A Practical Introduction to Data Structures and Algorithm Analysis

548 Chap. 16 Patterns of Algorithms16.3.3 Matrix MultiplicationThe standard algorithm for multiplying two n × n matrices requires Θ(n 3 ) time.It is possible to do better than this by rearranging and grouping the multiplicationsin various ways. One example of this is known as Strassen’s matrix multiplicationalgorithm.For simplicity, we will assume that n is a power of two. In the following, Aand B are n × n arrays, while A ij and B ij refer to arrays of size n/2 × n/2. Usingthis notation, we can think of matrix multiplication using divide and conquer in thefollowing way:[ ][ ] [ ]A11 A 12 B11 B 12 A11 B= 11 + A 12 B 21 A 11 B 12 + A 12 B 22.A 21 A 22 B 21 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22Of course, each of the multiplications and additions on the right side of thisequation are recursive calls on arrays of half size, and additions of arrays of halfsize, respectively. The recurrence relation for this algorithm isT (n) = 8T (n/2) + 4(n/2) 2 = Θ(n 3 ).This closed form solution can easily be obtained by applying the Master Theorem14.1.Strassen’s algorithm carefully rearranges the way that the various terms aremultiplied and added together. It does so in a particular order, as expressed by thefollowing equation:[ ][ ] [ ]A11 A 12 B11 B 12 s1 + s=2 − s 4 + s 6 s 4 + s 5.A 21 A 22 B 21 B 22 s 6 + s 7 s 2 − s 3 + s 5 − s 7In other words, the result of the multiplication for an n × n array is obtained bya different series of matrix multiplications and additions for n/2 × n/2 arrays.Multiplications between subarrays also use Strassen’s algorithm, and the additionof two subarrays requires Θ(n 2 ) time. The subfactors are defined as follows:s 1 = (A 12 − A 22 ) · (B 21 + B 22 )s 2 = (A 11 + A 22 ) · (B 11 + B 22 )s 3 = (A 11 − A 21 ) · (B 11 + B 12 )s 4 = (A 11 + A 12 ) · B 22s 5 = A 11 · (B 12 − B 22 )s 6 = A 22 · (B 21 − B 11 )s 7 = (A 21 + A 22 ) · B 11 .