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(d) Conclude therefore that for any classical circuit C there is an equivalent quantum circuit Q using just NOT and C-SWAP gates in the following sense: if C outputs y on input x, then Q outputs ∣ ∣ x, y, z 〉 on input ∣ ∣x, 0, 0 〉 . (Here z is some set of junk bits that are generated during this computation). (e) Now show that that there is a quantum circuit Q −1 that outputs ∣ ∣ x, 0, 0 〉 on input ∣ ∣x, y, z 〉 . (f) Show that there is a quantum circuit Q ′ made up of NOT, CNOT, and C-SWAP gates that outputs ∣ ∣ x, y, 0 〉 on input ∣ ∣x, 0, 0 〉 . 10.8. In this problem we will show that if N = pq is the product of two odd primes, and if x is chosen uniformly at random between 0 and N − 1, such that gcd(x, N) = 1, then with probability at least 3/8, the order r of x mod N is even, and moreover x r/2 is a nontrivial square root of 1 mod N. (a) Let p be an odd prime and let x be a uniformly random number modulo p. Show that the order of x mod p is even with probability at least 1/2. (Hint: Use Fermat’s little theorem (Section 1.3).) (b) Use the Chinese remainder theorem (Exercise 1.37) to show that with probability at least 3/4, the order r of x mod N is even. (c) If r is even, prove that the probability that x r/2 ≡ ±1 is at most 1/2. 312
Historical notes and further reading Chapters 1 and 2 The classical book on the theory of numbers is G. H. Hardy and E. M. Wright, Introduction to the Theory of Numbers. Oxford University Press, 1980. The primality algorithm was discovered by Robert Solovay and Volker Strassen in the mid- 1970’s, while the RSA cryptosystem came about a couple of years later. See D. R. Stinson, Cryptography: Theory and Practice. Chapman and Hall, 2005. for much more on cryptography. For randomized <strong>algorithms</strong>, see R. Motwani and P. Raghavan, Randomized Algorithms. Cambridge University Press, 1995. Universal hash functions were proposed in 1979 by Larry Carter and Mark Wegman. The fast matrix multiplication algorithm is due to Volker Strassen (1969). Also due to Strassen, with Arnold Schönhage, is the fastest known algorithm for integer multiplication: it uses a variant of the FFT to multiply n-bit integers in O(n log n log log n) bit operations. Chapter 3 Depth-first search and its many applications were articulated by John Hopcroft and Bob Tarjan in 1973—they were honored for this contribution by the Turing award, the highest distinction in Computer Science. The two-phase algorithm for finding strongly connected components is due to Rao Kosaraju. Chapters 4 and 5 Dijkstra’s algorithm was discovered in 1959 by Edsger Dijkstra (1930-2002), while the first algorithm for computing minimum spanning trees can be traced back to a 1926 paper by the Czech mathematician Otakar Boruvka. The analysis of the union-find data structure (which is actually a little more tight than our log ∗ n bound) is due to Bob Tarjan. Finally, David Huffman discovered in 1952, while a graduate student, the encoding algorithm that bears his name. Chapter 7 The simplex method was discovered in 1947 by George Danzig (1914-2005), and the min-max theorem for zero-sum games in 1928 by John von Neumann (who is also considered the father of the computer). A very nice book on linear programming is V. Chvátal, Linear Programming. W. H. Freeman, 1983. And for game theory, see Martin J. Osborne and Ariel Rubinstein, A course in game theory. M.I.T. Press, 1994. 313
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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.
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Figure 1.6 A simple extension of Eu
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We say x is the multiplicative inve
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Figure 1.7 An algorithm for testing
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Figure 1.8 An algorithm for testing
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Randomized algorithms: a virtual ch
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Alice’s encryption function is th
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Figure 1.9 RSA. Bob chooses his pub
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There is nothing inherently wrong w
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Exercises 1.1. Show that in any bas
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the depth of the circuit or the lon
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(c) Give an example of positive int
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x = x L x R = 2 n/2 x L + x R y = y
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Figure 2.2 Divide-and-conquer integ
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Binary search The ultimate divide-a
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An n log n lower bound for sorting
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The three sublists S L , S v , and
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to be the best running time possibl
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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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§¦ ¥¤ £¢ ©¨ #"
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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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Page 279 and 280: 9.2.3 TSP The triangle inequality p
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Page 283 and 284: iterations will be needed: whether
Page 285 and 286: Figure 9.8 Local search. Figure 9.7
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Page 289 and 290: Figure 9.10 The search space for a
Page 291 and 292: Exercises 9.1. In the backtracking
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Page 295 and 296: Chapter 10 Quantum algorithms This
Page 297 and 298: Figure 10.2 Measurement of a superp
Page 299 and 300: Figure 10.3 A quantum algorithm tak
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Page 303 and 304: The Fourier transform of a periodic
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Page 307 and 308: 10.6 Factoring as periodicity We ha
Page 309 and 310: Figure 10.6 Quantum factoring. 0 QF
Page 311: Exercises 10.1. ∣ ψ 〉 = 1 √2
Page 315 and 316: Index O(·), 13 Ω(·), 14 Θ(·),
Page 317 and 318: interior-point method, 217 interpol
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