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Exercises 0.1. In each of the following situations, indicate whether f = O(g), or f = Ω(g), or both (in which case f = Θ(g)). f(n) g(n) (a) n − 100 n − 200 (b) n 1/2 n 2/3 (c) 100n + log n n + (log n) 2 (d) n log n 10n log 10n (e) log 2n log 3n (f) 10 log n log(n 2 ) (g) n 1.01 n log 2 n (h) n 2 / log n n(log n) 2 (i) n 0.1 (log n) 10 (j) (log √ n) log n n/ log n (k) n (log n) 3 (l) n 1/2 5 log 2 n (m) n2 n 3 n (n) 2 n 2 n+1 (o) n! 2 n (p) (log n) log n 2 (log 2 ∑ n)2 n (q) i=1 ik n k+1 0.2. Show that, if c is a positive real number, then g(n) = 1 + c + c 2 + · · · + c n is: (a) Θ(1) if c < 1. (b) Θ(n) if c = 1. (c) Θ(c n ) if c > 1. The moral: in big-Θ terms, the sum of a geometric series is simply the first term if the series is strictly decreasing, the last term if the series is strictly increasing, or the number of terms if the series is unchanging. 0.3. The Fibonacci numbers F 0 , F 1 , F 2 , . . . , are defined by the rule F 0 = 0, F 1 = 1, F n = F n−1 + F n−2 . In this problem we will confirm that this sequence grows exponentially fast and obtain some bounds on its growth. (a) Use induction to prove that F n ≥ 2 0.5n for n ≥ 6. (b) Find a constant c < 1 such that F n ≤ 2 cn for all n ≥ 0. Show that your answer is correct. (c) What is the largest c you can find for which F n = Ω(2 cn )? 0.4. Is there a faster way to compute the nth Fibonacci number than by fib2 (page 13)? One idea involves matrices. We start by writing the equations F 1 = F 1 and F 2 = F 0 + F 1 in matrix notation: ( ) ( ) ( ) F1 0 1 F0 = · . F 2 1 1 F 1 Similarly, ( F2 ) = F 3 ( ) ( ) 0 1 F1 · = 1 1 F 2 ( ) 2 ( ) 0 1 F0 · 1 1 F 1 18
and in general ( Fn ) = F n+1 ( ) n ( ) 0 1 F0 · . 1 1 F 1 So, in order to compute F n , it suffices to raise this 2 × 2 matrix, call it X, to the nth power. (a) Show that two 2 × 2 matrices can be multiplied using 4 additions and 8 multiplications. But how many matrix multiplications does it take to compute X n ? (b) Show that O(log n) matrix multiplications suffice for computing X n . (Hint: Think about computing X 8 .) Thus the number of arithmetic operations needed by our matrix-based algorithm, call it fib3, is just O(log n), as compared to O(n) for fib2. Have we broken another exponential barrier? The catch is that our new algorithm involves multiplication, not just addition; and multiplications of large numbers are slower than additions. We have already seen that, when the complexity of arithmetic operations is taken into account, the running time of fib2 becomes O(n 2 ). (c) Show that all intermediate results of fib3 are O(n) bits long. (d) Let M(n) be the running time of an algorithm for multiplying n-bit numbers, and assume that M(n) = O(n 2 ) (the school method for multiplication, recalled in Chapter 1, achieves this). Prove that the running time of fib3 is O(M(n) log n). (e) Can you prove that the running time of fib3 is O(M(n))? (Hint: The lengths of the numbers being multiplied get doubled with every squaring.) In conclusion, whether fib3 is faster than fib2 depends on whether we can multiply n-bit integers faster than O(n 2 ). Do you think this is possible? (The answer is in Chapter 2.) Finally, there is a formula for the Fibonacci numbers: ( ) n ( F n = √ 1 − √ 1 5 5 1 + √ 5 2 1 − √ ) n 5 . 2 So, it would appear that we only need to raise a couple of numbers to the nth power in order to compute F n . The problem is that these numbers are irrational, and computing them to sufficient accuracy is nontrivial. In fact, our matrix method fib3 can be seen as a roundabout way of raising these irrational numbers to the nth power. If you know your linear algebra, you should see why. (Hint: What are the eigenvalues of the matrix X?) 19
Page 1: Algorithms Copyright c○2006 S. Da
Page 4 and 5: 4 Paths in graphs 109 4.1 Distances
Page 7 and 8: List of boxes Bases and logs . . .
Page 9 and 10: Preface This book evolved over the
Page 11 and 12: Chapter 0 Prologue Look around you.
Page 13 and 14: The first question is moot here, as
Page 15 and 16: for addition, which works on one di
Page 17: 4. Likewise, any polynomial dominat
Page 22 and 23: Bases and logs Naturally, there is
Page 24 and 25: Figure 1.1 Multiplication à la Fra
Page 26 and 27: 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
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The equivalence of the two polynomi
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Figure 2.6 The complex roots of uni
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A matrix reformulation To get a cle
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The orthogonality property can be s
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FFT n (input: a 0 , . . . , a n−1
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The slow spread of a fast algorithm
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(e) T (n) = 8T (n/2) + n 3 (f) T (n
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2.21. Mean and median. One of the m
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(c) In fact, squaring matrices is n
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Chapter 3 Decompositions of graphs
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Therefore, each edge appears in exa
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Figure 3.3. 1 The previsit and post
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Figure 3.6 (a) A 12-node graph. (b)
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Tree edges are actually part of the
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the same thing. Since a dag is line
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Figure 3.10 The reverse of the grap
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Exercises 3.1. Perform a depth-firs
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(a) Model this as a graph problem:
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For instance, in the graph below (w
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3.30. On page 97, we defined the bi
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Chapter 4 Paths in graphs 4.1 Dista
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Figure 4.4 The result of breadth-fi
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Figure 4.6 Breaking edges into unit
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into account. This viewpoint gives
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Which heap is best? The running tim
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Figure 4.11 (a) A binary heap with
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Figure 4.13 The Bellman-Ford algori
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Figure 4.15 A single-source shortes
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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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