wherea ≥b are the lengths of the sides of a rectangle and ?_crit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary ofn bow ties (see Figure 1) isO(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.     

Identifying rectangles in laser range data for urban scene reconstruction

In urban scenes, many of the surfaces are planar and bounded by simple shapes. In a laser scan of such a scene, these simple shapes can still be identified. We present a one-parameter algorithm that can identify point sets on a plane for which a rectangle is a fitting boundary. These rectangles have a guaranteed density: no large part of the rectangle is empty of points. We prove that our algorithm identifies all angles for which a rectangle fits the point set of size n in O(nlogn) time. We evaluate our method experimentally on 13 urban data sets and we compare the rectangles found by our algorithm to the α‐shape as a surface boundary.     

Cluster scheduling for real-time systems: utilization bounds and run-time overhead

Cluster scheduling, where processors are grouped into clusters and the tasks that are allocated to one cluster are scheduled by a global scheduler, has attracted attention in multiprocessor real-time systems research recently. In this paper, assuming that an optimal global scheduler is adopted within each cluster, we investigate the worst-case utilization bounds for cluster scheduling with different task allocation/partitioning heuristics. First, we develop a lower limit on the utilization bounds for cluster scheduling with any reasonable task allocation scheme. Then, the lower limit is shown to be the exact utilization bound for cluster scheduling with the worst-fit task allocation scheme. For other task allocation heuristics (such as first-fit, best-fit, first-fit decreasing, best-fit decreasing and worst-fit decreasing), higher utilization bounds are derived for systems with both homogeneous clusters (where each cluster has the same number of processors) and heterogeneous clusters (where clusters have different number of processors). In addition, focusing on an efficient optimal global scheduler, namely the boundary-fair (Bfair) algorithm, we propose a period-aware task allocation heuristic with the goal of reducing the scheduling overhead (e.g., the number of scheduling points, context switches and task migrations). Simulation results indicate that the percentage of task sets that can be scheduled is significantly improved under cluster scheduling even for small-size clusters, compared to that of the partitioned scheduling. Moreover, when comparing to the simple generic task allocation scheme (e.g., first-fit), the proposed period-aware task allocation heuristic markedly reduces the scheduling overhead of cluster scheduling with the Bfair scheduler.     

Branch-and-bound method for minimizing the weighted completion time scheduling problem on a single machine with release dates

In this paper, we consider a single-machine scheduling problem with release dates. The aim is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose two new lower bounds that can be, respectively, computed in O(n²) and in O(nlogn) time where n is the number of jobs. We prove a sufficient and necessary condition for local optimality, which can also be considered as a priority rule. We present an efficient heuristic using such a condition. We also propose some dominance properties. A branch-and-bound algorithm incorporating the heuristic, the lower bounds and the dominance properties is proposed and tested on a large set of instances.     

New resolution algorithm and pretreatments for the two-dimensional bin-packing problem

The two-dimensional bin-packing (2BP) problem involves packing a given set of rectangles A into a minimum number of larger identical rectangles called bins. In this paper, we introduce the concept of dependent orientation items that have special characteristics, and give the formulation that characterizes these items. Then we propose three pretreatments for the non-oriented version of the problem. These pretreatments allow finding optimal packing of some items subsets of the given instance. They enable increasing the total area of the items and consequently the continuous lower bound. Finally, we propose a new heuristic method based on a best-fit algorithm adapted to the 2BP problem. Numerical experiments show that this method is competitive with the heuristic and metaheuristic algorithms proposed in the literature for the considered problem in respect of both the quality of the solution and the computing time.     

Labeling points with given rectangles

In this paper, we consider the following problem: Given n pairs of a point and an axis-parallel rectangle in the plane, place each rectangle at each point in order that the point lies on the corner of the rectangle and the rectangles do not intersect. If the size of the rectangles may be enlarged or reduced at the same factor, maximize the factor. This paper generalizes the results of Formann and Wagner [Proc. 7th Annual ACM Symp. on Comput. Geometry (SoCG'91), 1991, pp. 281-288]. They considered the uniform squares case and showed that there is no polynomial time algorithm less than 2-approximation. We present a 2-approximation algorithm of the non-uniform rectangle case which runs in O(n²logn) time and takes O(n²) space. We also show that the decision problem can be solved in O(nlogn) time and space in the RAM model by transforming the problem to a simpler geometric problem.     

Heuristic search for scheduling flexible manufacturing systems using lower bound reachability matrix

For scheduling flexible manufacturing systems efficiently, we propose new heuristic functions for A^* algorithm that is based on the T-timed Petri net. In minimizing makespan, the proposed heuristic functions are usually more efficient than the previous functions in the required number of states and computation time. We prove that these heuristic functions are all admissible and one of them is more informed than that using resource cost reachability matrix. We also propose improved versions of these heuristic functions that find a first near-optimal solution faster. In addition, we modify the heuristic function of Yu, Reyes, Cang, and Lloyd (2003b) and propose an admissible version in all states. The experimental results using a random problem generator show that the proposed heuristic functions perform better as we expected.     

Heuristic search for scheduling flexible manufacturing systems using lower bound reachability matrix

For scheduling flexible manufacturing systems efficiently, we propose new heuristic functions for A^* algorithm that is based on the T-timed Petri net. In minimizing makespan, the proposed heuristic functions are usually more efficient than the previous functions in the required number of states and computation time. We prove that these heuristic functions are all admissible and one of them is more informed than that using resource cost reachability matrix. We also propose improved versions of these heuristic functions that find a first near-optimal solution faster. In addition, we modify the heuristic function of Yu, Reyes, Cang, and Lloyd (2003b) and propose an admissible version in all states. The experimental results using a random problem generator show that the proposed heuristic functions perform better as we expected.     

Addressing the N/1 scheduling problem—A heuristic approach

The problem of sequencing n-jobs on one machine (n/1) to minimize maximum job lateness has been the subject of much prior research. Most of this research has been directed at identifying optimal solutions to the problem via algorithmic search techniques. A weakness in employing an algorithm for solving the problem, however, is that lengthy computational times may result because of the necessity of searching n! sequences. By employing a multiple heuristic approach this limitation can be avoided. An optimal or near optimal schedule can be identified in a finite number of steps.This paper describes a multiple heuristic model that is effective more than eighty-ninety percent of the time in providing an optimal schedule for the N/l/L max scheduling program. Ten separate heuristics are described, and the results of testing the heuristics over fifteen hundred and sixty randomly generated problems is presented. Three of the heuristics are combined to form the heuristic-scheduling model.     

A new algorithm for the largest empty rectangle problem

A rectangleA and a setS ofn points inA are given. We present a new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior. The computational complexity of the presented algorithm isO(n logn + s), where s is the number of possible restricted rectangles considered. Moreover, the expected performance isO(n · logn).     

The build of n-Bits Binary Coding ICBP Ensemble System

To optimize the construct of an ensemble of many relatively small and simple classifiers may be more realistic than to optimize the design of an individual large and complex classifier. Problems of local minima and slow convergence may be mitigated within an ensemble system where the decisions obtained from locally optimal components classifiers are integrated together. However, it is very difficult to design the structure of individual neural networks (NNs) in an ensemble and the architecture of a whole ensemble. In n-Bits Binary Coding ICBP Ensemble System (nBBC-ICBP-ES), the crucial parameter that is required to be set a priori is an appropriate number of hidden nodes of the corresponding improved circular back-propagation (ICBP) root model. Thereby, both the number of individual ICBPs and the architecture of each ICBP component in an nBBC-ICBP-ES can be decided straightly. nBBC-ICBP-ES is computationally more efficient, with relatively fewer user specified parameters, and does not need any manual division to the training data set for the purpose of its construction. It is rather easy to be understood and implemented, while inheriting the benefits of ICBP root model in a natural manner. Simulation and t-test results on four large-scale benchmark classification data sets demonstrated that, in most cases, nBBC-ICBP-ES significantly improved the classification and generalization performances of the two typically large single ICBPs, Same-2^N_h-ICBP-ES (i.e., an ensemble system with the same 2^N_h ICBP components), conventional Bagging and AdaBoost ensemble. We conclude that, for NN applications in pattern recognition, assembling many small NNs might be better than just utilizing an individual large one, and further, assembling many heterogeneous small NNs might be better than assembling many homogeneous ones. The proposed nBBC-ICBP-ES is simple but efficient and effective, and also potentially significant for the applications of NN ensemble.     

二维矩形条带装箱问题的底部左齐择优匹配算法

针对二维矩形条带装箱问题提出了一种启发式布局算法,即底部左齐择优匹配算法(lowest-level left align best fit,简称LLABF). LLABF算法遵循最佳匹配优先原则,该原则综合考虑完全匹配优先、宽度匹配优先、高度匹配优先、组合宽度匹配优先及可装入优先等启发式规则.与BL(bottom-left),IBL(improved-bottom-left)与BLF(bottom-left-fill)等启发算法不同的是,LLABF能够在矩形装入过程中自动选择与可装区域匹配的下一个待装矩形     

Simple extractors via constructions of cryptographic pseudo-random generators

Trevisan has shown that constructions of pseudo-random generators from hard functions (the Nisan-Wigderson approach) also produce extractors. We show that constructions of pseudo-random generators from one-way permutations (the Blum-Micali-Yao approach) can be used for building extractors as well. Using this new technique we build extractors that do not use designs or polynomial-based error-correcting codes and that are very simple and efficient. For example, one extractor produces each output bit separately in O(log²n) time. These extractors work for weak sources with min-entropy λn, for arbitrary constant λ>0, have seed length O(log²n), and their output length is ≈n^λ/3.     

Petri net based decision system modeling in real-time scheduling and control of flexible automotive manufacturing systems

This paper presents the design and the implementation of a Petri net (PN) model for the control of a flexible manufacturing system (FMS). A flexible automotive manufacturing system used in this environment enables quick cell configuration, and the efficient operation of cells. In this paper, we attempt to propose a flexible automotive manufacturing approach for modeling and analysis of shop floor scheduling problem of FMSs using high-level PNs. Since PNs have emerged as the principal performance modeling tools for FMS, this paper provides an object-oriented Petri nets (OOPNs) approach to performance modeling and to implement efficient production control. In this study, we modeled the system as a timed marked graph (TMG), a well-known subclass of PNs, and we showed that the problem of performance evaluation can be reduced to a simple linear programming (LP) problem with m − n + 1 variables and n constraints, where m and n represent the number of places and transitions in the marked graph, respectively. The presented PN based method is illustrated by modeling a real-time scheduling and control for flexible automotive manufacturing system (FAMS) in Valeo Turkey.     

A variable neighborhood search approach for planning and scheduling of jobs on unrelated parallel machines

In this paper, we study a planning and scheduling problem for unrelated parallel machines. There are n jobs that have to be assigned and sequenced on m unrelated parallel machines. Each job has a weight that represents the priority of the corresponding customer order, a given due date, and a release date. An Automated Guided Vehicle is used to transport at maximum Load _max jobs into a storage space in front of the machines in a given period of time. We consider t _max consecutive periods. We are interested in minimizing the total weighted tardiness of the jobs across the periods. This measure is important when we are interested in a good on-time delivery performance. We present an appropriate mixed integer program. To solve this NP-hard problem, we develop a heuristic methodology based on decomposition and variable neighborhood search (VNS). The proposed approaches are assessed using randomly generated problem instances. We compare them with a simple heuristic based on decomposition and list scheduling using the Apparent Tardiness Cost dispatching rule. The results demonstrate that the heuristic approach based on VNS performs comparably to the mixed integer program while having reasonable solution times and outperforms the simple heuristic and a genetic algorithm (GA) from previous research.     

An adaptable vertical partitioning method in distributed systems

Vertical partitioning is a process of generating the fragments, each of which is composed of attributes with high affinity. The concept of vertical partitioning has been applied to many research areas, especially databases and distributed systems, in order to improve the performance of query execution and system throughput. However, most previous approaches have focused their attention on generating an optimal partitioning without regard to the number of fragments finally generated, which is called best-fit vertical partitioning in this paper. On the other hand, there are some cases that a certain number of fragments are required to be generated by vertical partitioning, called n-way vertical partitioning in this paper. The n-way vertical partitioning problem has not fully investigated.In this paper, we propose an adaptable vertical partitioning method that can support both best-fit and n-way vertical partitioning. In addition, we present several experimental results to clarify the validness of the proposed algorithm.     

On the numerical solution of the eigenvalue problem of the laplace operator by a capacitance matrix method

The problem of finding several eigenfunctions and eigenvalues of the interior Dirichlet problem for Laplace's equation on arbitrary bounded plane regions is considered. Two fast algorithms are combined: an iterative Block Lanczos method and a capacitance matrix method. The capacitance matrix is generated and factored only once for a given problem. In each iteration of the Block Lanczos method, a discrete Helmholtz equation is solved twice on a rectangle at a cost of the order ofn ² log₂ n operations wheren is the number of mesh points across the rectangle in which the region is imbedded.     

Covering a set of points in a plane using two parallel rectangles

In this paper we consider the problem of finding two parallel rectangles in arbitrary orientation for covering a given set of n points in a plane, such that the area of the larger rectangle is minimized. We propose an algorithm that solves the problem in O(n³) time using O(n²) space. Without altering the complexity, our approach can be used to solve another optimization problem namely, minimize the sum of the areas of two arbitrarily oriented parallel rectangles covering a given set of points in a plane.     

Optimal algorithm for a special point-labeling problem

A special class of map labeling problem is studied. Let P={p₁,p₂,…,p_n} be a set of point sites distributed on a 2D map. A label associated with each point p_i is an axis-parallel rectangle r_i of specified width. The height of all are same. The placement of r_i must contain p_i at its top-left or bottom-left corner, and it does not obscure any other point in P. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. We first consider a simple model for this problem. Here, for each point p_i, the corner specification (i.e., whether the point p_i would appear at the top-left or bottom-left corner of the label) is known a priori. We show that the time complexity of this problem is , and then propose an algorithm for this problem which runs in O(nlogn) time. If the corner specifications of the points in P are not known, our algorithm is a 2-approximation algorithm. Here we propose an efficient heuristic algorithm that is easy to implement. Experimental evidences show that it produces optimal solutions for most of the randomly generated instances and for all the standard benchmarks available in http://www.math-inf.uni-greifswald.de/map-labeling/.