System Of Uniform Recurrence Equations Research Articles

Space-Time Equations for Non-Unimodular Mappings

The class of systems of uniform recurrence equations (UREs) is closed under uni-modular transformations. As a result, every systolic array described by a unimodular mapping can be specified by a system of space-time UREs, in which the time and space coordinates are made explicit. As non-unimodular mappings are frequently used in systolic designs, this paper presents a method that derives space-time equations for systolic arrays described by non-unimodular mappings. The space-time equations for non-unimodular mappings are known elsewhere as sparse UREs (SUREs) because the domains of their variables are sparse and their data dependences are uniform. Our method is compositional in that space-time SUREs can be further transformed by unimodular and non-unimodular mappings, allowing a straightforward implementation in systems like ALPHA. Specifying a systolic design by space-time equations has two advantages. First, the space-time equations exhibit all useful properties about the design, allowing the design to be formally verified. Second, depending on the application area and performance requirement, the space-time equations can be realised as custom VLSI systems, FPGAs, or programs to be run on a parallel computer.

Algorithm transformations for the data broadcast elimination method

VLSI technology has had tremendous success in revolutionizing computer design with processor arrays. Local communication and interconnection is a constraint that dictates the design of processor arrays. The shared bus and global access to memory are now no longer used, since they lower the speed. Consequently, parallel algorithms must be designed according to these constraints. One of the problems that must be resolved for the above mentioned constraints is data broadcast elimination. Algorithms must be transformed into a form that uses data propagation instead of data broadcast. Here systems of affine recurrence equations are analyzed and data broadcast is denned in context of the definition of data dependence and affine recurrence equations. A method for data broadcast elimination is introduced in [1] and expands the system of affine recurrence equations into new recurrence equations, that define data propagation and eliminates the data dependences where data broadcast occurs. Parallel algorithms are usually given as a set of similar tasks repetitively performed on different data. The iteration form of presenting the algorithms is most common. Several techniques are introduced to transform the algorithm to a single assignment form of recurrence equations. Some improvements of these techniques are presented to make the application of the data broadcast elimination method easier and more straight forward. The presented techniques are classified as the transformation of iterative algorithms to a recurrence form, the transformation of recurrence form to a single assignment form, and fulfilling the index forms of the algorithms. A system of affine recurrence equations with the data broadcast property is always obtained by applying these procedures. The method of data broadcast elimination successfully transforms this system of affine recurrence equations into a system of uniform recurrence equations which can be used for parallel implementation on VLSI processor arrays.

Computations of Uniform Recurrence Equations Using Minimal Memory Size

We consider a system of uniform recurrence equations of dimension 1 and we show how its computation can be carried out using minimal memory size with several synchronous processors. This result is then applied to register minimization for digital circuits and parallel computation of task graphs.

Localized algorithms for vlsi processor arrays

In this paper we analyze the algorithms expressed as a system of recurrence equations. The algorithms are called 2*1 output algorithms if two output values of one function (variable identification) are specified by the system of recurrence equations for each index point in the algorithm. The algorithm is in free form if the indexes of these two values are not dependent. Two standard classes are determined by this criteria: the nearest neighbour and the all pair form. For example the sorting algorithm can be expressed in the all pair form i.e., the linear insertion algorithm or in the nearest neighbour form i.e., the bubble sort algorithm. However these algorithms are different in their nature. A procedure to eliminate the computational broadcast for the all pair 2*1output algorithm has been proposed by the authors in [1]. The result obtained by implementing this procedure was a localized form of the algorithm and a system of uniform recurrence equations by eliminating the computational and data broadcast. So he data dependence method can be efficiently used for parallel implementations. The proposed procedure cannot be implemented directly on the nearest neighbour form algorithms. Here we show how the algorithm can be restructured into a form where the computational and data broadcast can be eliminated. These transformations result in localized algorithms. A few examples show how these algorithms can be implemented on processor arrays. For example, the Gentleman Kung triangular array [2] can be used for solving the QR decomposition algorithm for both forms of the algorithm. The implementations differ in the order of the data flow and the processor operation. We show that the implementation of the nearest neighbour algorithm is even better than the standard one.

NEW RESULTS ON COMPUTABILITY OF RECURRENCE EQUATIONS

Systems of uniform recurrence equations were proposed by Karp et al.13 as a mean of automatically deriving programs for parallel architectures. Extensions of this formalism are used by many authors in systolic array synthesis. The computability of a system of recurrence equations is therefore of primary importance, and is considered as the first point to be examined when trying to implement an algorithm. This paper investigates the computability of recurrence equations and especially new results obtained in this area. We first recall the definitions of computability proposed by Karp et al.,13 Rao,31 Joinnault12 and Saouter et al.32 Then we correct and generalize those results. Finally, we give a new original proof for the undecidability of computability of non-uniform, non-conditional systems of recurrence equations.

Determination of processor allocation in the design of processor arrays

This paper deals with the design of processor arrays for algorithms which can be represented as systems of uniform recurrence equations. We present an approach to determine allocation functions as part of this design. The objective of our approach is to generate allocation functions minimizing the necessary chip area for a hardware implementation of a processor array. At first, we propose an algorithm leading to a small number of processors. Then, the algorithm is extended to include the chip area necessary to implement the processors in silicon. The arising optimization problems can be solved using integer linear programming.

Affine scheduling on bounded convex polyhedric domains is asymptotically optimal

We address the scheduling problem for algorithms which can be described by a system of uniform recurrence equations, when the computation domain is a bounded convex polyhedron. We study an affine schedule, first introduced by Darte and Robert, and we show that it is asymptotically time-optimal. Moreover we study the difference between its makespan and the optimal one, and we show that, in a special case, it is bounded by a logarithmic function of thedomain size.

Algorithm transformations for computational and data broadcast

Problems that must be resolved for the constraints imposed by VLSI technology are computational and data broadcast. Algorithms must be transformed into a form that uses data propagation instead of data broadcast. The techniques introduced here are classified as the transformation of iterative algorithms to a recurrence form, the transformation of recurrence form to a single assignment form, the transformation of the recurrences in iterative algorithms and fulfilling the index forms of the algorithms. Systems of affine recurrence equations are analyzed here and computational and data broadcast are defined in context of the definition of data dependence and affine recurrence equations. A method for data broadcast elimination is introduced in [1, 2] and expands the system of affine recurrence equations into new recurrence equations, that define data propagation and eliminates the data dependencies where computational or data broadcast occurs. Some improvements of these techniques are presented to make the applic ion of the broadcast elimination method easier and more straight forward. A system of affine recurrence equations with broadcast property is always obtained by applying these procedures. The method of broadcast elimination successfully transforms this system of affine recurrence equations into a system of uniform recurrence equations which can be used for parallel implementation on VLSI processor arrays.

Optimal fine and medium grain parallelism detection in polyhedral reduced dependence graphs

This paper presents an optimal algorithm for detecting line or medium grain parallelism in nested loops whose dependences are described by an approximation of distance vectors by polyhedra. In particular, this algorithm is optimal for the classical approximation by direction sectors. This result generalizes, to the case of several statements. Wolf and Lam's algorithm which is optimal for a single statement. Our algorithm relies on a dependence uniformization process and on parallelization techniques related to system of uniform recurrence equations. It can also be viewed as a combination of both Allen and Kennedy's algorithm and Wolf and Lam's algorithm.

REVISITING THE DECOMPOSITION OF KARP, MILLER AND WINOGRAD

This paper is devoted to the construction of multi-dimensional schedules for a system of uniform recurrence equations. We show that this problem is dual to the problem of computability of a system of uniform recurrence equations. We propose a new study of the decomposition algorithm first proposed by Karp, Miller and Winograd: we base our implementation on linear programming resolutions whose duals give exactly the desired multi-dimensional schedules. Furthermore, we prove that the schedules built this way are optimal up to a constant factor.

Converting affine recurrence equations to quasi-uniform recurrence equations

Most work on the problem of synthesizing a systolic array from a system of recurrence equations is restricted to systems of uniform recurrence equations. Recently, researchers have begun to relax this restriction to include systems of affine recurrence equations. A system of uniform recurrence equations typically can be embedded in spacetime so that the distance between a variable and a dependent variable does not depend on the problem size. Systems of affine recurrence equations that are not uniform do not enjoy this property. A method is presented for converting a system of affine recurrence equations to an equivalent system of recurrence equations that is uniform, except for points near the boundaries of its index sets. Necessary and sufficient conditions are given for an affine system to be amenable to such a conversion, along with an algorithm that checks for these conditions, and a procedure that converts those affine systems which can be converted.The characterization of convertible systems brings together classical ideas in algebraic geometry, number theory, and matrix representations of groups. While the proof of this characterization is complex, the characterization itself is simple, suggesting that the mathematical ideas are well chosen for this difficult problem in array design.

New linear systolic arrays for digital filters and convolution

The algorithm transformation techniques described in [1] and the data broadcast elimination method from [2] are used to represent the algorithm as a system of uniform recurrence equations. Usually the algorithms are given in iterative form, i.e. as a repetitive operation on data. The procedure A transforms these algorithms into a recursive form. The algorithm is then transformed to a single assignment code presentation by procedure B. The problem of recurrences defined in the iteration is solved by procedure C. Then the full index form of the algorithms are obtained by applying procedure D. By applying these algorithm transformation techniques a system of affine recurrence equations with the data broadcast property is obtained. The data broadcast elimination method is used to transform the algorithm into a system of uniform recurrence equations. The reader is referred to [3–9] for a description of the data dependence method and linear transformations. In this paper we will use the notation proposed in [1,9]. The folding transformations, described in [10], are used to obtain more efficient processor array implementations for both the MA and AR filters. The ARMA filter is a composition of these two implementations that have an interlocking property. However the systolic array designed by H.T. Kung in [11] is rather straight forward solution resulting in cells with double complexity according to the inner product step processors. A much better solution is obtained by using a translation and the interlocking property of the partial implementations in the composition.

New linear systolic arrays for the string comparison algorithm

VLSI technology has achieved tremendous success in revolutionizing computer design with processor arrays. Foster and Kung, in [1] propose a linear systolic array for the string comparison algorithm. This solution is based on a regular movement of the pattern and the input string in opposite directions keeping the results (1 bit values) resident in the array. A special technique, requiring two wires is applied to unload the result array in a pipeline with the input string so no extra time is required. Disadvantages of this approach are 50% maximum processor efficiency and intensive memory access when string sizes are greater then the array size. We use several algorithm transformation techniques to transform the algorithm into a system of Affine Recurrence Equations (ARE), [2] and then apply the data broadcast elimination method [3]. A system of Uniform Recurrence Equations (URE) is resulted and then mapped [4] on VLSI processor arrays. The folding transformation [5], is then applied to establish the optimal implementations by selecting the plane achieving maximum symmerty and the vector of interlocking property with minimal norm. The implementations obtained result in more efficient VLSI processor arrays.

Computability of recurrence equations

Systems of uniform recurrence equations were proposed by Karp et al. (1967) as a means to derive automatically programs for parallel architectures. Since then, extensions of this formalism were used by many authors, in particular, in the fields of systolic array synthesis. The computability of a system of recurrence equations is, therefore, of primary importance, and is considered as the first point to be examined when trying to implement an algorithm. This paper investigates the computability of recurrence equations. We first recall the results established by Karp et al. (1967) on the computability of systems of uniform recurrence equations, by Rao (1985) on regular iterative arrays, and Joinnault's (1987) undecidability result on the computability of conditional systems of uniform recurrence equations with nonbounded domain. Then we consider systems of parameterized affine recurrence equations, that is to say, systems of recurrence equations whose domains depend linearly on a size parameter, and establish that the computability of such system is also undecidable.

Mapping uniform recurrences onto small size arrays

Given a regular application described by a system of uniform recurrence equations, systolic arrays are commonly derived by means of an affine transformation; an affine schedule determines when the computations are performed and an affine processor allocation where they are performed. Circuit transformations are then applied on the resulting circuit when the application needs to be mapped onto a smaller size array. This method is in two steps and thus can hardly be optimized globally. We hereafter present a different method for designing small size arrays. We derive them in one step by means of an affine schedule and a near-affine processor allocation. By doing so, we can generalize the optimization technique for affine mapping to be applicable here. The method is illustrated on the band-matrix multiplication and on the convolution algorithms.

Decoupling the dimension of a system of affine recurrence equations

Most work on the problem of synthesizing a systolic array from a system of recurrence equations is restricted to systems of uniform recurrence equations. In this paper, this restriction is relaxed to include systems of affine recurrence equations. A system of uniform recurrence equations typically can be embedded in space-time so that the distance between a variable and a dependent variable does not depend on the problem size. Systems of affine recurrence equations which are not uniform do not enjoy this property. However, a method has been presented in another paper for converting a system of affine recurrence equations to an equivalent system of recurrence equations that is uniform, except for points near the boundaries of its index sets. In this paper a procedure is presented for decoupling the dimensions of the system of affine recurrence equations, thereby simplifying the conversion to an equivalent system that is uniform (except for points near the boundaries).

SPECIFYING CONTROL SIGNALS FOR SYSTOLIC ARRAYS BY UNIFORM RECURRENCE EQUATIONS

Starting with a system of uniform recurrence equations (source UREs) for a systolic design, we provide a method that allows a specification of control signals by another system of UREs (control UREs), which has no notion of time and space. Necessary and sufficient conditions for the correctness of the control UREs are stated. The standard space-time mapping technique is extended so that systolic arrays with a description of both data and control flow are directly synthesised from the source and control UREs. Optimisations of control signals are discussed. An example is provided.

Scheduling a system of nonsingular affine recurrence equations onto a processor array

Most work on the problem of scheduling computations onto a systolic array is restricted to systems of uniform recurrence equations. In this paper, this restriction is relaxed to include systems of affine recurrence equations. In this broader class, a sufficient condition is given for the system to be computable. Necessary and sufficient conditions are given for the existence of an affine schedule, along with a procedure that constructs the schedule vector, when one exists.