Algorithm Design

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234 Chapter 5 Divide and Conquer a three-way branching one, with a running time that satisfies T(n)
we’ll just give a simple example here to suggest one way in which the convolution arises. am-l) which represents a sequence of measurements, such as a temperature or a stock price, sampled at m consecutive points in time. Sequences like this are often very noisy due to measurement error or random fluctuations, and so a common operation is to "smooth" the measurements by averaging each value ai with a weighted sum of its neighbors within k steps to the left and right in the sequence, the weights decaying quickly as one moves away from ai. For example, in Gaussian smoothing, one replaces ai with for some "width" parameter k, and with Z chosen simply to normalize the weights in the average to add up to 1. (There are some issues With boundary conditions--what do we do when i - k < 0 or i + k > m?--but we could dea! with these, for example, by discarding the first and last k entries from the smoothed signal, or by scaling them differently to make up for the missing terms.) To see the connection with the convolution operation, we picture this smoothing operation as follows. We first define a "mask" Wk_D consisting of the weights we want to use for averaging each point with e -1, 1, e -1 ..... e-(k-l)2, e-k2) in the Gaussian case above.) We then iteratively position this mask so it is centered at each possible point in the sequence a; and for each positioning, we compute the weighted average. In other words, ~ k we replace ai with a, = ~s=-k Wsai+s" This last expression is essentially a convolution; we just have to warp the notation a bit so that this becomes clear. Let’s define b = b2k ) by setting be = tvk_g. Then it’s not hard to check that with this definition we have the smoothed value a i ~,~):]+~=i+k albg. In other words, the smoothed sequence is just the convolution of the original signal and the reverse of the mask (with some meaningless coordinates at the beginning and end). We mention one final application: the problem of combining histograms. Suppose we’re studying a population of people, and we have the fol!owing two histograms: One shows the annual income of all the men in the population, and one shows the annual income of all the women. We’d now like to produce a new histogram, showing for each k the number of pairs (M, W) for which man M and woman W have a combined income of k. This is precisely a convolution. We can write the first histogram as a ara-1), to indicate that there are a~ men with annual income equal to i. We can similarly write the second histogram as a bn_~). Now, let c~ denote the number of pairs (m, w) with combined income k; this is the number of ways of choosing a man with income ai and a woman with income hi, for any pair (i, j) where i + j = k. In other words, c~= ~ aib~. (i,]):i+j=k so the combined histogram c = (c o ..... cra+n-2) is simply the convolution of a and b. (Using terminology from probability that we will develop in Chapter 13, one can view this example as showing how convolution is the underlying means for computing the distribution of the sum of two independent random variables.) Computing the Convolution Having now motivated the notion of convolution, let’s discuss the problem of computing it efficiently. For simplicity, we will consider the case of equal length vectors (i.e., m = n), although everything we say carries over directly to the case of vectors of unequal lengths. Computing the convolution is a more subtle question than it may first appear. The definition of convolution, after al!, gives us a perfectly valid way to compute it: for each k, we just calculate the sum E aibj (i,j):i+j=k and use this as the value of the k th coordinate. The trouble is that this direct way of computing the convolution involves calculating the product a~bj for every pair (i,j) (in the process of distributing over the sums in the different terms) and this is ®(n 2) arithmetic operations. Spending O(n 2) time on computing the convolution seems natural, as the definition involves O(n 2) multiplications azbj. However, it’s not inherently clear that we have to spend quadratic time to compute a convolution, since the input and output both only have size O(n). 237
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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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6 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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Before swapping: After swapping: d
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Page 117 and 118: Chapter Divide artd Cortquer Divide
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Page 139 and 140: 252 Chapter 6 Dynamic Programming b
Page 141 and 142: 256 Chapter 6 Dynamic Programming 6
Page 143 and 144: 260 Index 1 2 3 4 5 6 L~ I = 2 Chap
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336 Chapter 6 Dynamic Programming h
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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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4O8 Chapter 7 Network Flow Why are
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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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Chapter 7 Network Flow 44. 45. Give
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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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680 Chapter 12 Local Search saw ear
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684 Chapter 12 Local Search utting
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688 Chapter 12 Local Search this di
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692 Chapter 12 Local Search sides a
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696 Chapter 12 Local Search Of cour
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7OO Chapter 12 Loca! Search and for
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7O4 Chapter 12 Local Search A previ
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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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