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4 Paths in graphs 109 4.1 Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Breadth-first search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Lengths on edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.4 Dijkstra’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.5 Priority queue implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.6 Shortest paths in the presence of negative edges . . . . . . . . . . . . . . . . . . . 122 4.7 Shortest paths in dags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5 Greedy algorithms 133 5.1 Minimum spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2 Huffman encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3 Horn formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.4 Set cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6 Dynamic programming 161 6.1 Shortest paths in dags, revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2 Longest increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.3 Edit distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.4 Knapsack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.5 Chain matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.6 Shortest paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.7 Independent sets in trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7 Linear programming and reductions 189 7.1 An introduction to linear programming . . . . . . . . . . . . . . . . . . . . . . . . 189 7.2 Flows in networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.3 Bipartite matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7.5 Zero-sum games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.6 The simplex algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.7 Postscript: circuit evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8 NP-complete problems 233 8.1 Search problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.2 NP-complete problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.3 The reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4
9 Coping with NP-completeness 269 9.1 Intelligent exhaustive search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.2 Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.3 Local search heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10 Quantum algorithms 295 10.1 Qubits, superposition, and measurement . . . . . . . . . . . . . . . . . . . . . . . 295 10.2 The plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 10.3 The quantum Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 10.5 Quantum circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 10.6 Factoring as periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 10.7 The quantum algorithm for factoring . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Historical notes and further reading 313 Index 315 5
Page 1: Algorithms Copyright c○2006 S. Da
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 and 18: 4. Likewise, any polynomial dominat
Page 19: and in general ( Fn ) = F n+1 ( ) n
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
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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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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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