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Figure 3.9 (a) A directed graph and its strongly connected components. (b) The meta-graph. (a) A B C (b) D E F A B,E C,F G H D G,H,I J,K,L I J K L 3.4.2 An efficient algorithm The decomposition of a directed graph into its strongly connected components is very informative and useful. It turns out, fortunately, that it can be found in linear time by making further use of depth-first search. The algorithm is based on some properties we have already seen but which we will now pinpoint more closely. Property 1 If the explore subroutine is started at node u, then it will terminate precisely when all nodes reachable from u have been visited. Therefore, if we call explore on a node that lies somewhere in a sink strongly connected component (a strongly connected component that is a sink in the meta-graph), then we will retrieve exactly that component. Figure 3.9 has two sink strongly connected components. Starting explore at node K, for instance, will completely traverse the larger of them and then stop. This suggests a way of finding one strongly connected component, but still leaves open two major problems: (A) how do we find a node that we know for sure lies in a sink strongly connected component and (B) how do we continue once this first component has been discovered? Let’s start with problem (A). There is not an easy, direct way to pick out a node that is guaranteed to lie in a sink strongly connected component. But there is a way to get a node in a source strongly connected component. Property 2 The node that receives the highest post number in a depth-first search must lie in a source strongly connected component. This follows from the following more general property. Property 3 If C and C ′ are strongly connected components, and there is an edge from a node 98
Figure 3.10 The reverse of the graph from Figure 3.9. A B C D E F A B,E C,F G H D G,H,I J,K,L I J K L in C to a node in C ′ , then the highest post number in C is bigger than the highest post number in C ′ . Proof. In proving Property 3, there are two cases to consider. If the depth-first search visits component C before component C ′ , then clearly all of C and C ′ will be traversed before the procedure gets stuck (see Property 1). Therefore the first node visited in C will have a higher post number than any node of C ′ . On the other hand, if C ′ gets visited first, then the depthfirst search will get stuck after seeing all of C ′ but before seeing any of C, in which case the property follows immediately. Property 3 can be restated as saying that the strongly connected components can be linearized by arranging them in decreasing order of their highest post numbers. This is a generalization of our earlier algorithm for linearizing dags; in a dag, each node is a singleton strongly connected component. Property 2 helps us find a node in the source strongly connected component of G. However, what we need is a node in the sink component. Our means seem to be the opposite of our needs! But consider the reverse graph G R , the same as G but with all edges reversed (Figure 3.10). G R has exactly the same strongly connected components as G (why?). So, if we do a depth-first search of G R , the node with the highest post number will come from a source strongly connected component in G R , which is to say a sink strongly connected component in G. We have solved problem (A)! Onward to problem (B). How do we continue after the first sink component is identified? The solution is also provided by Property 3. Once we have found the first strongly connected component and deleted it from the graph, the node with the highest post number among those remaining will belong to a sink strongly connected component of whatever remains of G. Therefore we can keep using the post numbering from our initial depth-first search on G R 99
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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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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
Page 56 and 57: Binary search The ultimate divide-a
Page 59 and 60: An n log n lower bound for sorting
Page 61 and 62: The three sublists S L , S v , and
Page 63 and 64: to be the best running time possibl
Page 66 and 67: qp t AB CD EF GH IJ KL MN OP QR Why
Page 68 and 69: The equivalence of the two polynomi
Page 70 and 71: Figure 2.6 The complex roots of uni
Page 72 and 73: A matrix reformulation To get a cle
Page 74 and 75: The orthogonality property can be s
Page 76 and 77: FFT n (input: a 0 , . . . , a n−1
Page 78 and 79: The slow spread of a fast algorithm
Page 80 and 81: (e) T (n) = 8T (n/2) + n 3 (f) T (n
Page 82 and 83: 2.21. Mean and median. One of the m
Page 84 and 85: (c) In fact, squaring matrices is n
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: the same thing. Since a dag is line
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
Page 117 and 118: §¦ ¥¤ £¢ ©¨ #"
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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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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