CA2076293A1 - Multiprocessor computer for solving sets of equations - Google Patents

Abstract

A Multiprocessor Computer for Solving Sets of Equations Abstract A multiple-instruction stream-multiple-data stream computer system and method for solving a set of equations which can be represented in matrix form.
An illustrative embodiment of the invention combines a novel method for solving a set of equations represented as a matrix with a novel multiprocessor computer architecture that is capable of exploiting the parallelism created by the method. The method partitions the task of solving a matrix into quasi-distinct tasks such that as many of the quasi-distinct tasks can be executed in parallel as possible. The multiprocessor architecture comprises a number of Von Neumann computers that areinterconnected via an interprocessor communications network which is controlled by special purpose network processor. The network processor assures that each Von Neumann computer is efficiently utilized executing quasi-distinct tasks and routes whatever information needs to be communications between the respective computers.

FILE: agrp006for.1 agrp006for.CL agrp006for.ab

Description

~ !~ 7 A Multiprocessor Computer for Solvirlg Sets of Equations Back~round of the Invention Field of the Invention The invention relates to computer systems in general and, more 5 particularly, to computer methods and systems for solving sets of equations that can be represented in matrix form.
Description of the Related Art Solving sets of equations is useful in, inter alia, mechanics, economics, fluid dynamics and electric circuit simulation. Circuit simulation, for example,10 typically involves assembling a set of equations that model the behavior of an electric circuit and then solving such set of equations to learn the value of unknown voltages and currents in the circuit. The equations that describe the circuit being simulated are often nonlinear and, for this reason, an iterative technique, such as the Newton Raphson method has typically been used to solve such set of equations.
15 Various iterative techniques, including dhe Newton Raphson method, are well known to persons having ordinary skill in the art.
One part of the Newton Raphson method involves solving a set of linear equations. Empirically, the time needed to solve the set of equations dominates the time needed to simulate a large circuit. Using conventional circuit simulation 20 techniques, even dle most powerful supercomputers take many hours to complete a simulation and therefore, the simulation of large integrated circuits is still impractical.
Although a number of methods can be used to solve a set of linear equations, Lower Upper Decomposition ("LUD"), a method well known to persons 25 having ordinary skill in the art, is generally preferred because of its accuracy and stability. LUD, however, cannot be parallelized easily.
A number of efforts have been made to parallelize LUD. J. Huang and O. Wing, "Optimal Parallel Triangulation of a Sparse Matrix, I.E.E.E. Trans. on Circuits and Systems, Vol. CAS-26, pp. 726-732 (Sept. 1979); N. Karmarkar, "A
30 New Parallel Computer for Sparse Matrix Computation," Siam Conference on Discrete Mathematics, Adanta (June 1990); and O. Wing and J. Huang, "A
Computation Model of Parallel Solution of Linear Equations," I.E.E.E. Trans. on Computers, Vol. C-29, pp. 632-638 (July 1980). These methods concentrate on precompiling each step required in LUD and constructing a task graph which is dhen -2- 2~&~

scheduled to run on a multiprocessor. These procedures yield a large number of tasks that need to be scheduled and require too much computer memory for realistic circuits.
An alternative approach, proposed by P. Sadayappan and s V. Visvanathan, "Circuit Simulation on Shared Memory Multiprocessors," I.E.E.E.
Transactions on Computers, Vol. C-37, pp. 1634-1642 (Dec. 1988), uses a partially compiled approach which has been shown to be very effective in extracting a highdegree of parallelism from LUD of sparse matrices while minimizing the complexity of the task graph that has to be scheduled. This approach is a good compromise 10 between the generic LUD approach and the compiled approach.
The Sadayappan-Visvanathan approach was originally developed for a shared memory multiprocessor and extended for use on a distributed memory multiprocessor in J. Trotter and P. Agrawal, "Circuit Simulation Methods on a Distributed Memory Multiprocessor System," I.E.E.E. Proceedings ICCAD, Santa 15 Clara, pp.438-441 (Nov. 1990). This paper is hereby incorporated by reference into this specification.
Another approach proposed by T. Nakata, N. Tanabe, H. Onozuka, T. Kurobe and N. Koike in "A Multiprocessor System for Modular Circuit Simulation," I.E.E.E. ICCAD, pp.364-367 (1987), used LUD without compilation on 20 a distributed memory multiprocessor. In the Nakata et. al. system each processor can access the memory of other processors using a shared bus.

Summary of the Invention The invention provides a mechanism for a solving set of equations represented in matrix form by using an approach different from the prior art. A set 25 of equations can be solved by the present invention more quickly than with previous mechanisms.
An illustrative embodiment of the invention comprises several processing elements that are interconnected via a communications network. The scheduling of the communications between the processors is dictated by a special30 purpose network processor.

Brief Description of the Drawin~
FIG. 1 shows a schematic diagram of a typical electric circuit to be analyzed by an illustrative embodiment of the invention.

3 2~7t3,~33 FIG. 2 shows a set of equations that model the electrical characteristics of the electric circuit in FIG. 1.
FIG. 3 shows a matrix representation of the set of equations shown in FIG. 2.
FIG. 4 shows an augmented matrix corresponding to the matrix shown in FIG. 3.
FIG. 5 shows the zero/non-zero structure of the augmented matrix shown in Fig. 4.
FIG. 6 shows a task graph for solving the augmented matrix in FIG. 5.
FIG. 7 shows an overview of an illustrative embodiment of the present invention.
FIG. 8 shows an overview of a processing element as shown in Fig. 7.
FIG. 9 shows the mechanism for communicating between the processing elements as shown in Fig. 7.
IS FIG. 10 shows an overview of the communications processor as shown in Fig. 8.
FIG. 11 shows a temporal representation of a task to be performed by the illustrative embodiment.
FIG. 12 shows options for inserting tasks into the task schedule.
FIG. 13 shows a task graph for solving the augmented matrix in FIG. 3.
FIG. 14 shows a schedule for task execution corresponding to the task graph in FIG. 13.
FIG. 15 shows the timing of a pipelined bus operation on the illustrative embodiment.
2s FIG. 16 shows the matrix of Fig. 4 after the first task has been completed.
FIG. 17 shows the matrix of Fig. 4 after three tasks have been completed.
FIG. 18 shows the solution matrix corresponding to the circuit of Fig. 1.

30 Detailed Description 1. INTRODUCTION
The invention provides a computer system and method for modeling or simulation. It is believed that the illustrative embodiment of the invention to be described will be more easily understood by presenting it in the context of solving an 2~7~,2- .

illustrative problem. Although the illustrative embodiment can solve problems arising in diverse fields, the illustrative problem is from electrical engineering. In the next section, the illustrative problem will be presented. Next, an illustrative embodiment of the invention will be presented and finally it will be shown how the s illustrative problem can be solved with the illustrative embodiment.
2. THE ILLUSTRATIVE PROBLEM
2.1. Electric Circuit Design Electric c*cuits, such as the one represented by the schematic diagram in Fig. l, are typically designed to perform a specific function. While a great deal is l0 known about designing electric c*cuits, it remains substantially emp*ical such that fabricated c*cuits often do not perform as intended by the* designers. When the cost of fabrication and testing is low, a circuit can be iteratively designed, fabricated, tested and re-designed until it performs as des*ed. But when the cost of fabrication and/or testing is high, it is advantageous to verify that the c*cuit design will perform 15 as intended before it is fabricated.
One means of predicting the electrical characteristics of a design is through a technique known as "circuit simulation." Circuit simulation involves building a mathematical model of a c*cuit based on mathematical models of the c*cuit elements and on the topology of the c*cuit. Once the model of the c*cuit is 20 created, its behavior can be accurately predicted through well-known techniques.
2.2. An Illustrative Electric Circuit As is well known to persons having ordinary skill in the art, the electric circuit represented by the schematic diagram in Fig. 1 comprises a voltage source and five resistors. For purposes of illustration, suppose that the voltage source is 10 2s volts and that the designer of the c*cuit intends that in the fabricated c*cuit the voltage at V2 is 2.0 volts and the voltage at V3 is 0.175 volts. To assure that the design will perform as des*ed, it should be simulated.
2.3. Modeling the Illustrative Electric Circuit As is well known to persons having ordinary skill in the art, the electric 30 c*cuit represented in Fig. 1 can be modeled, according to Kirchhoff's Current Law and Kirchhoff's Voltage Law, and according to the well known technique known as Modified Nodal Analysis, by the set of five equations shown in Fig. 2.

-5- 2~7~3~

In general, circuit simulation requires that the equations modeling a circuit be intensively manipulated. To permit them to be more easily manipulated, especially by computer, the equations are typically represented in matrix form. As is well known to persons having ordinary skill in the art, the set of equations in Fig. 2 5 can be represented in matrix form as shown in Fig. 3.
2.4. Simulating the Circuit As is well known to persons having ordinary skill in the art, the matrix of Fig. 3 can be solved using a number of well known techniques. J. Vlach and K. Singhal, "Computer Methods for Circuit Analysis and Design," Van Nostrand Reinhold, 1983, and P.C. Shields, Elementarv Linear Algebra, 3rd Ed., 1980 are hereby incorporated by reference into this specification. In circuit simulation a high degree of accuracy is required and therefore techniques which give exact solutions are preferred. One such technique is Lower Upper Decomposition ("LUD") followed by Forward Elimination ("FE") and Back Substitution ("BS").
As is well known to persons having ordinary skill in the art, LUD
involves two operations: normalization and updating. In summary: normalization involves dividing each after-diagonal element of a row by its diagonal element and updating involves manipulating the rows below the diagonal element of the most recently normalized row. The procedure starts with the top left diagonal element and 20 proceeds down the diagonal. The LUD is complete when all of the diagonal elements have been normalized. According to one technique, after LUD is completed, the FE and BS is performed to yield the solution to V2 and V3.
Empirically, some matrices have a high percentage of zero elements and are therefore said to be "sparse." In contrast, matrices with a low percentage of zero 25 elements are said to be "dense." While the LUD of a sparse matrix can be identical to the LUD of a dense matrix, the presence of numerous zero elements in the sparse matrix means that many of the operations carried out in performing the LUD are unnecessary.
In the preferred embodiment, Forward Elimination is carried out 30 concuIrently with the LUD in a manner that is well known to persons having ordinary skill in the art. In summary, the matrix representing the coefficients is augmented with the vector representing the right hand side of the equations in Fig. 3.
Fig. 4 presents the augmented matrix corresponding to the matrix of Fig. 3.

- 6 ~ 2 Fig. S shows a representation of the matrix of Fig. 4 where an "X"
represents the position of a non-zero value in the matrix. As is well known to persons having ordinary skill in the art, during LUD additional non-zero elements are introduced into the matrix in determinable positions. These are represented by s "F"s in Fig. 5. This is significant because it affects the scheduling of the tasks needed for LUD.
2.5. A Task Graph for Solving the Matrix In general, LUD, FE and BS require that each row be successively normalized and that each row that has a non-zero element (i.e., an "X" or an "F") 10 under the diagonal of the row normalized be updated. As is well known to persons having ordinary skill in the art, based solely on the position of the Xs and Fs in a matrix, a schedule can be generated of the operations needed to solve the set ofequations represented by the matrix. A representation of the operations to be performed and their interdependencies can be drawn. Such a representation is 1S known as a "task graph."
The construction of task graphs from matrices is well known to persons having ordinary skill in the art. O. Wing and J. Huang, "A Computation Model of Parallel Solution of Linear Equations," I.E.E.E. Trans. on Computers, Vol. C-29, pp.
632-638 (July 1980), and P. Sadayappan and V. Visvanathan, "Circuit Simulation on 20 Shared Memory Multiprocessors," I.E.E.E. Transactions on Computers, Vol. C-37, pp. 1634-1642 (Dec. 1988) are hereby incorporated by reference into this specification. A task graph can be derived from the pattern of non-zero and fill-in elements in a matrix. Fig. 6 presents the task graph associated with the matrix shown in Fig. 5. The task graph can be derived by representing each row 2s normalization, row update and back-substitution operation as a task and by noting the interdependencies of the tasks.
A task is represented in the task graph by a circle circumscribing a description of the operation to be performed. A description prefaced with an "N"represents a normalization operation. A description prefaced with a "U" represents 30 an update operation and a description prefaced with a "B" represents a back substitution operation. The number following the prefixes "N" and "U" designate the row to be operated on. For example, the task containing the description "Nl" means normalize row 1 and the task containing the description "U2Wl" means update row 2 with information from row 1. The pair of numbers following the prefix "B" is the 35 row and column, respectively, of the element in the matrix involved in back -7- ~7 ~2~ `~

substitution. A given task can only be performed when all of the preceding tasksthat feed information to it have been performed.

3. An Illustrative Embodiment of the Invention 3.1. In General When, as in Fig. 6, the task graph indicates that two or more tasks are independent, the task graph can be completed more rapidly by performing those s tasks in parallel. An illustrative embodiment of the invention, presented in Fig. 7, permits the tasks to be performed largely in parallel.
3.2. The Organization of the Illustrative Embodiment Fig. 7 presents an overview of the illustrative embodiment of the invention. It is based on a distributed memory multiprocessor system and consists of o three processing elements 707 ("PE"s), a communication network 703 ("CN") that is controlled by a communication network controller 705 ("CNC") and a host 701 processor. In this embodiment there are three PEs but it will be obvious to persons having ordinary skill in the art how to modify the design to have a different number of PEs. In this embodiment, the CNC provides an interface to the host but in other S embodiments the host may interface directly with either a PE or a portion of the CN.
In this embodiment, the CNC controls the CN and the communication processor 809 within each PE.
Each PE has a communication processor that provides an interface to the communication network, an integer unit that is responsible for address calculation, a 20 floating point unit, a cache for holding the rows just normalized, and random access memory to hold a portion of the matrix. The host processor executes the preprocessing necessary to construct the task graph and a task schedule.
3.3 Operation An initialization program is executed on the host. Each of the 2s processing elements in the system has an initialization control signal that forces it to read in data from the communication network into memory. Once the program and data have been loaded into the processing elements they can begin to perform tasks.
3.4 The Communication Network Controller (CNC) As shown in Fig. 9, the CNC 903 handles communication between the 30 host 909 and the processing elements 905 as well as controlling a sequencing unit.
The sequencing unit is within the CNC and takes a communication schedule that was generated during preprocessing. The communication schedule is used to control communication between the processing elements.

2 ~

3.5 Communication Network ~CN) Recall that the row operations and communications are predetermined prior to an execution of the task graph. A processing element 905 can therefore assume that information that it transmits to the comrnunication network 901 will be s routed correctly by the communication network controller 903. More specifically, the communication processor 907 within the processing element 905 holds the information until the communication network controller indicates that it may be placed on the communication network.
There are many ways in which the communication network can be 10 implemented. One option includes using a crossbar switch. Other options use ring communication systems or a bus. Simulation studies presented by R. Telichevesky,P. Agrawal and J. Trotter in "Fast Scheduling Schemes for Efficient Sparse Matrix Factorization on Multiprocessors," I E.E.E. Proceedings ICCD 91, Boston (1991), indicate that a bus system running at half the processing elements speed is adequate s for 16 processors when as in one embodiment of the invention the Intel I860 microprocessor is used as the processor in the processing element. The bus scheme has the advantage of being easy to implement and being able to broadcast to multiple processing elements.
3.6. The Processing Element (PE) The Processing Element 707 is the main compute engine for performing tasks. Hence, it contains a number of resources that are used to communicate and to evaluate the various functions. It has a wide data bus (64 bits) that allows a double precision floating point value to be read in quickly. It contains an 8K data cache, which is enough to hold a source row and its index values. The instruction cache can 25 be used to hold the code for the normalization and the update. The main memory only holds the precompiled instruction list that tells the processor which rows to normalize and update. The integer unit can be operated in parallel with the floating point unit for address calculation. The floating point unit can be used in a pipelined multiply-add mode (this is the core operation of the update), increasing its 30 throughput. The PE must be able to manipulate integers for computing the addresses of the various elements in the data structure. High speed memory or cache is used for storing matrix }ows and their indices. A custom VLSI integrated circuit could be specifically designed for these tasks. In another embodiment a special purpose processor with a microcontroller and a high speed specialized floating point unit 35 could be used. Alternately, any of the general purpose microprocessors that have _ 10 _ 2 ~ 3 ~' cache storage and a floating point unit could be used.
3.7.1. Integer unit The integer unit 803 calculates the addresses for the memory system and implements the basic branch and control instructions of the program. The use of the S integer unit to calculate the various addresses should not adversely affect the speed of the floating point processor.
3.7.2. Cache The processing element has a cache 805 large enough to hold the data representing one or more matrix rows. This allows for efficient storage of the source lo row that is used to update other rows of the matrix with higher row numbers.
3.7.3. Floating Point Unit The processing element includes a high speed floating point unit 807 to handle all of the necessary arithmetic computations. Double precision is preferred for accurate circuit simulation.
3.7.4. Memory The memory system 811 allows random access to the data representing the rows in the matrix. Because the memory is accessed randomly and at high speed, static memory is preferred to achieve the fastest possible speed.
3.7.5. Communication processor (CP) The communication processor 809 is the part of the processing element 707 that provides the interface to the communication network 703. The communication processor contains a bi-directional data buffer which allows the processing element to transmit data to the communication processor and vice versa.
The advantage of the buffer is that the processing element can perform computations 2s while the data is queued for transmission to the communication network and while data is being received. The data representing elements in the matrix is transmitted over the communication network in untagged form. Therefore, the processing element relies on the context of the data to indicate its meaning.
Thus the buffer is implemented as a first-in first-out (FIFO) queue and is 30 shown in Figure 10. In order to achieve high speed communication, the communication bus is 96 bits wide. In other embodiments the communication bus can have a different width. The processing elements' internal bus is 64 bits wide but can be other widths equal to or not equal to the communication bus width.

4. Schedulin~ Techniques The computer described above can be used to solve matrices. Because the computer has three processing elements, each with its own memory, the rows of the matrix are distributed among the various processing elements. Although the 5 operations on each row are somewhat independent there is some information thatneeds to be communicated between the processing elements. Thus a communication takes place when a processor needs the result of a calculation perforrned by another processor.
Fig. 13 presents a task graph indicating that different tasks are to be 10 executed on different processors. A task performed on the first processing element is represented by a circle. A task performed by the second processing element is represented by a square and a task performed on the third processing element is represented by a triangle. An inter-processing element communication over the communication network is represented in Fig. 13 by a dashed line between tasks.
A task can be executed when all of the tasks that it is dependent on have been completed. Thus, it is possible to schedule a subset of the tasks to be performed by each of the processing elements. In addition, it is possible to schedule a list of communications that have to occur between processing elements. The set of equations can be solved by providing each processor with a list of tasks it must20 execute and providing the scheduler with a list of communications, both of which can be obtained from the task graph. Fig. 14 shows the schedule obtained from the task graph of Fig. 13.
The remainder of this section details how the task graph of Fig. 13 and the task schedule of Fig. 14 are constructed given the illustrative problem and the 2s computer of Fig. 7.

4.1 Notations and Definitions All circuit analysis procedures require the solution of a system of linear equations of the form ~ = b, where A is the coefficient matrix containing elements a ~ is the unknown vector of circuit variables, and b is the excitation vector. The 30 size of the coefficient matrix is directly proportional to the size of the circuit. Most of the entries of A are zero resulting from an extremely sparse set of equations.
We will study here the direct solution of the system of equations = b. A represents a sparse matrix of rank n, b the right hand side and~is the solution vector. In R. Telichevesky, P. Agrawal and J. Trotter in "Efficient 35 Partitioning Schemes for Circuit Simulation Matrix Factorization on -12- 2~ 7~ -~

Multiprocessors," Int. Conf. on Applications Specific Array Processors, Barcelona, Spain (Sept. 1991), we present a brief explanation of the basic operations that are necessary to accomplish this objective and a discussion on the level of granularity in which these basic operations should be represented.
The LU decomposition of the matrix A can be represented as a series of operations of three basic types:
A division, denoted by Dk, is defined as: For k = 1, 2, ..., n:

Pk ~ -- (l) akk A row-wise normalization, denoted by Nk, is defined as: For all non-10 zero akj, akj ~ akj ' Pk ~ j>k (2) and the row-wise update, denoted by k ~ i, corresponding to row i being updated by row k, is defined as:
For all non-zerO a kj ~

aij ~ aij--aik akj . i,j>k (3) We shall refer to k as a source row and to i as a destination or target orfanout row in Equation 3.
In the literature, the division usually has been considered as the initial 20 phase of the normalization. However, state-of-the-art hardware can perform a floating point division (usually a very costly operation) as a background process, concurrent with multiply-add operations, justifying its treatment as a separate task.
Each row-wise normalization or update is implemented in a real processor as a series of elemental multiply-add operations, usually pipelined in order to achieve 25 maximum performance.
These operations can be executed in any sequence, provided the following rules are respected:
i. Nk cannot be started until Dk is completed.
ii. Dk cannot be started unless all update operations that modify akk are completed.

- 1 3 ~

iii. k ~ i cannot start before all updates that modify a ik have been completed and iv. Element a ij in k--~ i cannot be updated before a kj has been normalized.
Rules (i) through (iii) establish dependencies in the strong sense with respect to row-wise operations, i.e. one operation cannot start fetching operands S before the other has finished storing all results, which usually is a bad characteristic for pipelined systems. Rule (iv), however, permits a row update task to start as soon as the first result of the normalize operation is available, provided both pipelines run at the same speed. This rule establishes a new type of dependency which we shallrefer to as a mild dependency.
These tasks and the dependencies among them could be conveniently expressed as a generalized multitask graph G(V,Es ,Em), consisting of a node set V
and two arc sets Es and Em (corresponding to both strong and mild dependencies).This task graph has no cycles and is a Direct Acyclic Graph (DAG). In order to properly exploit pipelining in a multiprocessor vector machine we suggest the 15 following task model.
Each node 1); ~ V corresponds to a pipelined task, as shown in Figure 14. Figure 14 depicts the execution of the task in a real processor. The first piece of dataentersthepipelineatt = tif. Afteratimedelaypi(t = tof = tif+pi) corresponding to the pipeline latency the result of the first elemental operation is 20 available. After a time c i (t = t il = t if + c i ), which corresponds to the pipeline throughput times the number of elemental operations the last piece of data is fed into the pipeline and the processor is ready to start feeding the pipeline some data corresponding to a new task. At t = tol = t jf + c i +P i the result of the last elemental operation is ready and task 1) i is finished. If a task Vj depends strongly on 1~ i, it can 25 only start at t = tol, but in the case of a mild dependency, it can start in another processor as early as t = tof or in the same processor as early as t - til. Strong dependencies are represented by e~si ~ while mild ones are represented the e jmj . -Figure 6 shows the task graph that represents the LU Decomposition of the sparse matrix depicted in Figure 5. We assumed in this case that the pipeline 30 latency is 1 cycle and the throughput is 1 cycle for multiply-add operations and a division (nonpipelined) takes 1 cycle.
A task is called an initial or a- task if it does not depend on any previous operations and is called a terminal or ~-task if no task depends on it. We say that 1~ i depends on l)j if and only if there is at least one directed path Rjk i that 35 starts at l)j and passes through or terminates at vi. Every task l)i has an associate set 14- ~ t~

of dependency paths R a~i Each element Rka i ~ R ~ i is itself a set of verticesthat are traversed in the path that connects v i to any a-node (and includes them).
Furthermore, we can associate with each set Rka i an integer function S (Rka i ) that represents the cardinality of Rka i . A task v i is said to be at level h i if among all the 5 paths Rka~i connecting the initial tasks to vi, the largest one contains hi elements, or hi = max{S(Rka~i)} ~Rka.i ~ Ra,i k The height H of a task graph is defined as the level of the vertex at the highest level, or H = max{hi} ~i I ui~ V is a terminal node. (S) i Clearly, the height of the task graph is associated with the minimal completion time of the matrix decomposition. Task graphs that are short and stout are expected to exploit parallelism better than those that are tall and skinny. In the example in Figure 6, the height of the task graph is seven. However, just the level of the task is not enough to properly model some hardware features and different task lS sizes, so we introduce the concept of the completion time di of a given task v i, which is defined as follows:
di = max{ ~ (cost(v~ Uj I Vj ~ R.,aC~i} (6) K j=a where Rka~i is any path starting at an a-node and finishing at v i and cost(vj ) can be either c; (resource cost), pj (pipeline latency time) or even c; +pj, depending on the 20 type of the dependencies on Vj that occur while traversing the path Rk i. Theearliest completion time D for the decomposition is defined as the completion time of the node with the largest di, or D = max{di3 ~i I vi ~ V is a terminal node. (7) i We can also define the critical path RCa~i as the path whose total cost 2s di =D. Notice that in the case of a mild dependency the cost of a parent node is only its latency. The earliest completion time establishes the first barrier on the available i s 2 ~ ~ ~ 2 q3 3 concurrency for a given matrix and cost function. The partitioning bound is usually more restrictive than the critical path bound for row-wise operations. Nevertheless, the critical path plays an important role in the scheduling algorithm, as it seems natural that tasks that belong to the critical path should be executed before other less s important tasks. The last definition we need for our scheduling algorithm is the remaining completion time, ~i, for a given task vi. Assuming l)j is a terminal task and there is at least one path Rklj~ we define i as:

i = max{ (cost(~p)) ~ p 1 1~p ~ Rk~j and ~3p ~ (8) ~c P=i The remaining completion time is actually a measure of how important 10 is the early execution of a given task, as it is known that once the task 1) i is finished, the minimum time for completion of all the tasks that depend on 1) i is o i . Clearly o for ,B-tasks is zero. In the following we shall describe an algorithm for computing the remaining completion time for the entire task graph in linear time and a quasi-linear time scheduling scheme based on o-techniques.

5 4.2 SchedulingSchemes In a multiprocessor environment there are two major factors that heavily influence the utilization of computational resources during the sparse matrix factorization. The first is the allocation of data to the processors and the second is the task scheduling within each processor and in the communication network.
20 Earlier in R. Telichevesky, P. Agrawal and J. Trotter in "Efficient Partitioning Schemes for Circuit Simulation Matrix Factorization on Multiprocessors," Int. Conf.
on Applications Specific Array Processors, Barcelona, Spain (Sept. 1991), we studied several partitioning algorithms and empirically demonstrated that the load balancing scheme yields nearly optimal partitions with respect to overall efficiency.
2s Two scheduling techniques were used in conjunction with various partitioning techniques. One of them is a simple-minded level-based technique presented by J.Trotter and P. Agrawal in "Circuit Simulations Algorithms on a Distributed Memory Multiprocessor System," I E.E.E. Int. Conf. on Computer Aided Desi~n, Santa Clara, CA, pp. 438-441 (Nov. 1990~ and the other a greedy scheduling 30 technique. In the level-based technique, all tasks scheduled at a level are completed before proceeding to execute the tasks at the next level. The processors operate in a lock-step fashion. Obviously, some performance is lost as processors may have to - 16~ ?J Sl '~7 wait although there are tasks ready for execution. A simple greedy scheduling mechanism, described later, produces results that are better than the level-based technique. However, it too falls short of the theoretical maximum performance imposed by the partitioning algorithm.
In order to improve the utilization of the system, a good scheduling algorithm should be able to choose among the enabled tasks the most critical andschedule it to be executed first. V. Sarkar in "Partitioning and Scheduling Parallel Programs for Execution on Multicomputers," The MIT Press, 1989, proposed an algorithm based on critical path scheduling that produces very good results. In his 10 scheme, the first scheduled tasks are in the critical path. Next, each branch is analyzed and the local critical path is scheduled in a recursive fashion.
Unfortunately, the time complexity of this algorithm is O(E(V+E)) where E and V
are, the number of edges and the number of vertices in the task graph, respectively.
E represents the dependencies and V, the number of row-wise operations present in 15 the task graph. For large task graphs the scheduling time quickly becomes prohibitive.
Two new scheduling techniques are based on the remaining completion time (o). In these schemes, tasks with larger o are scheduled to be executed before the others, as tasks in the critical path or in heavier branches tend to have a larger 20 than others. The net effect of this scheduling is not as good as Sarkar's, but at a much smaller cost, as o can be computed for the entire task graph in linear time. In the following, we show how to compute ~, then introduce two scheduling techniques based on ~-techniques and finally present experimental results. For the sake of comparison, we begin by describing the greedy scheduling technique described in R.
25 Telichevesky, P. Agrawal and J. Trotter in "Efficient Partitioning Schemes for Circuit Simulation Matrix Factorization on Multiprocessors," Int. Conf. on Applications Specific Array Processors, Barcelona, Spain (Sept. 1991).

4.2.1 A Greedy O(E) + O(V) Scheduling Heuristic The greedy technique works as follows. Each processor will execute all 30 assigned tasks that are ready for execution, i.e., all tasks on which it depends on have already finished. Otherwise, the processor will stall. This technique is very fast and yields in almost all cases better results than the level approach. However, it requires a more complex synchronization mechanism among processors. In the following, we give a brief description of the technique.

- 17 - 2 ~ 9 ~`

We associate with each task l)j the number of dependencies dj that must be satisfied before l~j is enabled. Clearly, for initial tasks dj = O. We also associate with 1~j a timestamp tj which represents the earliest time the task can be executed considering its data dependencies and the processor allocation. In the beginning of 5 the technique all tj = O. When the technique finishes, each tj will contain the starting time for the execution of the task Vj. Each processorp has a local timerfp which holds the time in which the last scheduled task was completed. The technique starts after the task graph is built, all dj are properly set, allfp = 0 and the data is properly distributed using the load balancing partitioning technique described in R.
lo Telichevesky, P. Agrawal and J. Trotter in "Efficient Partitioning Schemes for Circuit Simulation Matrix Factorization on Multiprocessors," Int. Conf. on Applications Specific Array Processors, Barcelona, Spain (Sept. 1991).
1. For each processorp visit all tasks ~ Qp such that dj = O. For each visited task l)j do:
a. Schedule ~j at time tj = max(tj,fp).
b. Set the local timer in processor p atfp = tj + C (~j ).
c. Visit all tasks 1) i that depend on the completion of ~j, i.e. 3 e ij. For each visited l~ i do:
i. Set ti=max(ti,fp)-ii. Setdi=di--1.
2. Repeat set 1 until all the tasks have been scheduled in all processors.
We define the comple~ion timefor the greedy algorithm tG, as the maximum value offp when the scheduling finishes. We can define also the greedy scheduling utilization rl G as:
T

Il G = t P (9) 4.2.2 Computing Remaining Completion Time in O(V)+O(E) We assume that in the beginning of the technique the task graph is levelized and each node v i contains a list of tasks ~j on which it depends. Thedependencies correspond to the arc directions that are opposite to those shown in 30 Figure 6. Initially, we set D=0 and each node 1) i will contain o i = 0. The levels in -18- ~7~ 3 the task graph range from 1 to H, H being the highest level. The technique proceeds as:
l . At each level k starting fiom k = H towards k = 1 visit all nodes v i such that h i = k. For each node 1) i do:
s a. Set ~5=~i+ci+pi.
b. Traverse all arcs ejSi in the reverse direction and set ~j = max ( ~j ~s ) c. Traverse all arcs ejmi in the reverse direction and:
i. Set m = s--Cj.
ii. Set oj =max(om ,~i ,~j) 2. Set D = max (D, ~s) At the end of the technique execution all nodes ~ i will be set to o i and D will be set to the earliest completion time. If desired, we could annotate in 1 (b) and l(c) the edge that caused ~j to be maximum as the critical edge eic. This set of critical edges form the critical path and its branches.

5 4.2.3 Vertical O(V log V) Scheduling Heuristic We associate with each task l)i the remaining completion time i as computed in Section 4.2.2. We also associate with each l) i a timestamp t i which represents the earliest time the task can be executed considering its data dependencies and the processor allocation. Finally, each node contains information 20 about the number of dependencies ndi that must be satisfied before 1) i is enabled.
Each processor p has a local timerfp which holds the time the last scheduled task is completed. An auxiliary tv register will contain the total time necessary to complete the decomposition. The technique described here is divided into two phases, the first is the actual scheduling and the second is the computation of timestamps for 25 simulation purposes only. The technique starts after the task graph is built, all o and ndi are properly set, allfp =0, all ti =, tv=O and the data is distributed according to some partitioning technique.
l. In each processor sort the tasks in decreasing order of o (O(V log V) step). The scheduling is done. Theorem 1 shows that the resulting schedule is deadlock-free.

] 9 ~ ?3 2. For each processor p visit the next scheduled task ~ L. For each visited task u do:
a. If ndi~0 go to another processor, else proceed.
b. Set ti = t jf = max(ti ,fp).
S c. SettOI=tif+ci+Pi d. Set tof =tif +ci+Pi.
e. Setfp =til =tif +c f. Traverse all arcs eS; and for each visited Vj that depends strongly on do:
lo i. Set ndj = nd~
ii. Set tj=max(tj,tOl) g. Traverse all arcs e~J and for each visited ~ j that depends mildly on do:
i. Set ndj-ndj- 1.
ls ii. Set tj=max(tj,tOf).
h. Set tV-max(tOl~tv)-3. Repeat Step 2 until all tasks are visited.
The scheduling scheme proposed in Step 1 is so simple that one might ask if it is correct, as it is not obvious that it will generate deadlock-free code. The 20 following proof, by contradiction, provides necessary and sufficient conditions for a deadlock-free code:

Theorem 1 This proof is divided into three parts, the f~rst corresponding to theuniprocessor case and the remaining two treat interprocessor deadlocks.
a. Suppose two tasks ~ i and ~ j are in the same processor and l) i depends on ~ j.
2s A deadlock can occur if and only if ~ i is scheduled to be executed before v;.
However, if ~ i depends on 1) j there must be at least one directed path Rjk i and therefore j = i + ~ k jcost(u p ) provided l) p ~ R jk i and l) p . ~ j . Assuming cost (~ p) > O, otherwise not physically possible, ~ j > ~ i end ~ j will be ~ - 2 ~ 7 s,~

scheduled to be executed before 1) i, which contradicts the initial deadlock hypothesis.
b. Suppose ~ i and u; are in a processor p r and l) m is in processor p 5 . An interprocessor deadlock can occur if and only if ~ m depends on l)j and ~ i depends on l)mJ but 1) i is scheduled to happen before l)j. Using the same arguments as in part (a) we can say oj > m and m > i . Using transitivity j > i, whichfinishes the proof of the theorem by the same arguments as in (aJ
c. By induction on (b) we can prove that there will be no deadlocks in any number of processors.
Even though the above technique produces deadlock-free schedules, there is one obvious drawback. This is because there is a possibility that some processors may be idle waiting for data to arrive from another processor in spite of enabled tasks being present. The next Section will present a modified version of the 15 vertical scheduling technique which will reduce processor idle time by introducing some greediness.

4.2.4 o-Greedy O(V log V) Scheduling Scheme This technique is based on the same assumptions we made in Section 4.2.3, but we add a greedy element in this scheme. Since the greedy and the o 20 techniques are conflicting, we need to create a compromise between them, quantitatively in the form of a scheduling elasticity K, which is discussed next.
In Step 2(b) in Section 4.2.3 we set tif = max(ti,fp). This code means that 1) i cannot start before it is enabled by the completion of tasks it depends on (ti ), or before the processor is ready to execute it (fp), whichever occurs last. If ti > fp 2s the processor stalls. However, there might be another task Vj such that tj < t i, but j ~ i and ndj = 0. In this case, we could execute 1)j before 1) i in order to reduce the slack. Figure 12 depicts four possible timing situations, which might occur if we try to insert oj before O i-In (a) and (c) tj > fp and therefore there is still a little slack left, while in 30 (b) and (d) l)j causes no slack. In (a) and (b) the insertion of 11j causes the beginningof 1~ i to be delayed, while in (c) and (d) there is no delay involved, since the inserted task finishes before 1) i is enabled. The technique used to decide whether or not insert a task ~j depends on K, SO if the condition - 21 - 2 ~

slackj + Cj ( 10) slack i is satisfied, we insert l)j before ~ i. If K<O no insertion is permitted and thetechnique yields the same results as the vertical scheduling. If 1C< 1 no conflict happens between the ~-technique and the greedy technique. In this case, the s insertions depicted in Figure 11 parts (a) and (b) are not allowed. If K> 1 there is a compromise between the greedy technique and the o-based scheme.
The technique is basically the same as described in Section 4.2.3, with exception of line 2(b) which is substituted by the following procedure:
1. If ti<fp set tif=fp andreturn.
10 2. Set~s =l)i, slackS = slacki = ti - fp-3. Visit all tasks ~j that are enabled in processor p (ndj = 0). For each visited task do:
a. Set slackj = max(tj -fp, O).
b. If (slackj < slackS) and (slack j + c; < K slacki) do:
i. Setslacks=slackj.
ii. Set Vs =i iii. If slackS =0 finish the search.
4. If l~s=~i settif=ti andreturn (searchfailed).
5. Schedule u s before u i. set t if -fp + slack s and return.

20 4.2.5 Simulation Results and Discussion In section 4.2.3 we computed tv, the completion time for the vertical technique. A similar procedure could be used for computing t~G for the o-greedy technique. If P is the number of processors and T is the sequential execution time, we represent the scheduling utilizations for the vertical and o-greedy techniques and 2s for the optimal partitioning by 1l v . ~ G and r~ Opl, respectively.
T
~v= t P (11) ~a~3~;~J~j T (12) t~G P
T

opt = -- (13) top~ .p top~ is computed as follows:
topt = max (~, C(~j)) (14) s where ~;; j C(l~j ) represents the total execution time of the tasks 1)j that have row i as destination. topt represents the barrier to the amount of parallelism available during the LU decomposition and corresponds to the minimum completion time due to partitioning. This can be easily computed for each row of the matrix by determining the dependencies and the number of operations needed to update a given 10 row.
Table 1 lists the simulation results for the LU decomposition using the techniques previously described. In the case of the level-based and greedy scheduling we reproduce the results reported in our previous work. R. Telichevesky, P. Agrawal and J. Trotter in "Efficient Partitioning Schemes for Circuit Simulation 15 Matrix Factorization on Multiprocessors," Int. Conf. on Applications Specific Array Processors, Barcelona, Spain (Sept. 1991). In order to produce a fair comparison of the techniques, we used a very simple cost function, in which each elemental operation takes 1 cycle and we assume no pipelining (p ~ = 0 for all i).
The vertical scheduling yields poor results in most cases, since the code 20 generated results in too much slack among the tasks. However, when we approach the bound of the minimum completion time using many processors (in this case there is a lot of slack anyways), the technique tends to generate nearly optimal results, as shown in the table for feb and omega for more than 8 processors.
In contrast, the o-greedy scheduling yields better results in 80% of the 2s cases, as it combines the advantages of both the vertical and the greedy schemes.
The only two cases in which the bare greedy technique outperforms the ~-greedy is for iirl2 and mfr with a moderate number of processors. In these examples the critical path does not play an important role, as efficiency is dominated by the greedy allocation of tasks with a minimum slack. In all cases we computed K according to:

K D P ( 15 ) As P increases we shall have more slack available, thus we shall permit less changes in the vertical ordering. As the ratio between the total cost and the size of the critical path increases we shall pay less attention to the vertical ordering and s increase the chances for insertions.
The results presented in this section are valuable for the comparison of different techniques. However, the cost functions that they represent are not realistic. In the next section we present the results of several simulations involving real cost functions in order to compare the performance of state-of-the-art 10 microprocessors connected through a high bandwidth network.

4.3 Architecture Evaluation If the processors are connected through a high speed bus a simple modification of our task graph allows its simulation. First, we add a broadcast task Bi, that mildly depends on the normalizations, provided the throughput of the bus is 1S the same as the normalization pipeline. Then we change the dependencies of the updates; the dependency on the proper source row normalization will be replaced by the proper source row broadcast. Finally, we force the processor that is allocated to the row in question to broadcast the row using the bus. There are several advantages for the usage of this model. First, it is consistent with the general task model.
20 Hence, any scheduling technique can be executed with no modifications. Second, the bus transactions could be scheduled automatically according to their importance, as they have information on the remaining completion time as any other tasks.
Finally, if the system is carefully designed we can use the bus as a pipelined device avoiding complex buffering schemes and the overhead they - 24 ~ ~ f ~ b ~

matrix Level 11 G ¦ 11 V ~1~ _ 2 0.88 0.91 0.85 --0.95 1. 0 4 0.710.87 0.74 0.87 1.00 feb 8 0.510.61 0.63 0.82 1.00 16 0.330.38 0.60 0.60 0.60 (n= 32 0.190.22 0.30 0.30 0.30 10060) 64 0.110.13 0.15 0.15 0.15 2 0.970.99 0.93 1.00 1.00 4 0.910.99 0.85 1.00 1.00 iirl2 8 0.830.89 0.74 0.87 1.00 16 0.700.77 0.53 0.75 1.00 (n= 32 0.520.64 0.43 0.68 1.00 7310) 64 0.370.50 0.32 0.56 1.00 2 0.960.99 0.90 1.00 1.00 4 0.850.89 0.79 0.93 1.00 mfr 8 0.720.87 0.66 0.82 1.00 16 0.530.74 0.55 0.72 1.00 (n= 32 0.360.52 0.45 0.60 0.85 5496) 64 0.220.30 0.32 0.42 0.42 2 0.850.78 0.82 0.92 1.00 4 0.610.70 0.71 0.84 1.00 omega 8 0.390.44 0.65 0.74 0.74 16 0.230.27 0.37 0.37 0.37 (n= 32 0.120.16 0.18 0.18 0.18 3~ 4212) 64 0.070.08 0.09 0.09 0.09 40 Table 1: Comparison of Different Scheduling Schemes i860 Wp W B p B ~
4s 1 151.8 233.4 49.1~1 46.41 9.841 2 75.91 120.0 24.56 23.54 - 4.904 4 39.18 61.49 12.32 12.10 2.442 8 20.24 32.85 6.205 6.460 1.226 16 11.12 18.75 3.569 3.703 0.694 32 6.233 9.958 1.992 1.903 0.401 64 4.781 6.155 1.615 1.238 0.334 6s Table 2: Matrix Factorization Time for DRAM (ms) introduce in the system. In order to better illustrate this concept, consider the tirning diagram depicted in Figure 15. Assume that processorp 1 is executing a normalization Ni while P 2 needs the data generated by Ni in order to perform a row update i~ j. If these tasks are critical, the bus can be scheduled to act as a pipe 70 between the two tasks, getting data out of the pipeline on processor p 1 and feeding it to the pipeline on processor p 2. This model not only provides us accurate - 2s - 2 ~1 u~

information on the actual performance of the system, but also improves the usage of the communication resources.
Once we have properly modeled the hardware characteristics of hypothetical architectures we compared their performance using a benchmark circuit s matrix that represents part of a dynamic ram memory. Table 2 contains the results, in mili-seconds for different hardware configurations. i860 represents a system built with Intel i860 processors running at 40Mhz. Wg and Ws represent systems built with a Weitek FPU and general purpose controller or a special purpose controllerrunning at 20Mhz. B g and B s represent systems that contain a Bit FPU and a general 10 or special purpose controller running at a 100Mhz, if it could be built. For the Bit FPU we changed the scheduling algorithm in order to properly model the Division Unit available in the chip. In all cases we model the bus as having the same throughput as the floating point pipeline within the processor. For the sake of comparison, the factorization of the same matrix on a Cray X-MP Supercomputer 15 takes 59.8 ms, while in a Sun 4 Workstation takes 19420 ms. The results obtained here are an incentive to the design and construction of such a machine.

5. Solving the Illustrative Problem The circuit of Fig. 1 can be modeled by the set of equations shown in Fig. 2 which are also represented in matrix form as shown in Fig. 3. The æro/non-20 æro structure of the matrix, shown in Fig. 5, is used to determine the normalization,update and back-substitution tasks necessary to solve the set of equations represented by the matrix. These tasks can be scheduled to allow more than one processing element in a multiprocessing system to execute the tasks in parallel. Recall that the tasks are based on row level operations and therefore whole rows of the matrix are 25 distributed amongst the respective processing elements.
This partitioning, is based on methods taught by R. Telichevesky, P. Agrawal and J. Trotter in "Efficient Partitioning Schemes for Circuit Simulation Matrix Factorization on Multiprocessors," Int. Conf. on Applications Specific Array Processors, Barcelona, Spain (Sept. 1991). The method fixes the set of tasks which 30 to be executed by an individual processing element. Once this task partitioning is done the tasks can be scheduled on the computer. In the task graph of Fig. 13, adashed line between tasks indicates that a data communication has to take place over the communication network between the respective processing elements.

2, An aspect of the illustrative embodiment of the invention, ("the scheduler"), first assigns a cost to each task in the task graph. Fig. 13 depicts a task graph with cost values of p and c associated with each task. For example, the cost, p, of the N1 task is 1 because the results of task N1 become available for use by other 5 tasks after one time unit. The total cost of a task is equal to p plus c which is represented by the length in time of the associated rectangle as shown in Fig. 14.
Based on these costs it is possible to calculate a metric, o, called the "earliest completion time." This metric indicates the least amount of time in which the rest of the task graph could be computed were there an infinite amount of processing 10 resources available. A value of ~ is assigned to each task in the graph beginning at the terminal tasks (e.g., task B2,3) and proceeding toward the initial tasks (e.g., task N1 and task N4). The value of o for a given task is determined by adding the maximum of the o values of the tasks which depend on the given task to the cost of the given task. For example, as shown in Fig. 13, o for the task U5W2 is calculated 15 by adding its cost (c+p = 3+0 = 3) to the maximum ~ value of the tasks dependent on it, the USW3 task, which has a ~ of 2.
The o of task N1 can be determined by taking the maximum of the ~s for tasks U2W1 (o = 10), U5W1 (o = 8), B1,4 (o = 0) and B1,2 (o = 0), the maximum being equal to 10, and adding this to the cost of the N1 task which is 1 because the 20 tasks U5W1 and U2W1 depend mildly the N 1 task giving a o for N 1 of 11. Once the os have been calculated for all of the tasks in the task graph they are sorted in decreasing value of o. The schedule is then derived by taking the set of tasks that are to be executed on each processing element, and for that processing element scheduling the tasks to be executed in reverse order of o. When there is time 25 between the completion of a given task and the next task could be scheduled, then an attempt is made to schedule a another task to be executed in the interim. This procedure continues until all of the tasks on all the processing elements have been scheduled. Additionally, a schedule of communications is determined which is used by the communication network controller to sequence the communications on the 30 communication network. Fig. 14 shows a schedule of the tasks to be executed for each of the three processors and the sequence of communications that is necessary to communicate the data between processing elements.
Once the task schedule has been generated for each of the processors the computer is used to solve the equations. A list of the tasks and the instructions 35 necessary to execute those tasks is transmitted from the host to each of the processing elements. The rows of the matrix are transmitted from the host to the - 27 - ~ ~3 7 ~ 2 .J ~

respective processing elements responsible for operating on the row. A list representing the sequence of communications to be carried by the communication network is transmitted from the host to the communication network controller.
As is shown in Fig. 14, the first task executed is Nl (i.e., the s normalization of row 1) on processing element Pl (represented by a circle). The values in the matrix after this task has been completed are shown in Fig. 16. The result of the Nl task is transmitted by Pl to processing element P2, represented by a square. Note that the scheduling of the transmittal is dictated by the communication network controller. Next, task U2Wl depends on the data generated by task Nl and10 can start executing as the data becomes available. The task USWl is executed next on P2 and it uses the same data transmitted from Pl previously. The values of the matrix after these three tasks have been executed is shown in Fig. 17.
Note that the data from task N4, generated on processing element Pl is needed by both P2 and P3 to execute tasks B2,4, B3,4 and U5W4. In this case the s communication network controller causes the data to be broadcast to both the processing elements P2 and P3. Eventually, all of the tasks are completed and the memory of the processing elements contains, in aggregate, the solution vector (i.e., the solution to the set of equations). The matrix of Fig. 18 shows the matrix after all the tasks have been executed and the solution vec~or is then the last column of the 20 matrix. This is read from the computer by the host.
As shown in Fig. 18 in the last column, the voltages for V2 and V3 are determined to be 1.95 and 1.77 volts respectively.

Claims (2)

1. A multiple instruction stream-multiple data stream computer system CHARACTERIZED BY:
(1) communications means for connecting a plurality of processing elements;
(2) said plurality of processing elements comprising (a) a memory for holding a set of data signals, a set of instruction signals anda set of result signals, (b) means for processing said set of data signals based on said set of instruction signals to produce said set of result signals, and (c) a communications processor for receiving said set of instruction signals and said set of data signals from said communications means and for transmitting said set of result signals to one or more of said processing elements via said communications means in response to a communications network controller; and (3) said communications network controller comprising (a) means for controlling when each of said processing elements receives said set of instruction signals and said set of data signals from said communications means, and (b) means for controlling when each of said processing elements transmits said set of results onto said communications means.

2. On a multiple instruction stream-multiple data stream computer comprising a plurality of processing elements, a method for modeling physical phenomena represented by a manipulation of a set of linear equations, said method CHARACTERIZED BY:
resolving said manipulation of said set of linear equations into a plurality of tasks;
distributing said tasks to said processing elements;
for each of said tasks, computing a resource metric based on the amount of time needed by one of said processing elements to manipulate said task;
for each of said tasks, computing a completion metric based on the amount of time needed by said processing elements to complete manipulation of said set of linear equations after said task has been manipulated;
processing each of said tasks in an order based on said resource metric and said completion metric.