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4.19. Generalized shortest-paths problem. In Internet routing, there are delays on lines but also, more significantly, delays at routers. This motivates a generalized shortest-paths problem. Suppose that in addition to having edge lengths {l e : e ∈ E}, a graph also has vertex costs {c v : v ∈ V }. Now define the cost of a path to be the sum of its edge lengths, plus the costs of all vertices on the path (including the endpoints). Give an efficient algorithm for the following problem. Input: A directed graph G = (V, E); positive edge lengths l e and positive vertex costs c v ; a starting vertex s ∈ V . Output: An array cost[·] such that for every vertex u, cost[u] is the least cost of any path from s to u (i.e., the cost of the cheapest path), under the definition above. Notice that cost[s] = c s . 4.20. There is a network of roads G = (V, E) connecting a set of cities V . Each road in E has an associated length l e . There is a proposal to add one new road to this network, and there is a list E ′ of pairs of cities between which the new road can be built. Each such potential road e ′ ∈ E ′ has an associated length. As a designer for the public works department you are asked to determine the road e ′ ∈ E ′ whose addition to the existing network G would result in the maximum decrease in the driving distance between two fixed cities s and t in the network. Give an efficient algorithm for solving this problem. 4.21. Shortest path <strong>algorithms</strong> can be applied in currency trading. Let c 1 , c 2 , . . . , c n be various currencies; for instance, c 1 might be dollars, c 2 pounds, and c 3 lire. For any two currencies c i and c j , there is an exchange rate r i,j ; this means that you can purchase r i,j units of currency c j in exchange for one unit of c i . These exchange rates satisfy the condition that r i,j · r j,i < 1, so that if you start with a unit of currency c i , change it into currency c j and then convert back to currency c i , you end up with less than one unit of currency c i (the difference is the cost of the transaction). (a) Give an efficient algorithm for the following problem: Given a set of exchange rates r i,j , and two currencies s and t, find the most advantageous sequence of currency exchanges for converting currency s into currency t. Toward this goal, you should represent the currencies and rates by a graph whose edge lengths are real numbers. The exchange rates are updated frequently, reflecting the demand and supply of the various currencies. Occasionally the exchange rates satisfy the following property: there is a sequence of currencies c i1 , c i2 , . . . , c ik such that r i1,i 2 · r i2,i 3 · · · r ik−1 ,i k · r ik ,i 1 > 1. This means that by starting with a unit of currency c i1 and then successively converting it to currencies c i2 , c i3 , . . . , c ik , and finally back to c i1 , you would end up with more than one unit of currency c i1 . Such anomalies last only a fraction of a minute on the currency exchange, but they provide an opportunity for risk-free profits. (b) Give an efficient algorithm for detecting the presence of such an anomaly. Use the graph representation you found above. 4.22. The tramp steamer problem. You are the owner of a steamship that can ply between a group of port cities V . You make money at each port: a visit to city i earns you a profit of p i dollars. Meanwhile, the transportation cost from port i to port j is c ij > 0. You want to find a cyclic route in which the ratio of profit to cost is maximized. To this end, consider a directed graph G = (V, E) whose nodes are ports, and which has edges between each pair of ports. For any cycle C in this graph, the profit-to-cost ratio is ∑ (i,j)∈C r(C) = p j ∑ (i,j)∈C c . ij 130
Let r ∗ be the maximum ratio achievable by a simple cycle. One way to determine r ∗ is by binary search: by first guessing some ratio r, and then testing whether it is too large or too small. Consider any positive r > 0. Give each edge (i, j) a weight of w ij = rc ij − p j . (a) Show that if there is a cycle of negative weight, then r < r ∗ . (b) Show that if all cycles in the graph have strictly positive weight, then r > r ∗ . (c) Give an efficient algorithm that takes as input a desired accuracy ɛ > 0 and returns a simple cycle C for which r(C) ≥ r ∗ − ɛ. Justify the correctness of your algorithm and analyze its running time in terms of |V |, ɛ, and R = max (i,j)∈E (p j /c ij ). 131
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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
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 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: 4.16. Section 4.5.2 describes a way
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 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
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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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