Dual Processor Scheduling with Dynamic

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1 Dual Processor Scheduling with Dynamic Reassig nment 341 SHAHID H. BOKHARI, MEMBER, IEEE Abstract-The problem of finding an optimal dynamic assignment of a modular program for a two-processor system is analyzed. Stone's formulation of the static assignment problem is extended to include the cost of dynamically reassigning a module from one processor to the other and the cost of module residence without execution. By relocating modules during the course of program execution, changes in the locality of the program can be taken into account. It is shown that network flow algorithms may be used to find a dynamic assignment that minimizes the sum of module execution costs, module residence costs, intermodule communication costs, and module reassignment costs. Techniques for reducing the size of the problem are described for the case where the costs of residence are negligible. Index Terms-Computer networks, cutsets, distributed computers, dynamic assignment, load balancing, maximum flows, network flow, partitioning. I. INTRODUCTION IN this paper we examine the problem of optimally partitioning a modular program over a dual processor system. This problem is of relevance to the numerous dual and multiple processor systems that now exist in both academic and industrial environments. The basic problem is to find an assignment that minimizes the total running cost of the program, where cost may be measured in time, dollars, or other resource units. This problem has been analyzed by Stone [I1, [2] for the case where the total cost has two components. The first component is the cost of executing each module. The execution cost of a module is assumed, in general, to depend on the processor to which that module is assigned and the amount of computation performed by that module. The second component is the cost of intermodule communications between pairs of modules when they are resident on different processors. It is assumed that, should two modules be coresident on one processor, the cost for intermodule communica- Manuscript received July 1977; revised January 4, This work was supported by the National Science Foundation under Grant MCS Some of the results of this paper were presented at the Brown University Distributed Processing Workshop, August 3-5, This work is part of the author's doctoral dissertation at the University of Massachusetts. The author was with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA He is now with the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA tion between them is zero. The objective in making the assignment is to minimize execution costs by trying to assign modules to the processors on which they run with least cost while, at the same time, minimizing the communication costs by trying to keep pairs of modules that communicate with each other on the same processor. Stone has developed a mathematical model in which the modules to be assigned are represented by the nodes of a graph and the costs by weighted edges. It has been shown that an optimal assignment (i.e., one that minimizes the sum of execution and communication costs) corresponds to a partition of the graph imposed by a minimum weight cut. In the case of a two-processor system, the minimum weight cut ("mincut") may be found very efficiently using any one of several available network flow algorithms (3] -[6]. This model has successfully been used to find optimal assignments for the multiprocessor system at Brown University [71- [10], which is made up of an IBM System/360 computer linked to a microprogrammable minicomputer. This model yields an optimal static assignment because it assumes that once a module is assigned to a processor, it remains on that processor while the characteristics of the computation are constant. When these characteristics change, a new assignment must be computed. In order to make the best use of resources in a distributed system, it is essential to reassign modules dynamically during program execution, so as to take advantage of changes in the local reference patterns of the program. Clearly, the gains from relocating modules must outweigh the cost of relocation. In this paper we extend the mathematical model (as applied to a two-processor system) to permit relocation of modules from one processor to the other at certain points during the execution of the program. This relocation is assumed to incur a predefined relocation cost that contributes to the total cost of running the program. The model also includes module residence costs, these being the costs of modules residing on processors without executing. We show that an optimal dynamic assignment (i.e., one that minimizes the sum of execution, residence, relocation, and communication costs) corresponds to a "dynamic" partition imposed by a minimum weight cut in an extended graph. This mincut may be found using network flow algorithms as in the static assignment problem. The models for both the static and the dynamic assignment /79/ $ IEEE

2 342 problems assume serial execution of the modular program being considered, i.e., although there are two processors and several modules, only one module executes at one time. One or both of the processors may be multiprogrammed so that they execute concurrently on different programs, but not on the same program. This paper is organized as follows. The next section presents our definitions and assumptions. In Sections III and IV we present our mathematical model for, and the solution of the dynamic assignment problem. In Sections V and VI we present techniques for reducing the size of the problem when costs of residence are negligible. Section VII investigates upper and lower bounds on the cost of the dynamic assignment with respect to the static assignment. The final section of this paper contains a summary of the results presented, conclusions, and projections for further research. II. DEFINITIONS AND ASSUMPTIONS This section presents several definitions and assumptions that are relevant to the remainder of this paper. Consider the problem of finding an optimal assignment of a single program on a two-processor system. The program is assumed to be a collection of modules that are free, in general, to reside on either processor. A module may be a portion of executable code or a file of data. For the Brown University system [7]-[101, modules are subroutines that make up a large graphics program and control is transferred from module to module by subroutine calls. For the purposes of our mathematical model, the specific mechanism for transfer of control between modules is irrelevant-the modules may be subroutines or coroutines. It is assumed that the costs for executing each module on either processor are known. For each pair of modules, the cost of communication between them, should they not be coresident, is also assumed to be known (if two modules are coresident, the cost of communication between them is assumed to be zero). The net cost of an assignment of modules to processors is the sum of the module execution costs for all modules plus the sum of intermodule communication costs for all pairs of modules that are not coresident. The assignment that minimizes this cost may be found by creating a static assignment graph as shown in Fig. 1. The nodes P1 and P2 represent the two processors. Nodes A through F represent the modules. The weights on the edges joining pairs of module nodes represent intermodule communication costs. The weight on the edge joining a module node to processor node P1 represents the run cost of that module on P2 and vice versa. A full exposition of this model may be foundin Stone [1]. A cutset (or cut) in this graph is defined to be a set of edges such that when these edges are removed from the graph, nodes P1 and P2 are disconnected from each other. No proper subset of a cutset is also a cutset, i.e., a cutset is a minimal set. The weight of a cutset is the sum of the weights on the edges in the cutset. A cutset partitions the nodes of the assignment graph into two disjoint subsets such that the nodes in one subset are Fig. 1. A static assignment graph. reachable from PI and the nodes in the other are reachable from P2. Each node is reachable from either P1 or P2 - It has been shown by Stone [1] that each cutset in the static assignment graph corresponds in a one-to-one fashion to a module assignment and that the weight of each cutset equals the cost of the corresponding assignment. It follows that an optimal static assignment may be obtained by finding a minimum weight cutset in the graph. This may be done using network flow algorithms. The bold line in Fig. 1 indicates a minimum weight cut. The cost figures used in this model are the net execution and communication costs for the entire lifetime of the program. This model thus yields an optimal static assignment-a module assigned to a particular processor is assumed to stay on that processor for the lifetime of the program. The procedure is valid in a dynamic environment if applied at discrete intervals where the computational characteristics (i.e., the distribution of program activity over the modules) are constant over the interval, and if the cost of reassignment can be ignored. III. MATHEMATICAL MODEL FOR THE DYNAMIC ASSIGNMENT PROBLEM In this section we develop a mathematical model for the dynamic assignment problem. The basis for this model is the concept of the phase of a modular program. This is defined as follows. Definition: The phase of a modular program is a contiguous period of time during which only one module executes. During this period the executing module may communicate with any number of the remaining modules. A module may not be moved from one processor to another during a phase-it may be moved only between phases. With each phase is associated the following information. 1) The executing module during this phase. 2) Run cost of this module on each of the two processors. 3) Costs of residence of the remaining modules on each of the two processors. 4) Intermodule communication costs between the executing module and all other modules ifthey are on different processors. 5) Relocation cost for each module: the cost of reassigning

3 BOKHARI: DUAL PROCESSING SCHEDULING 343 TABLE 1 TABULATION OF COSTS FOR A DYNAMIC ASSIGNMENT PROBLEM INVOLVING THREE MODULES AND FIVE PHASES MODULES PHASE MODULE EXECUTION COST RESIDENCE COST P1 P2 P1 P2 1 A C A B INFINITY C A B C A B C A B C Fig. 2. PHASES An incomplete dynamic assignment graph. PHASE COST OF COMMUNICATION RELOCATION BETWEEN EXECUTING MODULE COSTS OF AND A B C A B C 1-3 [ _ _ each module from one processor to the other at the end of this phase. As an example, consider a hypothetical program made up of three modules (A, B, and C) and five phases. Complete information about this program is given in Table I. Consider Phase 3. The entries corresponding to this phase indicate that during this phase module A executes. The cost of running this module on P1 is 4; on P2 it is 17. The residence costs of the nonexecuting modules (B and C) are also listed. During this phase there is no communication between the executing module and module C. The cost of communication between modules A and B is 7. Costs for relocating modules A, B, and C at the end of this phase are 3, 2, and 3, respectively. Table I does not have any entries for the residence cost of the executing module during any phase. This may be assumed to be lumped into the execution cost. The relocation costs for the last phase are omitted as the program terminates at the end of the last phase. Fig. 2 shows how this information can be represented by a graph. The number of nodes in this graph equals the number of program modules multiplied by the number of phases. The nodes are arranged in a grid with the vertical "columns" of nodes representing the modules and the horizontal "rows" representing the phases. Each individual node represents the residence of a module during a specific phase, as will become clear in what follows. Each node is labeled with an uppercase Fig. 3. An arbitrary cut in the dynamic assignment graph. letter which identifies the module it represents and an integer which identifies the phase. The single module that executes during each phase is marked with an asterisk. The "vertical" edges in this graph connect successive residences of the same module and the weights of these edges represent the costs of relocating the module. The "horizontal" edges connect the executing module with other modules during the same phase and represent intermodule communication costs between the executing module and other modules during this phase. Note: The grid-like arrangement of the nodes of this graph and the use of the terms "column," "row," "horizontal," and "vertical" serve only as an aid in understanding the model. There is no graph-theoretic significance to these terms. In Fig. 3 we have added two nodes, marked P1 and P2, which represent the two processors. Edges representing the run costs have been drawn from P1 and P2 to each of the

4 344 nodes representing executing modules (those marked with an asterisk). As in the static assignment graph, the edge between a node and P1 represents its run cost on P2 and vice versa. In a similar fashion, edges representing residence costs have been drawn from PI and P2 to the remaining nodes. For clarity, some of the edges have been stopped short of P1 and P2. This completed graph is called a dynamic assignment graph in the remainder of this paper. A cutset in this graph gives a dynamic assignment of modules, i.e., it specifies which modules are to reside on which processor during each phase. The weight of a cutset gives the total cost of the corresponding dynamic assignment. These properties of the dynamic assignment graph are dealt with more formally in the following section. The bold line in Fig. 3 indicates an arbitrary cut. Focusing our attention on the first two phases we see that nodes Af, A2, and B1 are assigned to PI and nodes C1, B*, and C2 are assigned to P2. This means that during Phase 1, modules A and B are to reside on P1 and module C on P2. During this phase module A executes on Pi. The cost of this assignment (for Phase 1 alone) is given by the sum of the weights of the edges cut, viz., the edge between A* and Cl with weight 2, representing intermodule communication cost; the edge between Af and P2 with weight 4 representing A's run cost on PI; the edge between B1 and P2 with weight 1, representing B's residence cost on P1, and the edge between C1 and P1, representing C's residence cost on P2. The sum of the execution, residence, and communication costs for Phase 1 alone is 8. During Phase 2 module A resides on P1 and modules B and C on P2. It is clear that, in going from Phase 1 to Phase 2, there has been a relocation of module B from processor P1 to processor P2. The cost of this relocation is included in the total cost of the assignment because the cut includes the edge connecting B1 and B2, which represents relocation cost. IV. SOLUTION OF THE DYNAMIC ASSIGNMENT PROBLEM The solution of the dynamic assignment problem is formally presented in this section. We show that there is a one-to-one correspondence between dynamic assignments and cuts in the dynamic assignment graph. We prove a theorem that states that the weight of the cutset equals the cost of the corresponding assignment. Let us refer to those nodes in the assignment graph that represent residences of modules (i.e., all nodes other than P1 or P2) as interior nodes. Similarly, let us refer to those edges that represent communication costs and relocation costs (i.e., all edges not incident on P1 or P2) as interior edges. We will refer to the remaining edges (those that are incident on P1 or P2) as exterior edges. Every cutset in the dynamic assignment graph associates each interior node with either P1 or P2. As each interior node represents the residence of a module during a specific phase, this means that every cutset specifies the residence of each module during every phase. In other words, every cutset corresponds to a dynamic assignment. The converse is also true. Every dynamic assignment assigns each module to a specific processor during every phase. Since the dynamic assignment graph has exterior edges linking every interior node to P1 and to P2, it is clear that we can make any arbitrary set of interior nodes reachable from P1 (and the remaining reachable from P2) by removing a minimal number of edges. In other words, every dynamic assignment corresponds to a cutset. We can therefore conclude that there is a one-to-one correspondence between dynamic assignments and cutsets in the dynamic assignment graph. The following is an extension of Stone's theorem [1] to the dynamic assignment graph. Theorem: The weight of a cutset of a dynamic assignment graph equals the cost of the corresponding dynamic module assignment. Proof: By definition, a dynamic module assignment incurs four types of costs. These are 1) the cost of executing a module on a specific processor, 2) the cost of module residence (without execution) on a specific processor, 3) the cost of intermodule communications between two modules resident on different processors, and 4) the cost of relocating a module from one processor to the other. There are only four kinds of edges in the cutset and these correspond to the four types of costs mentioned above. Should a module that executes during a particular phase be assigned to a specific processor, say P1, then the edge connecting the node that represents this module and P2 must be cut. The weight of this edge is the run cost of the module on P1 (during the particular phase being considered). Similarly, should a module that does not execute during a particular phase be assigned to a specific processor, its cost of residence will be included in the weight of the cut. Should two modules that communicate with each other during a particular phase be assigned to different processors, the edge connecting the two nodes that represent these modules must be in the cut. The weight,of this edge is the cost of intermodule references during the specific phase being considered. Finally, should a module be assigned to different processors in two successive phases, the edge connecting the nodes representing successive residences of this module must be cut. The weight of this edge represents the relocation cost of the module. It follows that the weight of the cutset includes all costs due to the corresponding assignment. No other costs contribute to the weight. This proves the theorem. We have shown that a cut in the dynamic assignment graph gives us a dynamic assignment of modules. We have also shown that the weight of this cut gives us the cost of the corresponding assignment. It follows that an optimal (i.e., minimum cost) assignment corresponds to a minimum weight cut in the graph. This minimum weight cut may be found by running a network flow algorithm between nodes P1 and P2 in the dynamic assignment graph. Several network flow algorithms are available [3]-[6]. Of these the best is due to Karzanov [6] and has 0(n3) complexity, where n is the number of nodes in the graph. For a dynamic assignment problem with N modules and 0

5 BOKHARI: DUAL PROCESSING SCHEDULING 345 Fig. 4. A mincut in the dynamic assignment graph. Fig. 5. A zero residence cost dynamic assignment graph. phases, the number of nodes in the dynamic assignment graph is No and the problem is thus solvable in O(N3 3) time. A minimum weight cut for our running example is shown in Fig. 4. V. THE ZERO RESIDENCE COST DYNAMIC ASSIGNMENT PROBLEM In this section we consider the special case of the dynamic assignment problem where the costs of residence are negligible. We call this the zero residence cost dynamic assignment problem. This problem can, of course, be solved using the techniques described above by simply labeling the residence cost edges zero. If these zero weight edges are omitted from the graph, the one-to-one correspondence between cuts and assignments no longer holds. We show in this section that a minimum weight cut still corresponds to a minimum cost assignment, despite the fact that some assignments do not have any corresponding cuts. We are interested in working with graphs that have as few edges as possible because the amount of space required to store a graph when running a flow algorithm is proportional to the number of edges in the graph. Graphs that do not contain any residence cost edges can further be reduced as demonstrated in Section VI. A dynamic assignment graph that contains no edges for cost of residence is shown in Fig. 5. We call this the zero residence cost graph. In such a graph there is only one pair of exterior edges in any phase, representing the run costs of the executing module on P1 and P2. Every cutset in this graph associates each interior node with either P1 or P2 and, consequently, corresponds to a dynamic assignment. However, every dynamic assignment of modules to processors does not correspond to a cut in the graph. For example, in Fig. 5, an assignment that puts B4 on P2 and all remaining nodes on P1 does not correspond to any cutset (recall that a cutset is a minimal set of edges that disconnects P1 from P2). However, this presents no difficulties in view of the following theorem. Theorem: For each assignment in the zero residence cost Fig. 6. A condensed graph. problem that does not correspond to a cut in the zero residence cost graph, there exists another assignment with no greater cost that does correspond to a cut. Proof: Let A be an assignment that does not correspond to any cut. Of all assignments that do correspond to cuts, let B be one which most closely matches A, i.e., B corresponds to a cut that assigns the maximum number of modules to the same processor to which they are assigned by A. We will show that assignment B cannot have greater cost than assignment A. Let X(Y) be the subset of modules that are assigned to P1 (P2) by A and to P2 (P1) by B. X and Y thus represent the modules that B fails to assign to the same processor to which they are assigned by A. The zero residence cost graph corresponding to this problem can be condensed into the graph of Fig. 6. The nodes and edges in this graph represent collections of nodes and edges from the original graph. The nodes P' and P' of this graph represent the nodes P1 and P2 of the original graph plus the modules that are assigned to P1 and P2 by both A and B, respectively. X and Y represent the subsets of modules defined above. The edge joining nodes X and Y indicates that one or more nodes in subset X is linked to one or more nodes in subset Y in the original graph. The weights on the edges in this graph are the sums of the weights on the constituent edges in the original graph. Since X is the subset of modules that cut B fails to assign to P1, there can be no edge between X and P. Such an edge would permit one or more modules in X to be moved over to P' (without affecting the rest of cut B) thereby contradicting

6 346 Fig. 7. A zero residence cost graph that admits reduction. Fig. 8. Example of transformation procedure (2). our assumption that B assigns the maximum number of modules to the same processor to which they are assigned by A. Similarly, there can be no edge between Y and P'2. The costs contributing to assignment A are made up of the edges intersected by the dashed line. These do not form a cut because removal of a subset of these edges is sufficient to disconnect P'1 from P. The thick continuous line represents assignment B and does correspond to a cut. It is clear from the figure that, because of the absence of edges from X to P'1 and Y to P2, any edge included in B must also be included in A. Therefore, B, an assignment that does correspond to a cut, has no greater cost than A, an assignment that does not correspond to a cut. This proves the theorem. We have shown that all cuts in zero residence cost graph correspond to assignments. We have also proven that for each assignment that does not correspond to a cut, there exists an assignment with no greater cost that does correspond to a cut. It follows that a minimum weight cut corresponds to a minimum cost assignment regardless of whether there are any assignments that do not correspond to cuts. VI. REDUCTION OF ZERO RESIDENCE COST GRAPHS The zero residence cost graphs of the previous section may be further reduced to eliminate nodes that represent residences of modules during successive phases during which they neither execute nor are accessed by other modules. In this section we describe a procedure for removing such nodes from the graph so that the mincut in the reduced graph has the same weight and corresponds to the same dynamic assignment as the mincut in the original graph. In Fig. 7 we show a zero residence cost dynamic assignment graph in which module D executed in Phase 1, is accessed in Phase 2, and is thereafter not accessed until Phase 6. Module A, on the other hand, does not execute until Phase 5, and is not accessed before or after this phase. A minimum weight cut in this graph is shown by the bold line in Fig. 7. It should be clear that no minimum weight cut includes any of the edges representing the relocation costs of A. Since A executes only during one phase, and is not accessed during any other phase, it should be assigned to one processor for the lifetime of the program. The string of nodes A1 A4 and the leaf node A6 play no part in the assignment problem. Inclusion of any of the relocation cost edges incident on these nodes in the cutset contributes to the cost of the assignment without any reduction in execution or access costs. Examining the string of nodes D2... D6 we see that the mincut passes through the edge connecting D2 and D3, which has weight 5. We observe that the mincut can equally well pass through the edge connecting D3 and D4, which has the same weight, but can not pass through the edge connecting D5, D6 which has greater weight. We can see that if the mincut is to pass between any of the nodes in the stringd2 *D6, it must pass through the edge with the minimum weight. These observations prompt us to propose the following definition. Definition: Two zero residence cost graphs are said to be homeomorphic if one can be transformed into the other by repeated application of the following procedures. 1) If a node has degree one and the edge incident on it represents relocation cost, remove both the node and the edge from the graph. 2) If there exists a chain of nodes representing successive residences of the same module, such that the nodes at the extremities of this chain have degree two or more, and the nodes in between only have the two relocation costs incident on them-replace the entire chain by the pair of nodes in it which have the smallest relocation cost edge between them (Fig. 8). Procedure 1) serves to remove all leaf nodes and strings of those nodes that have only one path to the remainder of the graph. As has been explained above, the edges connecting such nodes can never be included in a mincut. Procedure 2) serves to replace all strings of nodes that rep-

7 BOKHARI: DUAL PROCESSING SCHEDULING 347 Fig. 9. The reduced graph. resent successive residences of a module during which it is neither executed nor accessed, by a pair of nodes representing the two successive phases between which relocation cost is minimum. If a mincut is to pass through such a string, it must pass through the edge that has the minimum weight. In view of the above discussion, it is clear that the weight and composition of the minimum weight cutset in a graph and its homeomorph are identical. The reduced graph in Fig. 9 is homeomorphic to the graph of Fig. 7. The mincuts in the two graphs are identical. The usefulness of the homeomorphism concept lies not so much in reducing an existing graph as in creating a reduced graph from observed statistics at the outset. We expect that in actual practice some modules in a modular program lie "dormant" (are not accessed or executed) for many phases at a time. We also expect relocation cost to remain constant over these dormant periods. (If a module is neither accessed nor executed, there does not seem to be much reason for expecting a change in relocation cost.) The homeomorphism concept permits us to take advantage of this by allowing us to model the situation with graphs that have fewer nodes and edges. VII. BOUNDS ON THE COST OF THE DYNAMIC ASSIGNMENT In this section we show how the dynamic graph may be condensed into a static graph. We also obtain upper and lower bounds on the costs of the dynamic assignment corresponding to the cases where relocation costs are all infinite and all zero, respectively. This comparison of static and dynamic assignments involves only zero residence cost dynamic graphs because there is no equivalent of residence cost in the static problem. The several phases represented by the rows of the dynamic assignment graph can be merged together to obtain the static assignment graph. This is done by simply merging together all pairs of nodes connected by edges representing relocation costs ("vertical" edges). After this, all edges connecting the same pair of nodes are merged into one edge. The weight of Fig. 10. I Condensing a dynamic graph into a static graph. Fig. 11. Relocation costs - 0. these merged edges is the sum of the weights of the component edges. This serves to combine together all the run costs and intermodule communication costs that were spread over several phases. In Fig. 10 we show an example of this condensing process. For clarity we have only labeled a few edges. The following analysis of upper and lower bounds on the cost of the dynamic assignment applies to zero residence cost graphs. In one extreme case the cut does not include any intermodule edges-it is exclusively "horizontal" and cuts only relocation cost edges and run cost edges (Fig. 11). This corresponds to a dynamic assignment where, during any phase, the executing module and all modules it accesses reside on the processor which runs the executing module with least cost. This assignment only incurs run costs and relocation costs.

8 348 Fig. 12. Relocation costs - -. Should the relocation costs be all zero, this assignment will have minimum cost equal to Z min(run cost on PI, run cost on P2). all modules This is a lower bound on the weight of the dynamic cut. An upper bound can be obtained corresponding to the case where the relocation costs approach infinity. The cut is exclusively "vertical" and does not include any relocation cost edge (Fig. 12). In this case the cut includes only run cost edges and communication cost edges. It corresponds to a fixed assignment of modules to processors, i.e., it corresponds to a static cut. A minimum weight dynamic cut will correspond to a minimum weight static cut in this case. We can see that in the worst case a minimum weight dynamic cut is no costlier than any minimum weight static cut. In the best case it eliminates all intermodule communication costs. VIII. CONCLUSIONS AND PROJECTIONS We have presented a solution to the dynamic assignment problem in this paper. Starting with the definition of the phase of a modular program, we have developed a mathematical model based on assignment graphs such that there is a one-to-one correspondence between minimum weight cuts and optimal assignments. We have shown how these assignment graphs may be reduced in size and have computed upper and lower bounds on the cost of the dynamic assignment. Stone's static assignment algorithm has worked successfully on test data derived from the dual processor system at Brown University [71 -[1O]. This system has facilities for gathering information on module execution costs and intermodule communication costs. The dynamic assignment problem presented in this paper requires information on essentially similar costs, but in greater detail-costs must be given for each program phase. Thus, the implementation of the dynamic algorithm, although requiring more information about the program, should not require any radical changes in the mechanism for gathering data. The following are some of the areas for further research. 1) When execution moves rapidly between modules, it is not feasible to relocate a module until a large number of phases have elapsed, and the locality of the program has changed drastically. This situation is depicted in Fig. 13. This points Fig. 13. Illustration of the working partition concept. to a "working partition" or "stable partition" property (vaguely similar to the "working set" property), where the partition change is indicated by the cutting of a relocation edge in the graph. Does this property simplify the assignment problem? Does it provide a technique for dynamic assignment without prior knowledge of program behavior? 2) Given that relocation costs are fixed or nearly constant for the lifetime of the program, is there a way to simplify the problem of gathering statistics or to simplify the solution of the problem itself? 3) How may we take into account a dynamically varying load on one or both of the two processors? This will cause the execution times to vary. Can the solution be extended to

9 include dynamic rescheduling based on the load on the computers? 4) How can the model be extended to include concurrency of execution? ACKNOWLEDGMENT The author is indebted to Prof. H. Stone for his valuable criticism and his encouragement of this research. Comrments of the referees have contributed significantly to this paper. 349 [81 A. van Dam, "Computer graphics and its applications," Brown Univ., Providence, RI, Final Rep., NSF Grant GJ-28410X, May [9] G. M. Stabler, "A system for interconnected processing," Ph.D. dissertation, Brown Univ., Providence, RI, Oct [10] J. Michel and A. van Dam, "Experience with distributed processing on a host/satellite graphics system," in Proc. SIGGRAPH '76, available as Computer Graphics (SIGGRAPH newslett.), vol. 10, no. 2, REFERENCES [1] H. S. Stone, "Multiprocessor scheduling with the aid of network flow algorithms," IEEE Trans. Software Eng., vol. SE-3, pp , Jan [2] -, "Critical load factors in distributed computer systems," IEEE Trans. Software Eng., vol. SE-4, pp , May [3] E. A. Dinic, "Algorithm for solution of a problem of maximum flow in a network with power estimation," Sov. Math. Dokl., vol. 11, no. 5, pp , [4] J. Edmonds and R. M. Karp, "Theoretical improvements in algorithmic efficiency for network flow problems," J. Ass. Comput. Mach., vol. 19, pp , Apr [5] L. R. Ford, Jr. and D. R. Fulkerson, F?ows in Networks. Princeton, NJ: Princeton Univ. Press, [6] A. V. Karzanov, "Determining the maximal flow in a network by the method of preflows," Sov. Math. Dokl., vol. 15, no. 2, pp , [71 A. van Dam, G. Stabler, and R. Harrington, "Intelligent satellites for interactive graphics," Proc. IEEE, vol. 62, pp , Apr Shahid H. Bokhari (S'75-M'78) was born in Lahore, Pakistan, in He received the B.Sc. degree in electrical engineering from the Pakistan University of Engineering and Technology, Lahore, in 1974, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Massachusetts, Amherst, in 1976 and 1978, respectively. In 1974 he worked as a Research Associate in the Department of Electrical Engineering at the Pakistan University of Engineering and Technology. From 1975 to 1978 he was a Research Assistant at the Department of Electrical and Computer Engineering, University of Massachusetts. Since July 1978 he has been a Research Scientist at the Institute for Computer Applications in Science and Engineering (ICASE) at NASA Langley Research Center, Hampton, VA. His current research interests include computer architecture, distributed processing, and performance evaluation. Dr. Bokhari is a member of the Association for Computing Machinery. Database Access Control in the Presence of Context Dependent Protection Requirements DAVID K. HSIAO, DOUGLAS S. KERR, MEMBER, IEEE, AND CHEN-JEN NEE, MEMBER, IEEE SENIOR MEMBER, IEEE, Abstract-Data items in a database are semantically related. Thus, the access control mechanism of a database system must be concerned with the possibility that access to one item may violate a denied access to another item. This study concentrates on two basic semantic relations for protection requirements. By utilizing a graph-theoretic approach, some of the fundamental properties of the protection relations can be readily identified. These properties can then be used as a basis for understanding more general context dependent protection requirements. Two fundamental properties of the two protection relations are Manuscript received October 3, 1977; revised November 27, This work was supported by the Office of Naval Research under Contract N C An earlier version of this work was presented by D. K. Hsiao at the International Federation of Information Processing WG. 2.2 Meeting, Pont a Mousson, France, The authors are with the Department of Computer and Information Science, The Ohio State University, Columbus, OH found. The first property addresses the question: given a database with a set of protection relations, is it possible to find a maximal subset of the database such that access to one item of the subset will not lead to any violation of a denied access to another item? The second property addresses the question: given a database with a set of protection relations, is it possible to find a sequence of accesses such that the protection requirement is enforced with no violation? Index Tenns-Access control, context protection, security, graphtheoretic approach. I. INTRODUCTION C ONVENTIONAL study of access control begins with an matrix A, with users identified with rows and entities of shareable resources as columns. The matrix entry Caccess /79/ $00.75 i 1979 IEEE