Algorithms by R. Sedgewick

By R. Sedgewick

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The running time of such an algorithm might be described by the recurrence M(N) = M(N/2) + 1. This is easier to solve than the one in the previous paragraph. We immediately have I14(2~) = n and, again, it turns out that M(N) z 1gN. Of course, it’s not always possible to get by with such trivial manipulations. For a slightly more difficult example, consider an algorithm of the type described in the previous paragraph which must somehow examine each element before or after the recursive step. The running time of such an algorithm is described by the recurrence M(N) = M(N/2) + N.

The “best” algorithm for matrix multiplication has still not been found, and this is one of the most famous outstanding problems of computer science. It is important to note that we have been counting multiplications only. Before choosing an algorithm for a practical application, the costs of the extra additions and subtractions for combining terms and the costs of the CHAPTER 4 recursive calls must be considered. These costs may depend heavily on the particular implementation or computer used.

Many of our best algorithms are designed according to this principle. In this section we’ll see how divide-and-conquer applies in particular to the polynomial multiplication problem. In the following section we’ll look at some analysis which gives a good estimate of how much is saved. One way to split a polynomial in two is to divide the coefficients in half: given a polynomial of degree N-l (with N coefficients) we can split it into two polynomials with N/2 coefficients (assume that N is even): by using the N/2 low-order coefficients for one polynomial and the N/2 high-order coefficients PoLMvoMIALs 49 for the other.

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