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3.17. Infinite paths. Let G = (V, E) be a directed graph with a designated “start vertex” s ∈ V , a set V G ⊆ V of “good” vertices, and a set V B ⊆ V of “bad” vertices. An infinite trace p of G is an infinite sequence v 0 v 1 v 2 · · · of vertices v i ∈ V such that (1) v 0 = s, and (2) for all i ≥ 0, (v i , v i+1 ) ∈ E. That is, p is an infinite path in G starting at vertex s. Since the set V of vertices is finite, every infinite trace of G must visit some vertices infinitely often. (a) If p is an infinite trace, let Inf(p) ⊆ V be the set of vertices that occur infinitely often in p. Show that Inf(p) is a subset of a strongly connected component of G. (b) Describe an algorithm that determines if G has an infinite trace. (c) Describe an algorithm that determines if G has an infinite trace that visits some good vertex in V G infinitely often. (d) Describe an algorithm that determines if G has an infinite trace that visits some good vertex in V G infinitely often, but visits no bad vertex in V B infinitely often. 3.18. You are given a binary tree T = (V, E) (in adjacency list format), along with a designated root node r ∈ V . Recall that u is said to be an ancestor of v in the rooted tree, if the path from r to v in T passes through u. You wish to preprocess the tree so that queries of the form “is u an ancestor of v?” can be answered in constant time. The preprocessing itself should take linear time. How can this be done? 3.19. As in the previous problem, you are given a binary tree T = (V, E) with designated root node. In addition, there is an array x[·] with a value for each node in V . Define a new array z[·] as follows: for each u ∈ V , z[u] = the maximum of the x-values associated with u’s descendants. Give a linear-time algorithm which calculates the entire z-array. 3.20. You are given a tree T = (V, E) along with a designated root node r ∈ V . The parent of any node v ≠ r, denoted p(v), is defined to be the node adjacent to v in the path from r to v. By convention, p(r) = r. For k > 1, define p k (v) = p k−1 (p(v)) and p 1 (v) = p(v) (so p k (v) is the kth ancestor of v). Each vertex v of the tree has an associated non-negative integer label l(v). Give a linear-time algorithm to update the labels of all the vertices in T according to the following rule: l new (v) = l(p l(v) (v)). 3.21. Give a linear-time algorithm to find an odd-length cycle in a directed graph. (Hint: First solve this problem under the assumption that the graph is strongly connected.) 3.22. Give an efficient algorithm which takes as input a directed graph G = (V, E), and determines whether or not there is a vertex s ∈ V from which all other vertices are reachable. 3.23. Give an efficient algorithm that takes as input a directed acyclic graph G = (V, E), and two vertices s, t ∈ V , and outputs the number of different directed paths from s to t in G. 3.24. Give a linear-time algorithm for the following task. Input: A directed acyclic graph G Question: Does G contain a directed path that touches every vertex exactly once? 3.25. You are given a directed graph in which each node u ∈ V has an associated price p u which is a positive integer. Define the array cost as follows: for each u ∈ V , cost[u] = price of the cheapest node reachable from u (including u itself). 104
For instance, in the graph below (with prices shown for each vertex), the cost values of the nodes A, B, C, D, E, F are 2, 1, 4, 1, 4, 5, respectively. 2 A 6 C 4 E B 3 D F 1 5 Your goal is to design an algorithm that fills in the entire cost array (i.e., for all vertices). (a) Give a linear-time algorithm that works for directed acyclic graphs. (Hint: vertices in a particular order.) Handle the (b) Extend this to a linear-time algorithm that works for all directed graphs. (Hint: Recall the “two-tiered” structure of directed graphs.) 3.26. An Eulerian tour in an undirected graph is a cycle that is allowed to pass through each vertex multiple times, but must use each edge exactly once. This simple concept was used by Euler in 1736 to solve the famous Konigsberg bridge problem, which launched the field of graph theory. The city of Konigsberg (now called Kaliningrad, in western Russia) is the meeting point of two rivers with a small island in the middle. There are seven bridges across the rivers, and a popular recreational question of the time was to determine whether it is possible to perform a tour in which each bridge is crossed exactly once. Euler formulated the relevant information as a graph with four nodes (denoting land masses) and seven edges (denoting bridges), as shown here. Northern bank Small island Big island Southern bank Notice an unusual feature of this problem: multiple edges between certain pairs of nodes. (a) Show that an undirected graph has an Eulerian tour if and only if all its vertices have even degree. Conclude that there is no Eulerian tour of the Konigsberg bridges. (b) An Eulerian path is a path which uses each edge exactly once. Can you give a similar if-and-only-if characterization of which undirected graphs have Eulerian paths? (c) Can you give an analog of part (a) for directed graphs? 3.27. Two paths in a graph are called edge-disjoint if they have no edges in common. Show that in any undirected graph, it is possible to pair up the vertices of odd degree and find paths between each such pair so that all these paths are edge-disjoint. 105
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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
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 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: (a) Model this as a graph problem:
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 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
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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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