Algorithm Design

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238 Chapter 5 Divide and Conquer Could one design an algorithm that bypasses the quadratic-size definition of convolution and computes it in some smarter way? In fact, quite surprisingly, this is possible. We now describe a method that computes the convolution of two vectors using only O(n log n) arithmetic operations. The crux of this method is a powerful technique known as the Fast Fourier Transform (FFT). The FFT has a wide range of further applications in analyzing sequences of numerical values; computing convolutions quickly, which we focus on here, is iust one of these applications. ~ <strong>Design</strong>ing and Analyzing the <strong>Algorithm</strong> To break through the quadratic time barrier for convolutions, we are going to exploit the connection between the convolution and the multiplication of two polynomials, as illustrated in the first example discussed previously. But rather than use convolution as a primitive in polynomial multiplication, we are going to exploit this connection in the opposite direction. Suppose we are given the vectors a = (ao, al ..... an_~) and b= (bo,, bn_l). We will view them as the polynomials A(x) = a o + alx + a2x 2 + ¯ . ¯ an_~x n-~ and B(x) = bo + b~x + b2 x2 ÷" "" bn-1 xn-1, and we’ll seek to compute their product C(x) = A(x)B(x) in O(n log rt) time. If c = (Co, c~ .... , c2n_2) is the vector of coefficients of C, then we recall from our earlier discussion that c is exactly the convolution a ¯ b, and so we can then read off the desired answer directly from the coefficients of C(x). Now, rather than multiplying A and B symbolically, we can treat them as functions of the variable x and multiply them as follows. (i) (ii) (iii) First we choose 2n values xl, x 2 ..... x2n and evaluate A(xj) and B(xj) for each of j = !, 2 ..... 2n. We can now compute C(xi) for each ] very easily: C(xj) is simply the product of the two numbers A(xj) and B(xj). Finally, we have to recover C from its values on x~, x2 ..... x2n. Here we take advantage of a fundamental fact about polynomials: any polynomial of degree d can be reconstructed from its values on any set of d + 1 or more points. This is known as polynomial interpolation, and we’ll discuss the mechanics of performing interpolation in more detail later. For the moment, we simply observe that since A and B each have degree at most n - !, their product C has degree at most 2n - 2, and so it can be C(x2n) that we computed reconstructed from the values C(xl), C(x2) ..... in step (ii). This approach to multiplying polynomials has some promising aspects and some problematic ones. First, the good news: step (ii) requires only : 5.6 Convolutions and the Fast Fourier Transform 239 O(n) arithmetic operations, since it simply involves the multiplication of O(n) numbers. But the situation doesn’t look as hopeful with steps (i) and (fii). In particular, evaluating the polynomials A and B on a single value takes S2 (n) operations, and our plan calls for performing 2n such evaluations. This seems to bring us back to quadratic time right away. The key idea that will make this all work is to find a set of 2n values x~, x2 ..... x2n that are intimately related in some way, such that the work in evaluating A and B on all of them can be shared across different evaluations. A set for which this wil! turn out to work very well is the complex roots o[ unity. The Complex Roots of Unity At this point, we’re going to need-to recal! a few facts about complex numbers and their role as solutions to polynomial equations. Recall that complex numbers can be viewed as lying in the "complex plane," with axes representing their real and imaginary parts. We can write a complex number using polar coordinates with respect to this plane as re °i, where e’r~= -1 (and e2’-n = 1). Now, for a positive integer k, the polynomial equation x k = 1 has k distinct complex roots, and it is easy to identify them. Each of the complex numbers wj,k = e 2’-qi/k (for] = 0, 1, 2 ..... k - 1) satisfies the equation, since and each of these numbers is distinct, so these are all the roots. We refer to these numbers as the k th roots of unity. We can picture these roots as a set of k equally spaced points lying on the unit circle in the complex plane, as shown in Figure 5.9 for the case k = 8. For our numbers x~ ..... x2n on which to evaluate A and B, we will choose the (2n) th roots of unity. It’s worth mentioning (although it’s not necessary for understanding the algorithm) that the use of the complex roots of unity is the basis for the name Fast Fourier Transform: the representation of a degree-d Figure 5.9 The 8 th roots of unity in the complex plane.
240 Chapter 5 Divide and Conquer polynomial P by its values on the (d ÷ 1) st roots of unity is sometimes referred to as the discrete Fourier transform of P; and the heart of our procedure is a method for making this computation fast. A Recursive Procedure for Polynomial Eualuatiou We want to design an algorithm for evaluating A on each of the (2n) th roots of unity recursively, so as to take advantage of the familiar recurrence from (5.1)--namely, T(n)
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Cornell University Boston San Franc
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Contents About the Authors Preface
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X Contents 9 10 Extending the Limit
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xiv Preface problems out in the wor
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Preface Chapters 2 and 3 also prese
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xxii Preface Preface xxiii Sue Whit
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2 Chapter 1 Introduction: Some Repr
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10 Chapter 1 Introduction: Some Rep
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14 Chapter ! Introduction: Some Rep
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30 Chapter 2 Basics of Algorithm An
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34 n= I0 n=30 n=50 = 100 = 1,000 10
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74 Chapter 3 Graphs 3.1 Basic Defin
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78 Chapter 3 Graphs Rooted trees ar
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82 Chapter 3 Graphs Current compone
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86 Chapter 3 Graphs Proof. Suppose
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90 Chapter 3 Graphs Two of the simp
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94 Chapter 3 Graphs most ~u nv = 2m
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98 Chapter 3 Graphs It is important
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102 Chapter 3 Graphs ordering, that
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106 Chapter 3 Graphs We can already
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110 Chapter 3 Graphs Exercises 111
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Greedy A~gorithms In Wall Street, t
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118 Chapter 4 Greedy Mgofithms o In
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122 Chapter 4 Greedy Algorithms Our
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126 Chapter 4 Greedy Algorithms Len
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134 Chapter 4 Greedy Algorithms wit
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Page 117 and 118: Chapter Divide artd Cortquer Divide
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Page 131: we’ll just give a simple example
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Page 139 and 140: 252 Chapter 6 Dynamic Programming b
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Page 143 and 144: 260 Index 1 2 3 4 5 6 L~ I = 2 Chap
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54O Chapter 7 Network Flow define f
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344 Chapter 7 Network Flow This aug
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348 Chapter 7 Network Flow Proof. ~
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352 Chapter 7 Network Flow In the n
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356 Chapter 7 Network Flow (7.19) T
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360 Chapter 7 Network Flow preflow
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364 Chapter 7 Network Flow Heights
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368 Chapter 7 Network Flow ~ The Pr
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372 Chapter 7 Network Flow that are
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376 Chapter 7 Network Flow I~low ar
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380 Chapter 7 Network Flow Figure 7
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384 Chapter 7 Network Flow Lower bo
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388 BOS 6 DCA 7 DCA 8 Chapter 7 Net
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392 Chapter 7 Network Flow natural
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396 Chapter 7 Network Flow 7.11 Pro
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400 Chapter 7 Network Flow 7.12 Bas
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404 Chapter 7 Network Flow to cross
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412 Chapter 7 Network Flow ProoL Th
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416 Chapter 7 Network Flow Figure 7
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420 Chapter 7 Network Flow 11. Now
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424 Figure 7.29 An instance of Cove
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428 Chapter 7 Network Flow 22. This
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432 Chapter 7 Network Flow generall
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436 Chapter 7 Network Flow (a) (b)
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440 Chapter 7 Network Flow going to
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448 Chapter 7 Network Flow 51. The
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452 Chapter 8 NP and Computational
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472 Chapter 8 NP and Computation!!
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5OO Chapter 8 NP and Computational
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532 Chapter 9 PSPACE: A Class of Pr
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556 Chapter 10 Extending the Limits
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58O Chapter 10 Extending the Limits
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596 Figure 10.12 A triangulated cyc
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600 Chapter 11 Approximation Algori
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628 Chapter 11 Approximation Mgorit
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636 Chapter !! Approximation Algori
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640 Chapter 11 Approximation Mgofit
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656 Chapter 11 Approximation Mgorit
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664 Chapter 12 Local Search basic p
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668 Chapter 12 Local Search moves b
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672 Chapter 12 Local Search of all
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676 Chapter 12 Local Search 12.4 Ma
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708 Chapter 13 Randomized Algorithm
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724 Chapter 13 Randomized Mgorithms
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736 Chapter 13 Randomized Mgofithms
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792 Chapter !3 Randomized Algorithm
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796 Epilogue: Algorithms That Run F
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800 Epilogue: Algorithms That Run F
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8O4 Epilogue: Algorithms That Run F
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808 References L. R. Ford. Network
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812 References M. Mitzenmacher and
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816 Index Approximation algorithms
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820 Cushions in packet switching, 8
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824 Index Graphs (cont.) queues and
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828 Index Mapping genomes, 279, 521
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832 Index Index 833 Preflow-Push Ma
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836 Index Stale items in randomized
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Algorithm Design

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