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Chapters 8 and 9 The notion of NP-completeness was first identified in the work of Steve Cook, who proved in 1971 that SAT is NP-complete; a year later Dick Karp came up with a list of 23 NP-complete problems (including all the ones proven so in Chapter 8), establishing beyond doubt the applicability of the concept (they were both given the Turing award). Leonid Levin, working in the Soviet Union, independently proved a similar theorem. For an excellent treatment of NP-completeness see M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman, 1979. And for the more general subject of Complexity see C. H. Papadimitriou, Computational Complexity. Addison-Wesley, Reading Massachusetts, 1995. Chapter 10 The quantum algorithm for primality was discovered in 1994 by Peter Shor. introduction to quantum mechanics for computer scientists see For a novel http://www.cs.berkeley.edu/∼vazirani/quantumphysics.html and for an introduction to quantum computation see the notes for the course “Qubits, Quantum Mechanics, and Computers” at http://www.cs.berkeley.edu/∼vazirani/cs191.html 314
Index O(·), 13 Ω(·), 14 Θ(·), 14 ∣ · 〉 , 291 addition, 19 adjacency list, 84 adjacency matrix, 84 advanced encryption standard (AES), 38 Al Khwarizmi, 9 amortized analysis, 137 ancestor, 90 approximation algorithm, 271 approximation ratio, 271 backtracking, 266 bases, 20 basic computer step, 12 Bellman-Ford algorithm, 119 BFS, see breadth-first search biconnected components, 103 big-O notation, 13–15 binary search, 54 binary tree complete, 20 full, 77, 142 bipartite graph, 98 Boolean circuit, 217, 255 Boolean formula, 146 conjunctive normal form, 231 implication, 146 literal, 146 satisfying assignment, 146, 231 variable, 146 branch-and-bound, 268 breadth-first search, 106 Carmichael numbers, 32, 34 Chinese remainder theorem, 47 circuit SAT, see satisfiability circuit value, 218 clique, 238, 247 clustering, 235, 273 CNF, see Boolean formula complex numbers, 67, 292 roots of unity, 65, 67 computational biology, 166 connectedness directed, 93 undirected, 88 controlled-NOT gate, 300 cryptography private-key, 37 public-key, 37, 39 cut, 132 s − t cut, 198 and flow, 198 balanced cut, 235 max cut, 288 minimum cut, 139, 235 cut property, 132 cycle, 91 dag, see directed acyclic graph Dantzig, George, 187 degree, 98 depth-first search, 85 back edge, 87, 90 cross edge, 90 forward edge, 90 tree edge, 87, 90 descendant, 90 DFS, see depth-first search digital signature, 48 Dijkstra’s algorithm, 111 directed acyclic graph, 92 longest path, 120 315
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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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Now that we know CIRCUIT SAT reduce
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Page 285 and 286: Figure 9.8 Local search. Figure 9.7
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Page 291 and 292: Exercises 9.1. In the backtracking
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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 and 312: Exercises 10.1. ∣ ψ 〉 = 1 √2
Page 313: Historical notes and further readin
Page 317 and 318: interior-point method, 217 interpol
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