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Figure 9.11 Simulated annealing. The trick is to start with T large and then gradually reduce it to zero. Thus initially, the local search can wander around quite freely, with only a mild preference for low-cost solutions. As time goes on, this preference becomes stronger, and the system mostly sticks to the lower-cost region of the search space, with occasional excursions out of it to escape local optima. Eventually, when the temperature drops further, the system converges on a solution. Figure 9.11 shows this process schematically. Simulated annealing is inspired by the physics of crystallization. When a substance is to be crystallized, it starts in liquid state, with its particles in relatively unconstrained motion. Then it is slowly cooled, and as this happens, the particles gradually move into more regular configurations. This regularity becomes more and more pronounced until finally a crystal lattice is formed. The benefits of simulated annealing come at a significant cost: because of the changing temperature and the initial freedom of movement, many more local moves are needed until convergence. Moreover, it is quite an art to choose a good timetable by which to decrease the temperature, called an annealing schedule. But in many cases where the quality of solutions improves significantly, the tradeoff is worthwhile. 290
Exercises 9.1. In the backtracking algorithm for SAT, suppose that we always choose a subproblem (CNF formula) that has a clause that is as small as possible; and we expand it along a variable that appears in this small clause. Show that this is a polynomial-time algorithm in the special case in which the input formula has only clauses with two literals (that is, it is an instance of 2SAT). 9.2. Devise a backtracking algorithm for the RUDRATA PATH problem from a fixed vertex s. To fully specify such an algorithm you must define: (a) What is a subproblem? (b) How to choose a subproblem. (c) How to expand a subproblem. Argue briefly why your choices are reasonable. 9.3. Devise a branch-and-bound algorithm for the SET COVER problem. This entails deciding: (a) What is a subproblem? (b) How do you choose a subproblem to expand? (c) How do you expand a subproblem? (d) What is an appropriate lowerbound? Do you think that your choices above will work well on typical instances of the problem? Why? 9.4. Given an undirected graph G = (V, E) in which each node has degree ≤ d, show how to efficiently find an independent set whose size is at least 1/(d + 1) times that of the largest independent set. 9.5. Local search for minimum spanning trees. Consider the set of all spanning trees (not just minimum ones) of a weighted, connected, undirected graph G = (V, E). Recall from Section 5.1 that adding an edge e to a spanning tree T creates an unique cycle, and subsequently removing any other edge e ′ ≠ e from this cycle gives back a different spanning tree T ′ . We will say that T and T ′ differ by a single edge swap (e, e ′ ) and that they are neighbors. (a) Show that it is possible to move from any spanning tree T to any other spanning tree T ′ by performing a series of edge-swaps, that is, by moving from neighbor to neighbor. At most how many edge-swaps are needed? (b) Show that if T ′ is an MST, then it is possible to choose these swaps so that the costs of the spanning trees encountered along the way are nonincreasing. In other words, if the sequence of spanning trees encountered is then cost(T i+1 ) ≤ cost(T i ) for all i < k. T = T 0 → T 1 → T 2 → · · · → T k = T ′ , (c) Consider the following local search algorithm which is given as input an undirected graph G. Let T be any spanning tree of G while there is an edge-swap (e, e ′ ) which reduces cost(T ): T ← T + e − e ′ return T Show that this procedure always returns a minimum spanning tree. At most how many iterations does it take? 291
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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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Which heap is best? The running tim
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Figure 4.11 (a) A binary heap with
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Figure 4.13 The Bellman-Ford algori
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Figure 4.15 A single-source shortes
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Input: Undirected graph G = (V, E)
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4.16. Section 4.5.2 describes a way
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Let r ∗ be the maximum ratio achi
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Chapter 5 Greedy algorithms A game
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Trees A tree is an undirected graph
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Figure 5.3 The cut property at work
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eturn x As can be expected, makeset
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Figure 5.7 The effect of path compr
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A randomized algorithm for minimum
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Figure 5.9 Top: Prim’s minimum sp
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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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Page 241 and 242: Figure 8.4 A more elaborate matchma
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Page 247 and 248: Figure 8.7 Reductions between searc
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Page 251 and 252: Figure 8.9 S is a vertex cover if a
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Page 255 and 256: Figure 8.10 Rudrata cycle with pair
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Page 269 and 270: Chapter 9 Coping with NP-completene
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Page 275 and 276: 9.2 Approximation algorithms In an
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Page 285 and 286: Figure 9.8 Local search. Figure 9.7
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Page 295 and 296: Chapter 10 Quantum algorithms This
Page 297 and 298: Figure 10.2 Measurement of a superp
Page 299 and 300: Figure 10.3 A quantum algorithm tak
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Page 303 and 304: The Fourier transform of a periodic
Page 305 and 306: Yet another basic gate, the control
Page 307 and 308: 10.6 Factoring as periodicity We ha
Page 309 and 310: Figure 10.6 Quantum factoring. 0 QF
Page 311 and 312: Exercises 10.1. ∣ ψ 〉 = 1 √2
Page 313 and 314: Historical notes and further readin
Page 315 and 316: Index O(·), 13 Ω(·), 14 Θ(·),
Page 317 and 318: interior-point method, 217 interpol
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