A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 17.1 Reductions 565SORTING:Integer ArrayTransform 1Integers1 to nIntegerArrayPAIRINGArray of PairsTransform 2SortedInteger ArrayFigure 17.3 The reduction of SORTING to PAIRING shown as a “blackbox”diagram.Our next example of reduction concerns the multiplication of two n × n matrices.For this problem, we will assume that the values stored in the matrices are simpleintegers and that multiplying two simple integers takes constant time (becausemultiplication of two int variables takes a fixed number of machine instructions).The standard algorithm for multiplying two matrices is to multiply each elementof the first matrix’s first row by the corresponding element of the second matrix’sfirst column, then adding the numbers. This takes Θ(n) time. Each of the n 2 elementsof the solution are computed in similar fashion, requiring a total of Θ(n 3 )time. Faster algorithms are known (see the discussion of Strassen’s Algorithm inSection 16.3.3), but none are so fast as to be in O(n 2 ).Now, consider the case of multiplying two symmetric matrices. A symmetricmatrix is one in which entry ij is equal to entry ji; that is, the upper-right triangleof the matrix is a mirror image of the lower-left triangle. Is there something aboutthis restricted case that allows us to multiply two symmetric matrices faster thanin the general case? The answer is no, as can be seen by the following reduction.Assume that we have been given two n × n matrices A and B. We can construct a2n × 2n symmetric matrix from an arbitrary matrix A as follows: