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epeat randomly choose a solution s ′ in the neighborhood of s if ∆ = cost(s ′ ) − cost(s) is negative: replace s by s ′ else: replace s by s ′ with probability e −∆/T . If T is zero, this is identical to our previous local search. But if T is large, then moves that increase the cost are occasionally accepted. What value of T should be used? 288
Figure 9.10 The search space for a graph with eight nodes. The space contains 35 solutions, which have been partitioned into seven groups for clarity. An example of each is shown. There are five local optima. 4 states, cost 6 8 states, cost 4 2 states, cost 4 8 states, cost 3 8 states, cost 3 4 states, cost 2 1 state, cost 0 289
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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
Page 237 and 238: cost as the budget. Fine, but how d
Page 239 and 240: Figure 8.3 What is the smallest cut
Page 241 and 242: Figure 8.4 A more elaborate matchma
Page 243 and 244: 8.2 NP-complete problems Hard probl
Page 245 and 246: I. These two translation procedures
Page 247 and 248: Figure 8.7 Reductions between searc
Page 249 and 250: Figure 8.8 The graph corresponding
Page 251 and 252: Figure 8.9 S is a vertex cover if a
Page 253 and 254: 2n − m pets will be left unmatche
Page 255 and 256: Figure 8.10 Rudrata cycle with pair
Page 257 and 258: est of the graph as shown in Figure
Page 259 and 260: RUDRATA CYCLE−→TSP Given a grap
Page 261 and 262: Now that we know CIRCUIT SAT reduce
Page 263 and 264: Exercises 8.1. Optimization versus
Page 265 and 266: Input: An undirected graph G = (V,
Page 267 and 268: • (w i , w i ′ ) for all i = 1,
Page 269 and 270: Chapter 9 Coping with NP-completene
Page 271 and 272: anching. We can expand either of th
Page 273 and 274: follow the same pattern. As before,
Page 275 and 276: 9.2 Approximation algorithms In an
Page 277 and 278: 2. It can be used to build a vertex
Page 279 and 280: 9.2.3 TSP The triangle inequality p
Page 281 and 282: Let’s consider the O(nV ) algorit
Page 283 and 284: iterations will be needed: whether
Page 285 and 286: Figure 9.8 Local search. Figure 9.7
Page 287: What can be done about such subopti
Page 291 and 292: Exercises 9.1. In the backtracking
Page 293 and 294: start with any S ⊆ V while there
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
Page 301 and 302: ¡ £¢ ¥¤ §¦ ©¨ #" %
Page 303 and 304: The Fourier transform of a periodic
Page 305 and 306: Yet another basic gate, the control
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 and 314: Historical notes and further readin
Page 315 and 316: Index O(·), 13 Ω(·), 14 Θ(·),
Page 317 and 318: interior-point method, 217 interpol
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