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

Sec. 12.2 Matrix Representations 409a 00 0 0 0a 10 a 11 0 0a 20 a 21 a 22 0a 30 a 31 a 32 a 33(a)a 00 a 01 a 02 a 030 a 11 a 12 a 130 0 a 22 a 230 0 0 a 33Figure 12.6 Triangular matrices. (a) A lower triangular matrix. (b) An uppertriangular matrix.the lower-indexed vertex. In this case, only the lower triangle of the matrix canhave non-zero values.We can take advantage of this fact to save space. Instead of storing n(n + 1)/2pieces of information in an n × n array, it would save space to use a list of lengthn(n + 1)/2. This is only practical if some means can be found to locate within thelist the element that would correspond to position [r, c] in the original matrix.We will derive an equation to convert position [r, c] to a position in a onedimensionallist to store the lower triangular matrix. Note that row 0 of the matrixhas one non-zero value, row 1 has two non-zero values, and so on. Thus, row ris preceded by r rows with a total of ∑ rk=1 k = (r2 + r)/2 non-zero elements.Adding c to reach the cth position in the rth row yields the following equation toconvert position [r, c] in the original matrix to the correct position in the list.matrix[r, c] = list[(r 2 + r)/2 + c].A similar equation can be used to convert coordinates in an upper triangular matrix,that is, a matrix with zero values at positions [r, c] such that r > c, as shown inFigure 12.6(b). For an n × n upper triangular matrix, the equation to convert frommatrix coordinates to list positions would be(b)matrix[r, c] = list[rn − (r 2 + r)/2 + c].A more difficult situation arises when the vast majority of values stored in ann × m matrix are zero, but there is no restriction on which positions are zero andwhich are non-zero. This is known as a sparse matrix.One approach to representing a sparse matrix is to concatenate (or otherwisecombine) the row and column coordinates into a single value and use this as a keyin a hash table. Thus, if we want to know the value of a particular position in thematrix, we search the hash table for the appropriate key. If a value for this positionis not found, it is assumed to be zero. This is an ideal approach when all queries tothe matrix are in terms of access by specified position. However, if we wish to findthe first non-zero element in a given row, or the next non-zero element below thecurrent one in a given column, then the hash table requires us to check sequentiallythrough all possible positions in some row or column.