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The slow spread of a fast algorithm In 1963, during a meeting of President Kennedy’s scientific advisors, John Tukey, a mathematician from Princeton, explained to IBM’s Dick Garwin a fast method for computing Fourier transforms. Garwin listened carefully, because he was at the time working on ways to detect nuclear explosions from seismographic data, and Fourier transforms were the bottleneck of his method. When he went back to IBM, he asked John Cooley to implement Tukey’s algorithm; they decided that a paper should be published so that the idea could not be patented. Tukey was not very keen to write a paper on the subject, so Cooley took the initiative. And this is how one of the most famous and most cited scientific papers was published in 1965, co-authored by Cooley and Tukey. The reason Tukey was reluctant to publish the FFT was not secretiveness or pursuit of profit via patents. He just felt that this was a simple observation that was probably already known. This was typical of the period: back then (and for some time later) algorithms were considered second-class mathematical objects, devoid of depth and elegance, and unworthy of serious attention. But Tukey was right about one thing: it was later discovered that British engineers had used the FFT for hand calculations during the late 1930s. And—to end this chapter with the same great mathematician who started it—a paper by Gauss in the early 1800s on (what else?) interpolation contained essentially the same idea in it! Gauss’s paper had remained a secret for so long because it was protected by an old-fashioned cryptographic technique: like most scientific papers of its era, it was written in Latin. 78
Exercises 2.1. Use the divide-and-conquer integer multiplication algorithm to multiply the two binary integers 10011011 and 10111010. 2.2. Show that for any positive integers n and any base b, there must some power of b lying in the range [n, bn]. 2.3. Section 2.2 describes a method for solving recurrence relations which is based on analyzing the recursion tree and deriving a formula for the work done at each level. Another (closely related) method is to expand out the recurrence a few times, until a pattern emerges. For instance, let’s start with the familiar T (n) = 2T (n/2) + O(n). Think of O(n) as being ≤ cn for some constant c, so: T (n) ≤ 2T (n/2) + cn. By repeatedly applying this rule, we can bound T (n) in terms of T (n/2), then T (n/4), then T (n/8), and so on, at each step getting closer to the value of T (·) we do know, namely T (1) = O(1). T (n) ≤ 2T (n/2) + cn A pattern is emerging... the general term is ≤ 2[2T (n/4) + cn/2] + cn = 4T (n/4) + 2cn ≤ 4[2T (n/8) + cn/4] + 2cn = 8T (n/8) + 3cn ≤ 8[2T (n/16) + cn/8] + 3cn = 16T (n/16) + 4cn . T (n) ≤ 2 k T (n/2 k ) + kcn. Plugging in k = log 2 n, we get T (n) ≤ nT (1) + cn log 2 n = O(n log n). (a) Do the same thing for the recurrence T (n) = 3T (n/2) + O(n). What is the general kth term in this case? And what value of k should be plugged in to get the answer? (b) Now try the recurrence T (n) = T (n − 1) + O(1), a case which is not covered by the master theorem. Can you solve this too? 2.4. Suppose you are choosing between the following three algorithms: • Algorithm A solves problems by dividing them into five subproblems of half the size, recursively solving each subproblem, and then combining the solutions in linear time. • Algorithm B solves problems of size n by recursively solving two subproblems of size n − 1 and then combining the solutions in constant time. • Algorithm C solves problems of size n by dividing them into nine subproblems of size n/3, recursively solving each subproblem, and then combining the solutions in O(n 2 ) time. What are the running times of each of these algorithms (in big-O notation), and which would you choose? 2.5. Solve the following recurrence relations and give a Θ bound for each of them. (a) T (n) = 2T (n/3) + 1 (b) T (n) = 5T (n/4) + n (c) T (n) = 7T (n/7) + n (d) T (n) = 9T (n/3) + n 2 79
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Algorithms Copyright c○2006 S. Da
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4 Paths in graphs 109 4.1 Distances
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List of boxes Bases and logs . . .
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Preface This book evolved over the
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Chapter 0 Prologue Look around you.
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The first question is moot here, as
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for addition, which works on one di
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4. Likewise, any polynomial dominat
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and in general ( Fn ) = F n+1 ( ) n
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Bases and logs Naturally, there is
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Figure 1.1 Multiplication à la Fra
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Figure 1.3 Addition modulo 8. 6 0 0
Page 28 and 29: Figure 1.4 Modular exponentiation.
Page 30 and 31: Figure 1.6 A simple extension of Eu
Page 32 and 33: We say x is the multiplicative inve
Page 34 and 35: Figure 1.7 An algorithm for testing
Page 36 and 37: Figure 1.8 An algorithm for testing
Page 38 and 39: Randomized algorithms: a virtual ch
Page 40 and 41: Alice’s encryption function is th
Page 42 and 43: Figure 1.9 RSA. Bob chooses his pub
Page 44 and 45: There is nothing inherently wrong w
Page 46 and 47: Exercises 1.1. Show that in any bas
Page 48 and 49: the depth of the circuit or the lon
Page 50 and 51: (c) Give an example of positive int
Page 52 and 53: x = x L x R = 2 n/2 x L + x R y = y
Page 54 and 55: Figure 2.2 Divide-and-conquer integ
Page 56 and 57: Binary search The ultimate divide-a
Page 59 and 60: An n log n lower bound for sorting
Page 61 and 62: The three sublists S L , S v , and
Page 63 and 64: to be the best running time possibl
Page 66 and 67: qp t AB CD EF GH IJ KL MN OP QR Why
Page 68 and 69: The equivalence of the two polynomi
Page 70 and 71: Figure 2.6 The complex roots of uni
Page 72 and 73: A matrix reformulation To get a cle
Page 74 and 75: The orthogonality property can be s
Page 76 and 77: FFT n (input: a 0 , . . . , a n−1
Page 80 and 81: (e) T (n) = 8T (n/2) + n 3 (f) T (n
Page 82 and 83: 2.21. Mean and median. One of the m
Page 84 and 85: (c) In fact, squaring matrices is n
Page 87 and 88: Chapter 3 Decompositions of graphs
Page 89 and 90: Therefore, each edge appears in exa
Page 91 and 92: Figure 3.3. 1 The previsit and post
Page 93 and 94: Figure 3.6 (a) A 12-node graph. (b)
Page 95 and 96: Tree edges are actually part of the
Page 97 and 98: the same thing. Since a dag is line
Page 99 and 100: Figure 3.10 The reverse of the grap
Page 101 and 102: Exercises 3.1. Perform a depth-firs
Page 103 and 104: (a) Model this as a graph problem:
Page 105 and 106: For instance, in the graph below (w
Page 107 and 108: 3.30. On page 97, we defined the bi
Page 109 and 110: Chapter 4 Paths in graphs 4.1 Dista
Page 111 and 112: Figure 4.4 The result of breadth-fi
Page 113 and 114: Figure 4.6 Breaking edges into unit
Page 115 and 116: into account. This viewpoint gives
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Page 119 and 120: Which heap is best? The running tim
Page 121 and 122: Figure 4.11 (a) A binary heap with
Page 123 and 124: Figure 4.13 The Bellman-Ford algori
Page 125 and 126: Figure 4.15 A single-source shortes
Page 127 and 128: Input: Undirected graph G = (V, E)
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4.16. Section 4.5.2 describes a way
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Let r ∗ be the maximum ratio achi
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Chapter 5 Greedy algorithms A game
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Trees A tree is an undirected graph
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Figure 5.3 The cut property at work
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eturn x As can be expected, makeset
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Figure 5.7 The effect of path compr
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A randomized algorithm for minimum
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Figure 5.9 Top: Prim’s minimum sp
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Figure 5.10 A prefix-free encoding.
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149
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5.3 Horn formulas In order to displ
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Figure 5.11 (a) Eleven towns. (b) T
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Exercises 5.1. Consider the followi
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(a) Code: {0, 10, 11} (b) Code: {0,
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Input: Undirected graph G = (V, E);
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Chapter 6 Dynamic programming In th
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Figure 6.2 The dag of increasing su
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Programming? The origin of the term
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Figure 6.4 (a) The table of subprob
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Common subproblems Finding the righ
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6.4 Knapsack During a robbery, a bu
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Memoization In dynamic programming,
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Figure 6.7 (a) ((A × B) × C) × D
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ecause it leads right back to the O
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d ij . We must pick the best such i
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Figure 6.11 I(u) is the size of the
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6.8. Given two strings x = x 1 x 2
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Figure 6.12 Two binary search trees
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6.26. Sequence alignment. When a ne
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Chapter 7 Linear programming and re
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It is a general rule of linear prog
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the feasible region. The point of f
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Figure 7.3 A communications network
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Reductions Sometimes a computationa
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Matrix-vector notation A linear fun
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Figure 7.5 An illustration of the m
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L e R s t e ′ We claim that size(
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Figure 7.6 Continued Current Flow R
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from the algorithm, which in such c
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Figure 7.10 A generic primal LP in
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Now suppose the two of them play th
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In LP form, this is min w −3y 1 +
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7.6.2 The algorithm On each iterati
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Figure 7.13 Simplex in action. Init
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close to one another? Unboundedness
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Gaussian elimination Under our alge
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output AND NOT OR AND OR NOT true f
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Exercises 7.1. Consider the followi
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7.12. For the linear program max x
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• f (i) is a valid flow from s i
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7.28. A linear program for shortest
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Chapter 8 NP-complete problems 8.1
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Satisfiability SATISFIABILITY, or S
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cost as the budget. Fine, but how d
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Figure 8.3 What is the smallest cut
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Figure 8.4 A more elaborate matchma
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8.2 NP-complete problems Hard probl
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I. These two translation procedures
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Figure 8.7 Reductions between searc
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Figure 8.8 The graph corresponding
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Figure 8.9 S is a vertex cover if a
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2n − m pets will be left unmatche
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Figure 8.10 Rudrata cycle with pair
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est of the graph as shown in Figure
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RUDRATA CYCLE−→TSP Given a grap
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Now that we know CIRCUIT SAT reduce
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Exercises 8.1. Optimization versus
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Input: An undirected graph G = (V,
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• (w i , w i ′ ) for all i = 1,
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Chapter 9 Coping with NP-completene
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anching. We can expand either of th
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follow the same pattern. As before,
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9.2 Approximation algorithms In an
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2. It can be used to build a vertex
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9.2.3 TSP The triangle inequality p
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Let’s consider the O(nV ) algorit
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iterations will be needed: whether
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Figure 9.8 Local search. Figure 9.7
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What can be done about such subopti
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Figure 9.10 The search space for a
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Exercises 9.1. In the backtracking
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start with any S ⊆ V while there
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Chapter 10 Quantum algorithms This
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Figure 10.2 Measurement of a superp
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Figure 10.3 A quantum algorithm tak
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¡ £¢ ¥¤ §¦ ©¨ #" %
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The Fourier transform of a periodic
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Yet another basic gate, the control
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10.6 Factoring as periodicity We ha
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Figure 10.6 Quantum factoring. 0 QF
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Exercises 10.1. ∣ ψ 〉 = 1 √2
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Historical notes and further readin
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Index O(·), 13 Ω(·), 14 Θ(·),
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interior-point method, 217 interpol
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