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Output: A tree T = (V, E ′ ), with E ′ ⊆ E, that minimizes weight(T ) = ∑ e∈E ′ w e . In the preceding example, the minimum spanning tree has a cost of 16: A 1 C E 4 2 5 B 4 D F However, this is not the only optimal solution. Can you spot another? 5.1.1 A greedy approach Kruskal’s minimum spanning tree algorithm starts with the empty graph and then selects edges from E according to the following rule. Repeatedly add the next lightest edge that doesn’t produce a cycle. In other words, it constructs the tree edge by edge and, apart from taking care to avoid cycles, simply picks whichever edge is cheapest at the moment. This is a greedy algorithm: every decision it makes is the one with the most obvious immediate advantage. Figure 5.1 shows an example. We start with an empty graph and then attempt to add edges in increasing order of weight (ties are broken arbitrarily): B − C, C − D, B − D, C − F, D − F, E − F, A − D, A − B, C − E, A − C. The first two succeed, but the third, B − D, would produce a cycle if added. So we ignore it and move along. The final result is a tree with cost 14, the minimum possible. The correctness of Kruskal’s method follows from a certain cut property, which is general enough to also justify a whole slew of other minimum spanning tree <strong>algorithms</strong>. Figure 5.1 The minimum spanning tree found by Kruskal’s algorithm. A 6 5 C E A C E 4 1 5 2 3 4 B 2 D 4 F B D F 134
Trees A tree is an undirected graph that is connected and acyclic. Much of what makes trees so useful is the simplicity of their structure. For instance, Property 2 A tree on n nodes has n − 1 edges. This can be seen by building the tree one edge at a time, starting from an empty graph. Initially each of the n nodes is disconnected from the others, in a connected component by itself. As edges are added, these components merge. Since each edge unites two different components, exactly n − 1 edges are added by the time the tree is fully formed. In a little more detail: When a particular edge {u, v} comes up, we can be sure that u and v lie in separate connected components, for otherwise there would already be a path between them and this edge would create a cycle. Adding the edge then merges these two components, thereby reducing the total number of connected components by one. Over the course of this incremental process, the number of components decreases from n to one, meaning that n − 1 edges must have been added along the way. The converse is also true. Property 3 Any connected, undirected graph G = (V, E) with |E| = |V | − 1 is a tree. We just need to show that G is acyclic. One way to do this is to run the following iterative procedure on it: while the graph contains a cycle, remove one edge from this cycle. The process terminates with some graph G ′ = (V, E ′ ), E ′ ⊆ E, which is acyclic and, by Property 1 (from page 133), is also connected. Therefore G ′ is a tree, whereupon |E ′ | = |V | − 1 by Property 2. So E ′ = E, no edges were removed, and G was acyclic to start with. In other words, we can tell whether a connected graph is a tree just by counting how many edges it has. Here’s another characterization. Property 4 An undirected graph is a tree if and only if there is a unique path between any pair of nodes. In a tree, any two nodes can only have one path between them; for if there were two paths, the union of these paths would contain a cycle. On the other hand, if a graph has a path between any two nodes, then it is connected. If these paths are unique, then the graph is also acyclic (since a cycle has two paths between any pair of nodes). 135
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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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Page 87 and 88: Chapter 3 Decompositions of graphs
Page 89 and 90: Therefore, each edge appears in exa
Page 91 and 92: Figure 3.3. 1 The previsit and post
Page 93 and 94: Figure 3.6 (a) A 12-node graph. (b)
Page 95 and 96: Tree edges are actually part of the
Page 97 and 98: the same thing. Since a dag is line
Page 99 and 100: Figure 3.10 The reverse of the grap
Page 101 and 102: Exercises 3.1. Perform a depth-firs
Page 103 and 104: (a) Model this as a graph problem:
Page 105 and 106: For instance, in the graph below (w
Page 107 and 108: 3.30. On page 97, we defined the bi
Page 109 and 110: Chapter 4 Paths in graphs 4.1 Dista
Page 111 and 112: Figure 4.4 The result of breadth-fi
Page 113 and 114: Figure 4.6 Breaking edges into unit
Page 115 and 116: 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: Chapter 5 Greedy algorithms A game
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 and 168: Figure 6.4 (a) The table of subprob
Page 169 and 170: Common subproblems Finding the righ
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
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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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