II. Notes on Data Structuring * - Cornell University

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154 c.A.R. HOARE to ensure that the sizes and location of the sequences and sections be chosen to correspond closely with the access characteristics of the storage medium. 10.1.4. Locally Dense Representation A special case of a sparse array encountered in numerical computer appli- cations is the sparse matrix. Quite frequently a sparse matrix can be split into submatrices, only a few of which contain significant non-zero entries. In this case, the matrix may be said to be locally dense, and should be represented and processed in a manner which takes advantage of this fact. One method of achieving this is to store with each significant submatrix its position and size, and to represent the whole matrix as a table or sequence off such submatrices, where each submatrix is stored contiguously in the usual way, using multiplicative address calculation. However, the submatrices will in general be of different sizes, and if the size varies during the processing of the matrix, the problems will be quite severe. A possible way of dealing with sparse matrices is to split them into submatrices of standard size, say sixteen by sixteen, and set up a table of pointers to each of these submatrices. A submatrix that is wholly zero is represented by a null pointer and occupies no additional storage; otherwise, the submatrix is stored in the usual way, using the following method of address calculation. Each access to the array involves first "interleaving" the bit values of the two subscripts, so that the least significant part of the result contains the least significant part of both subscripts. The more significant part of the result is then used to consult the table of addresses, to locate the desired submatrix, and the less significant part to find the position off the required element within the submatrix. This technique of interleaving subscripts may on some machines be more efficient than general multiplication. If some of the submatrices have to be held on backing store, this method of address calculation is particularly recommended, since it is equally efficient at processing the matrix by rows as by columns; and the method can then be recommended for all large arrays, whether sparse or not, particularly on a paged computer. The inventor of this method is Professor E. W. Dijkstra. 10.1.5. Grid Representation The phenomenon of cross-classification of files causes as many problems in a computer as it does in real life. It is usually solved by standardising on one of the classifications which is most convenient, and accepting the extra cost of processing in accordance with the other classification, even if this involves resorting the file. Thus the sparse mapping sparse array (i:D1 ; j" DE) of R is represented as: sparse array D1 of (sparse array D2 of R)
NOTES ON DATA STRUCTURING 155 However, it is also possible to deal with the two dimensions in a more symmetric fashion, using a method based on the chained representation of sequences. In this representation, each actually used value of D 1 is placed in one chain, and each actually used value of D2 is placed in another. These are called border chains. Each element of either border chain contains a base location pointing to a chained sequence of all elements with key values which fall into the class. Now each actual entry of the array has two addresses attached; one points to the next item of the sequence which has the same classification according to D1, and the other to the next item which has the same classification according to D2. Thus each item may be pictured as residing on an intersection of the lines of a two-dimensional grid, with pointers leading across and downwards to the next item on the same r6w or the same column. " j ], i • , r, i I d, ,I f ] I [d~,d2] i 1 [< ,dz]] i ........ I ! D 2 border chain Rest of this column Rest of :this column D2 border chain I Novolue of .......... Re~tofthis~o~ A[d~ ,d~] I I~1 ' . t .......... Rest of this row FIo. 9. Grid Representation of A : sparse array (dl ;D1 ;d2 :D2) of T This grid representation is unfortunately suitable only when the entire structure will fit into main store. If the main part of the sequences have to be held on backing store, some sort of blocking of adjacent elements would be desirable in the interests of efficiency. 11. EXAMPLE" EXAMINATION TIMETABLES In an educational establishment which offers students a wide choice of course combinations, there arises the problem of designing an examination time- table in which each examination is conducted in a single session, and yet
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II. Notes on Data Structuring * - Cornell University

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