Data Structures and Algorithm Analysis - Computer Science at ...

412 Chap. 12 Lists and Arrays Revisitedare non-zero. Different values for the relative sizes of data values, pointers, ormatrix indices can lead to a different break-even point for the two implementations.The time required to process a sparse matrix should ideally depend on NNZ.When searching for an element, the cost is the number of elements preceding thedesired element on its row or column list. The cost for operations such as addingtwo matrices should be Θ(n + m) in the worst case when the one matrix stores nnon-zero elements and the other stores m non-zero elements.Another representation for sparse matrices is sometimes called the Yale representation.Matlab uses a similar representation, with a primary difference beingthat the Matlab representation uses column-major order. 1 The Matlab representationstores the sparse matrix using three lists. The first is simply all of the non-zeroelement values, in column-major order. The second list stores the start positionwithin the first list for each column. The third list stores the row positions for eachof the corresponding non-zero values. In the Yale representation, the matrix ofFigure 12.7 would appear as:Values: 10 45 40 23 5 32 93 12 19 7Column starts: 0 3 5 5 7 7 7 7Row positions: 0 1 4 0 1 7 1 7 0 7If the matrix has c columns, then the total space required will be proportional toc + 2NNZ. This is good in terms of space. It allows fairly quick access to anycolumn, and allows for easy processing of the non-zero values along a column.However, it does not do a good job of providing access to the values along a row,and is terrible when values need to be added or removed from the representation.Fortunately, when doing computations such as adding or multiplying two sparsematrices, the processing of the input matrices and construction of the output matrixcan be done reasonably efficiently.12.3 Memory ManagementMost data structures are designed to store and access objects of uniform size. Atypical example would be an integer stored in a list or a queue. Some applicationsrequire the ability to store variable-length records, such as a string of arbitrarylength. One solution is to store in the list or queue fixed-length pointers to thevariable-length strings. This is fine for data structures stored in main memory.But if the collection of strings is meant to be stored on disk, then we might needto worry about where exactly these strings are stored. And even when stored inmain memory, something has to figure out where there are available bytes to holdthe string. We could easily store variable-size records in a queue or stack, where1 Scientific packages tend to prefer column-oriented representations for matrices since this thedominant access need for the operations to be performed.