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(e) T (n) = 8T (n/2) + n 3 (f) T (n) = 49T (n/25) + n 3/2 log n (g) T (n) = T (n − 1) + 2 (h) T (n) = T (n − 1) + n c , where c ≥ 1 is a constant (i) T (n) = T (n − 1) + c n , where c > 1 is some constant (j) T (n) = 2T (n − 1) + 1 (k) T (n) = T ( √ n) + 1 2.6. A linear, time-invariant system has the following impulse response: b(t) 1/t 0 ¢£ ¤¥ ¦§ ¨© ¡ t 0 ! "# $ $ % % &' ( () *+ ,- ./ t (a) Describe in words the effect of this system. (b) What is the corresponding polynomial? 2.7. What is the sum of the nth roots of unity? What is their product if n is odd? If n is even? 2.8. Practice with the fast Fourier transform. (a) What is the FFT of (1, 0, 0, 0)? What is the appropriate value of ω in this case? And of which sequence is (1, 0, 0, 0) the FFT? (b) Repeat for (1, 0, 1, −1). 2.9. Practice with polynomial multiplication by FFT. (a) Suppose that you want to multiply the two polynomials x + 1 and x 2 + 1 using the FFT. Choose an appropriate power of two, find the FFT of the two sequences, multiply the results componentwise, and compute the inverse FFT to get the final result. (b) Repeat for the pair of polynomials 1 + x + 2x 2 and 2 + 3x. 2.10. Find the unique polynomial of degree 4 that takes on values p(1) = 2, p(2) = 1, p(3) = 0, p(4) = 4, and p(5) = 0. Write your answer in the coefficient representation. 2.11. In justifying our matrix multiplication algorithm (Section 2.5), we claimed the following blockwise property: if X and Y are n × n matrices, and [ ] [ ] A B E F X = , Y = . C D G H where A, B, C, D, E, F , G, and H are n/2 × n/2 submatrices, then the product XY can be expressed in terms of these blocks: [ ] [ ] [ ] A B E F AE + BG AF + BH XY = = C D G H CE + DG CF + DH Prove this property. 80
2.12. How many lines, as a function of n (in Θ(·) form), does the following program print? Write a recurrence and solve it. You may assume n is a power of 2. function f(n) if n > 1: print_line(‘‘still going’’) f(n/2) f(n/2) 2.13. A binary tree is full if all of its vertices have either zero or two children. Let B n denote the number of full binary trees with n vertices. (a) By drawing out all full binary trees with 3, 5, or 7 vertices, determine the exact values of B 3 , B 5 , and B 7 . Why have we left out even numbers of vertices, like B 4 ? (b) For general n, derive a recurrence relation for B n . (c) Show by induction that B n is Ω(2 n ). 2.14. You are given an array of n elements, and you notice that some of the elements are duplicates; that is, they appear more than once in the array. Show how to remove all duplicates from the array in time O(n log n). 2.15. In our median-finding algorithm (Section 2.4), a basic primitive is the split operation, which takes as input an array S and a value v and then divides S into three sets: the elements less than v, the elements equal to v, and the elements greater than v. Show how to implement this split operation in place, that is, without allocating new memory. 2.16. You are given an infinite array A[·] in which the first n cells contain integers in sorted order and the rest of the cells are filled with ∞. You are not given the value of n. Describe an algorithm that takes an integer x as input and finds a position in the array containing x, if such a position exists, in O(log n) time. (If you are disturbed by the fact that the array A has infinite length, assume instead that it is of length n, but that you don’t know this length, and that the implementation of the array data type in your programming language returns the error message ∞ whenever elements A[i] with i > n are accessed.) 2.17. Given a sorted array of distinct integers A[1, . . . , n], you want to find out whether there is an index i for which A[i] = i. Give a divide-and-conquer algorithm that runs in time O(log n). 2.18. Consider the task of searching a sorted array A[1 . . . n] for a given element x: a task we usually perform by binary search in time O(log n). Show that any algorithm that accesses the array only via comparisons (that is, by asking questions of the form “is A[i] ≤ z?”), must take Ω(log n) steps. 2.19. A k-way merge operation. Suppose you have k sorted arrays, each with n elements, and you want to combine them into a single sorted array of kn elements. (a) Here’s one strategy: Using the merge procedure from Section 2.3, merge the first two arrays, then merge in the third, then merge in the fourth, and so on. What is the time complexity of this algorithm, in terms of k and n? (b) Give a more efficient solution to this problem, using divide-and-conquer. 2.20. Show that any array of integers x[1 . . . n] can be sorted in O(n + M) time, where M = max x i − min x i . i i For small M, this is linear time: why doesn’t the Ω(n log n) lower bound apply in this case? 81
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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
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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 78 and 79: The slow spread of a fast algorithm
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)
Page 129 and 130: 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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