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By Mary E. Mace

This venture had its beginnings within the Fall of 1980. at the moment Robert Wagner steered that I examine compiler optimi­ zation of information association, appropriate to be used in a parallel or vector desktop surroundings. We constructed a scheme within which the compiler, having wisdom of the machine's entry styles, does an international research of a program's operations, and immediately determines optimal association for the information. for instance, for sure architectures and sure operations, huge advancements in functionality may be attained through storing a matrix in row significant order. in spite of the fact that a next operation could require the matrix in column significant order. A decision needs to be made even if it's the most sensible answer globally to shop the matrix in row order, column order, or perhaps have copies of it, each one prepared otherwise. we now have built algorithms for making this decision. The method indicates promise in a vector laptop environ­ ment, really if reminiscence interleaving is used. Supercomputers comparable to the Cray, the CDC Cyber 205, the IBM 3090, in addition to superminis corresponding to the Convex are attainable environments for implementation.

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An extra cost of 10 units is charged for transposing an operand into another shape. Figure 4-3 illustrates a conflict free access of the M vector {a 11' a 12, a 13, a 14, a 15}' Accessing this vector has a cost of 10 units from rule 1 above. Figure 4-4 illustrates an M vector stored in such a way that accessing it causes four memory conflicts. Accessing this vector has a cost of 40 units from rule 2 above. <: o 1 2 3 4 all a l2 a22 a32 a42 aS2 a 13 a l4 ale; a25 a3S a45 aS5 a21 a31 a41 aSI U 23 a24 a43 aS3 a34 a44 aS4 a33 Figure 4-3 Conflict free access of M vector :::> 41 Collapsible Graph Algorithm o 1 2 3 a 13 a14 a23 a33 a43 a53 4 a24 a34 a44 a54 Figure 4-4 Accessing M vector with memory conflicts Cost functions for this program segment Figures 4-5 and 4-6 show the cost functions for accessing operands A and B just as input operations.

Ei corresponds to a set Aj in the set cover instance such that set Aj contains element ei. The cost for node ei is n otherwise. Call the cost table for element ei table Tei . ei. The shapes which are assigned to the shared node arcs are the variables of the problem. ei. e) = 0 if element ei E set Aj =n otherwise The cost of the shared node is the number of distinct shapes assigned to shared node arcs - 1.

Cq-l] + Tj [b , s/d), (Ti [a, S/1 ' C 1 ... 1 ])} The following chapter will illustrate a sequence of transformations A and B being applied to a graph with shapes and cost functions, resulting in an assignment of shapes to the program data. Chapter 4 ILLUSTRATION OF COLLAPSIBLE GRAPH ALGORITHM In this section we will present an example of how the collapsible graph algorithm works. Our illustration uses a short, simple code segment involving matrix operands, however the program graph of this code segment contains a shared node.

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