Linear Schedule Research Articles

Fiscal rules of income transformation

The purpose of this paper is to characterize the class of fiscal rules of income transformation which are equity improving and, at the same time, preserve the ranking of existing distributions. Contrary to the related literature individual well-being depends not just on income but also on prices. We show that, when the environment is restricted such that a general transformation class still can be defined, the only “desirable” fiscal rule is the simple redistributive linear taxation schedule, of the same type that is the rule in practice in most economies.

PROCESSOR LOWER BOUND FORMULAS FOR ARRAY COMPUTATIONS AND PARAMETRIC DIOPHANTINE SYSTEMS

Using a directed acyclic graph (dag) model of algorithms, we solve a problem related to precedence-constrained multiprocessor schedules for array computations: Given a sequence of dags and linear schedules parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n. In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions dn to a set of parametric linear Diophantine equations. We present an algorithm based on generating functions for constructing a formula for these numbers dn. The algorithm has been implemented as a Mathematica program. Example runs and the symbolic formulas for processor lower bounds automatically produced by the algorithm for Matrix-Vector Product, Triangular Matrix Product, and Gaussian Elimination problems are presented. Our approach actually solves the following more general problem: Given an arbitrary r× s integral matrix A and r-dimensional integral vectors b and c, let dn(n=0,1,…) be the number of solutions in non-negative integers to the system Az=nb+c. Calculate the (rational) generating function ∑n≥ 0dntn and construct a formula for dn.

Managerial incentives: on the Near Linearity of Optimal Compensation

Contracts are examined when outcomes depend on managers' choices as well as efforts. As the cost of effort shrinks relative to payoffs, the optimal contract converges to a linear payoff if the control space of the agent has full dimensionality, but not otherwise. Thus, when the agent can trade expected return for greater correlation with other returns, it is better to ignore relative performance when the cost of effort is small. When the choices include all fair gambles and hedges, the linear schedule is no more expensive than any other schedule that induces effort. Unfair gambles are examined as well.

Linear Scheduling Model: Development of Controlling Activity Path

There are several reasons why linear scheduling techniques have not been applied widely to construction projects even though they have been identified as very useful. If we consider the capabilities and the widespread use of critical-path method (CPM) in the construction industry, some of the reasons why linear scheduling is not popular become evident. There is an abundance of highly developed CPM computer applications available today. These applications provide a rich set of capabilities such as resource and cost allocation, progress reporting, and schedule updating, plus the generation of a multitude of reports and graphics. There are no comparable applications for linear scheduling. The reason for the shortage of applications is that linear scheduling typically has been regarded as a visual technique lacking the analytical qualities of the CPM of scheduling. Fundamental to linear scheduling's lack of an analytical perspective is its inability to determine a critical path. This paper documents a method by which a controlling activity path can be determined in a linear schedule. This fundamental ability further forms the basis for the determination of float for noncontrolling activities and provides a means of statusing and updating linear schedules.

Resource Leveling of Linear Schedules Using Integer Linear Programming

Since the early 1960s many techniques have been developed to plan and schedule linear construction projects. However, one, the critical path method (CPM), overshadowed the others. As a result, CPM developed into the powerful and effective tool that it is today. However, research has indicated that CPM is ineffective for linear construction. Linear construction projects are typified by activities that must be repeated in different locations such as highways, pipelines, and tunnels. Recently, there has been renewed interest in linear scheduling. Much of this interest has involved a technique called the linear scheduling method (LSM). Only recently has there been the ability to calculate the controlling activities of a linear schedule, independent of network analysis. Additional research needs to be done to develop some of the techniques available in CPM into comparable ones for linear scheduling. One of these techniques is resource leveling. This paper uses the vehicle of a highway construction project to present an integer linear programming formulation to level the resources of linear projects.

Loop parallelization algorithms: From parallelism extraction to code generation

In this paper, we survey loop parallelization algorithms, analyzing the dependence representations they use, the loop transformations they generate, the code generation schemes they require, and their ability to incorporate various optimizing criteria such as maximal parallelism detection, permutable loop detection, minimization of synchronizations, easiness of code generation, etc. We complete the discussion by presenting new results related to code generation and loop fusion for a particular class of multidimensional schedules called shifted linear schedules. We demonstrate that algorithms based on such schedules lead to simple codes.

Linear scheduling: past research efforts and future directions

Since the early 1960s, there have been different techniques to schedule linear projects, but for the most part, these have been overshadowed by the critical path method (CPM). Recently, there has been renewed interest in linear scheduling and in adapting some of the CPM techniques to linear scheduling. This necessitates a review of the research in the area of linear scheduling. The present paper provides an overview of linear scheduling, discusses the different approaches that have been used and expresses new avenues for research in the area of resource levelling of linear schedules.

Synthesis and sintering of nanocrystalline barium titanate powder under nonisothermal conditions. IV. Electron microscope study of the evolution of barium titanate powder morphology

The dependence of BaTiO3 powder morphology evolution during decomposition of barium titanyl-oxalate and subsequent barium titanate synthesis on heating rate is studied. It is established that the mechanisms of morphology development depend upon the heating mode and the necessity of a nonlinear heating schedule is proven. This schedule makes it possible to avoid some limitations of linear heating schedule and to use their advantages.

Kinetic parameters from complex thermal desorption spectrum

An method to study complex thermal desorption spectra in terms of energy states was developed and tested with computer simulated data. It is restricted to the case where the kinetic parameters of the each energy state are constants and the energy states are independents from each other. The method is valid for linear, exponential or hyperbolic temperature schedule. Furthermore, the use of the method does not require previous knowledge of the reaction orders or the number of energetically different surface sites. The method allows to resolve as individual states, states that are very close together in the original TPD spectrum.

On uniformization of affine dependence algorithms

The paper deals with the problem of transforming irregular data dependence structures of algorithms with nested loops into more regular ones. Algorithms under consideration are n-dimensional algorithms (algorithms with n nested loops) with affine dependences where dependences are affine functions of index variables of the loop. Methods are proposed to uniformize affine dependence algorithms, i.e., to transform affine dependence algorithms into uniform dependence algorithms where dependences are independent of the index variables (constant). Objectives are considered to guide the selection of feasible uniformizations. The first one is to reduce the number of dependences after uniformization. The second one is to maximize parallelism preserved by the uniformization. Some parallelism might be lost due to the uniformization. The parallelism preserved by the uniformization is measured by: the total execution time by the optimal linear schedule which assigns each computation in the algorithm an execution time according to a linear function of the index of the computation; and the size of the cone spanned by the dependence vectors after uniformization.

Optimal Time And Efficient Space Free Scheduling For Nested Loops

The most important issue when parallelizing sequential programs is the efficient assignment of computations into different processing elements. The most extensive, in terms of time execution, part of a program is the nested loops. Too many approaches have been devoted in parallelizing nested loops, and assigning the concurrent partitions of such a loop into different processors. In the past, all methods have been focused upon linear schedules produced by manipulating the reduced dependence graph, which in some cases achieve near optimal solutions. This paper presents a new method of free scheduling loop computations into time, based on task graph scheduling techniques. It will be shown that this schedule is optimal in terms of time, outperforming all linear schedules. Furthermore, in terms of total number of processors, the presented method includes a heuristic refinement of the free schedule which 'shuffles' computations into time, without loss of the optimal time performance, to augment the mean processor utilization. In all cases, the proposed method uses less number of processors, while preserving the optimal total execution time. The 'shuffling' of computations is based on graph theory approaches, and uses PERT problem techniques. Such scheduling is convenient for parallelizing tools (such as compilers), but has practical interest for shared memory multiprocessor systems, where the communication delay imposed by such non-regular scheduling is of no interest.

A unified partitioning and scheduling scheme for mapping multi-stage regular iterative algorithms onto processor arrays

This paper addresses the partitioning and scheduling problems in mapping multi-stage regular iterative algorithms onto fixed size distributed memory processor arrays. We first propose a versatile partitioning model which provides a unified framework to integrate various partitioning schemes such as "locally sequential globally parallel", "locally parallel globally sequential" and "multi-projection". To alleviate the run time data migration overhead--a crucial problem to the mapping of multi-stage algorithms, we further relax the widely adopted atomic partitioning constraint in our model such that a more flexible partitioning scheme can be achieved. Based on this unified partitioning model, a novel hierarchical scheduling scheme which applies separate schedules at different processor hierarchies is then developed. The scheduling problem is then formulated into a set of ILP problem and solved by the existing software package for optimal solutions. Examples indicate that our partitioning model is a superset of the existing schemes and the proposed hierarchical scheduling scheme can outperform the conventional one-level linear schedule.

Minimal data dependence abstractions for loop transformations: Extended version

Many abstractions of program dependences have already been proposed, such as the Dependence Distance, the Dependence Direction Vector, the Dependence Level or the Dependence Cone. These different abstractions have different precisions. The minimal abstraction associated to a transformation is the abstraction that contains the minimal amount of information necessary to decide when such a transformation is legal. Minimal abstractions for loop reordering and unimodular transformations are presented. As an example, the dependence cone, which approximates dependences by a convex cone of the dependence distance vectors, is the minimal abstraction for unimodular transformations. It also contains enough information for legally applying all loop reordering transformations and finding the same set of valid mono- and multi-dimensional linear schedules as the dependence distance set.

Design of space-optimal regular arrays for algorithms with linear schedules

The problem of designing space-optimal 2D regular arrays for N/spl times/N/spl times/N cubical mesh algorithms with linear schedule ai+bj+ck, 1/spl les/a/spl les/b/spl les/c, and N=nc, is studied. Three novel nonlinear processor allocation methods, each of which works by combining a partitioning technique (gcd-partition) with different nonlinear processor allocation procedures (traces), are proposed to handle different cases. In cases where a+b/spl les/c, which are dealt with by the first processor allocation method, space-optimal designs can always he obtained in which the number of processing elements is equal to N/sup 2c. For other cases where a+b>c and either a=b and b=c, two other optimal processor allocation methods are proposed. Besides, the closed form expressions for the optimal number of processing elements are derived for these cases. >

Rate-optimal schedule for multi-rate DSP computations

In this paper we present a novel framework ofmulti-rate scheduling of signal processing programs represented by regular stream flow graphs (RSFGs). The main contribution of this paper is translating the scheduling problem of RSFGs into an equivalent problem in the domain of Karp-Miller computation graphs. A distinct feature of our scheduling framework--called themulti-rate software pipelining--is to allow maximum overlapping of operations from successive iterations, subject only to precedence constraints caused by data dependences. p ]We demonstrate that the scheduling of regular stream flow graphs can be formulated as a mathematical problem by capturing data dependences between two actors as a precedence relation between the firing of these actors. Using linear schedules, the problem is further translated into a linear program formulation. An efficient solution for the linear programming problem is obtained by first constructing what is called theprecedence graph. A polynomial-time solution is obtained by observing that the optimal computation rate is theminimum cost-to-time ratio cycle (MCTRC) in the precedence graph and using the well-established solution methods for the MCTRC problem. Finally, to minimize the buffer requirement for the obtained rate-optimal schedule, a graph coloring method based on thecyclic interval graph representation has been proposed.

A NEW APPROACH TO FINDING OPTIMAL LINEAR SCHEDULES FOR UNIFORM DEPENDENCE ALGORITHMS†

A uniform dependence algorithm can be represented by an index set of index points and a finite set of data dependence vectors. Usually, the convex hull R of the index set is a nondegenerated convex polytope in R n, In this paper, we show that finding an optimal linear schedule for a uniform dependence algorithm with an arbitrary bounded convex index set is equivalent to finding a vector of smallest norm, where the vector norm is defined on a symmetric convex set R∗ (which is the dual of difference body R − R). A linear programming problem is derived for finding the smallest vector. This problem can be solved in the empirical average time complexity O(36n 3 + 12an 2 + a), where a is the number of the linear inequalities defining the convex hull R and n is the dimension of the index set. This time complexity is better than those of the existing methods.

On loop transformations for generalized cycle shrinking

This paper describes several loop transformation techniques for extracting parallelism from nested loop structures. Nested loops can then be scheduled to run in parallel so that execution time is minimized. One technique is called selective cycle shrinking, and the other is called true dependence cycle shrinking. It is shown how selective shrinking is related to linear scheduling of nested loops and how true dependence shrinking is related to conflict-free mappings of higher dimensional algorithms into lower dimensional processor arrays. Methods are proposed in this paper to find the selective and true dependence shrinkings with minimum total execution time by applying the techniques of finding optimal linear schedules and optimal and conflict-free mappings proposed by W. Shang and A.B. Fortes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Noisy Observation in Adverse Selection Models

We consider a principal-agent contracting problem under incomplete information where some of the agent's actions are imperfectly observable. Contracts take the form of reward schedules based on the noisy observation of the agent's action. We first review situations where the principal can reach the same utility as in the absence of noise. Then we focus on the use of linear reward schedules, which allow universal implementation, i.e. implementation of a given mechanism for any unbiased noise of observation, and on quadratic reward schedules, which only require the knowledge of the variance of the noise. We exhibit sufficient conditions under which linear reward schedules implement a given mechanism. Finally, we characterize necessary conditions for a mechanism to be implementable under noisy observation by a linear schedule, and by quadratic schedules. We give the geometric intuition behind all results.

Joint product signals of quality

The relationship between product quality, signals, and the firm's optimal pricing policy has been given much attention in economics. The literature is extended in this paper by considering the signaling problem of a firm that jointly produces two commodities—one of known quality to consumers (a search good) and one of unknown quality (an experience good). The model presented employs a stylized reputation function, a linear cost structure, and linear demand schedules to produce two interesting insights. First, the search good's price can potentially be used as a signal of the quality of the experience good. Second, the price of a search good will depend upon whether it is jointly produced with another search good or an experience good or whether it is produced in isolation by a single product firm. Furthermore, evidence from a paper on gasoline pricing seems to support this contention.

LINEAR SCHEDULING IS NEARLY OPTIMAL

This paper deals with the problem of finding optimal schedulings for uniform dependence algorithms. Given a convex domain, let Tf be the total time needed to execute all computations using the free (greedy) schedule and let Tl be the total time needed to execute all computations using the optimal linear schedule. Our main result is to bound Tl/Tf and Tl − Tf for sufficiently "fat" domains.