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Figure 6.5 The underlying dag, and a path of length 6. P O L Y N O M I A L E X P O N E N T I A L distances are given by shortest paths in the dag! To see this, set all edge lengths to 1, except for {(i − 1, j − 1) → (i, j) : x[i] = y[j]} (shown dotted in the figure), whose length is 0. The final answer is then simply the distance between nodes s = (0, 0) and t = (m, n). One possible shortest path is shown, the one that yields the alignment we found earlier. On this path, each move down is a deletion, each move right is an insertion, and each diagonal move is either a match or a substitution. By altering the weights on this dag, we can allow generalized forms of edit distance, in which insertions, deletions, and substitutions have different associated costs. 168
Common subproblems Finding the right subproblem takes creativity and experimentation. But there are a few standard choices that seem to arise repeatedly in dynamic programming. i. The input is x 1 , x 2 , . . . , x n and a subproblem is x 1 , x 2 , . . . , x i . x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 The number of subproblems is therefore linear. ii. The input is x 1 , . . . , x n , and y 1 , . . . , y m . A subproblem is x 1 , . . . , x i and y 1 , . . . , y j . x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 The number of subproblems is O(mn). iii. The input is x 1 , . . . , x n and a subproblem is x i , x i+1 , . . . , x j . x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 The number of subproblems is O(n 2 ). iv. The input is a rooted tree. A subproblem is a rooted subtree. If the tree has n nodes, how many subproblems are there? We’ve already encountered the first two cases, and the others are coming up shortly. 169
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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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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
Page 131 and 132: Let r ∗ be the maximum ratio achi
Page 133 and 134: Chapter 5 Greedy algorithms A game
Page 135 and 136: Trees A tree is an undirected graph
Page 137 and 138: Figure 5.3 The cut property at work
Page 139 and 140: eturn x As can be expected, makeset
Page 141 and 142: Figure 5.7 The effect of path compr
Page 143 and 144: A randomized algorithm for minimum
Page 145 and 146: Figure 5.9 Top: Prim’s minimum sp
Page 147 and 148: Figure 5.10 A prefix-free encoding.
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Page 151 and 152: 5.3 Horn formulas In order to displ
Page 153 and 154: Figure 5.11 (a) Eleven towns. (b) T
Page 155 and 156: Exercises 5.1. Consider the followi
Page 157 and 158: (a) Code: {0, 10, 11} (b) Code: {0,
Page 159 and 160: Input: Undirected graph G = (V, E);
Page 161 and 162: Chapter 6 Dynamic programming In th
Page 163 and 164: Figure 6.2 The dag of increasing su
Page 165 and 166: Programming? The origin of the term
Page 167: Figure 6.4 (a) The table of subprob
Page 171 and 172: 6.4 Knapsack During a robbery, a bu
Page 173 and 174: Memoization In dynamic programming,
Page 175 and 176: Figure 6.7 (a) ((A × B) × C) × D
Page 177 and 178: ecause it leads right back to the O
Page 179 and 180: d ij . We must pick the best such i
Page 181 and 182: Figure 6.11 I(u) is the size of the
Page 183 and 184: 6.8. Given two strings x = x 1 x 2
Page 185 and 186: Figure 6.12 Two binary search trees
Page 187 and 188: 6.26. Sequence alignment. When a ne
Page 189 and 190: Chapter 7 Linear programming and re
Page 191 and 192: It is a general rule of linear prog
Page 193 and 194: the feasible region. The point of f
Page 195 and 196: Figure 7.3 A communications network
Page 197 and 198: Reductions Sometimes a computationa
Page 199 and 200: Matrix-vector notation A linear fun
Page 201 and 202: Figure 7.5 An illustration of the m
Page 203 and 204: L e R s t e ′ We claim that size(
Page 205 and 206: Figure 7.6 Continued Current Flow R
Page 207 and 208: from the algorithm, which in such c
Page 209 and 210: Figure 7.10 A generic primal LP in
Page 211 and 212: Now suppose the two of them play th
Page 213 and 214: In LP form, this is min w −3y 1 +
Page 215 and 216: 7.6.2 The algorithm On each iterati
Page 217 and 218: 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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