By Robert Sedgewick
This ebook is meant to survey an important algorithms in use on desktops at the present time and educate the elemental innovations to the growing to be inhabitants drawn to changing into severe desktop clients.
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Extra info for Algorithms (Addison-Wesley series in computer science)
Sample text
Indeed, the particular recurrences that we’ve just solved are sufficient to describe the performance of most of the algorithms that we’ll be studying, and we’ll simply be referring back to them. Matrix Multiplication The most famous application of the divide-and-conquer technique to an arithmetic problem is Strassen’s method for matrix multiplication. We won’t go into the details here, but we can sketch the method, since it is very similar to the polynomial multiplication method that we have just studied.
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. For p(z) = po + pla: + . ‘, define f%(x) =p,, +pla:+“~+pN,2-~xN’2-1 Ph(x) = pN/2 + pN/2+15 -I- ’ ’ * + pN-1xN’2-? Then, splitting q(x) in the same way, we have: P(x) = Pi(x) + zN’2ph(x), q(x) = 41(x) + “N’2qh(x).
The evaluation problem is to find the values, given the polynomial and the points. ) The classic solution to the interpolation problem is given by Lagrange’s interpolation formula, which is often used as a proof that a polynomial of degree N - 1 is completely determined by N points: This formula seems formidable at first but is actually quite simple. s+13s5=j which simplifies to x2 +a:+ 1. For x from xl, x2, . . , XN, the formula is constructed so that p(xk) = yk for 1 5 k 5 N, since the product evaluates to 0 unless j = k, when it evaluates 48 CHAPTER 4 to 1.