A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 12.2 Matrix Representations 427a 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.12.2 Matrix RepresentationsSome applications must represent a large, two-dimensional matrix where many ofthe elements have a value of zero. One example is the lower triangular matrix thatresults from solving systems of simultaneous equations. A lower triangular matrixstores zero values at positions [r, c] such that r < c, as shown in Figure 12.6(a).Thus, the upper-right triangle of the matrix is always zero. Another example isthe representation of undirected graphs in an adjacency matrix (see Project 11.2).Because all edges between Vertices i and j go in both directions, there is no need tostore both. Instead we can just store one edge going from the higher-indexed vertexto 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.To derive an equation to do this computation, 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 store an upper triangular matrix, that is, a matrixwith zero values at positions [r, c] such that r > c, as shown in Figure 12.6(b). Foran n × n upper triangular matrix, the equation 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 × n matrix are zero, but there is no restriction on which positions are zero andwhich are non-zero. This is known as a sparse matrix.