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£¢¡ §¦ ¥¤ Figure 0.1 The proliferation of recursive calls in fib1. F n F n−1 F n−2 F n−4 F n−6 F n−2 F n−3 F n−3 F n−4 F n−4 F n−5 F n−3 F n−4 F n−5 F n−5 As with fib1, the correctness of this algorithm is self-evident because it directly uses the definition of F n . How long does it take? The inner loop consists of a single computer step and is executed n − 1 times. Therefore the number of computer steps used by fib2 is linear in n. From exponential we are down to polynomial, a huge breakthrough in running time. It is now perfectly reasonable to compute F 200 or even F 200,000 . 1 As we will see repeatedly throughout this book, the right algorithm makes all the difference. More careful analysis In our discussion so far, we have been counting the number of basic computer steps executed by each algorithm and thinking of these basic steps as taking a constant amount of time. This is a very useful simplification. After all, a processor’s instruction set has a variety of basic primitives—branching, storing to memory, comparing numbers, simple arithmetic, and so on—and rather than distinguishing between these elementary operations, it is far more convenient to lump them together into one category. But looking back at our treatment of Fibonacci algorithms, we have been too liberal with what we consider a basic step. It is reasonable to treat addition as a single computer step if small numbers are being added, 32-bit numbers say. But the nth Fibonacci number is about 0.694n bits long, and this can far exceed 32 as n grows. Arithmetic operations on arbitrarily large numbers cannot possibly be performed in a single, constant-time step. We need to audit our earlier running time estimates and make them more honest. We will see in Chapter 1 that the addition of two n-bit numbers takes time roughly proportional to n; this is not too hard to understand if you think back to the grade-school procedure 1 To better appreciate the importance of this dichotomy between exponential and polynomial algorithms, the reader may want to peek ahead to the story of Sissa and Moore, in Chapter 8. 14
for addition, which works on one digit at a time. Thus fib1, which performs about F n additions, actually uses a number of basic steps roughly proportional to nF n . Likewise, the number of steps taken by fib2 is proportional to n 2 , still polynomial in n and therefore exponentially superior to fib1. This correction to the running time analysis does not diminish our breakthrough. But can we do even better than fib2? Indeed we can: see Exercise 0.4. 0.3 Big-O notation We’ve just seen how sloppiness in the analysis of running times can lead to an unacceptable level of inaccuracy in the result. But the opposite danger is also present: it is possible to be too precise. An insightful analysis is based on the right simplifications. Expressing running time in terms of basic computer steps is already a simplification. After all, the time taken by one such step depends crucially on the particular processor and even on details such as caching strategy (as a result of which the running time can differ subtly from one execution to the next). Accounting for these architecture-specific minutiae is a nightmarishly complex task and yields a result that does not generalize from one computer to the next. It therefore makes more sense to seek an uncluttered, machine-independent characterization of an algorithm’s efficiency. To this end, we will always express running time by counting the number of basic computer steps, as a function of the size of the input. And this simplification leads to another. Instead of reporting that an algorithm takes, say, 5n 3 + 4n + 3 steps on an input of size n, it is much simpler to leave out lower-order terms such as 4n and 3 (which become insignificant as n grows), and even the detail of the coefficient 5 in the leading term (computers will be five times faster in a few years anyway), and just say that the algorithm takes time O(n 3 ) (pronounced “big oh of n 3 ”). It is time to define this notation precisely. In what follows, think of f(n) and g(n) as the running times of two algorithms on inputs of size n. Let f(n) and g(n) be functions from positive integers to positive reals. We say f = O(g) (which means that “f grows no faster than g”) if there is a constant c > 0 such that f(n) ≤ c · g(n). Saying f = O(g) is a very loose analog of “f ≤ g.” It differs from the usual notion of ≤ because of the constant c, so that for instance 10n = O(n). This constant also allows us to disregard what happens for small values of n. For example, suppose we are choosing between two algorithms for a particular computational task. One takes f 1 (n) = n 2 steps, while the other takes f 2 (n) = 2n + 20 steps (Figure 0.2). Which is better? Well, this depends on the value of n. For n ≤ 5, f 1 is smaller; thereafter, f 2 is the clear winner. In this case, f 2 scales much better as n grows, and therefore it is superior. This superiority is captured by the big-O notation: f 2 = O(f 1 ), because f 2 (n) f 1 (n) = 2n + 20 n 2 ≤ 22 for all n; on the other hand, f 1 ≠ O(f 2 ), since the ratio f 1 (n)/f 2 (n) = n 2 /(2n + 20) can get arbitrarily large, and so no constant c will make the definition work. 15
Page 1: Algorithms Copyright c○2006 S. Da
Page 4 and 5: 4 Paths in graphs 109 4.1 Distances
Page 7 and 8: List of boxes Bases and logs . . .
Page 9 and 10: Preface This book evolved over the
Page 11 and 12: Chapter 0 Prologue Look around you.
Page 13: The first question is moot here, as
Page 17 and 18: 4. Likewise, any polynomial dominat
Page 19: and in general ( Fn ) = F n+1 ( ) n
Page 22 and 23: Bases and logs Naturally, there is
Page 24 and 25: Figure 1.1 Multiplication à la Fra
Page 26 and 27: Figure 1.3 Addition modulo 8. 6 0 0
Page 28 and 29: Figure 1.4 Modular exponentiation.
Page 30 and 31: Figure 1.6 A simple extension of Eu
Page 32 and 33: We say x is the multiplicative inve
Page 34 and 35: Figure 1.7 An algorithm for testing
Page 36 and 37: Figure 1.8 An algorithm for testing
Page 38 and 39: Randomized algorithms: a virtual ch
Page 40 and 41: Alice’s encryption function is th
Page 42 and 43: Figure 1.9 RSA. Bob chooses his pub
Page 44 and 45: There is nothing inherently wrong w
Page 46 and 47: Exercises 1.1. Show that in any bas
Page 48 and 49: the depth of the circuit or the lon
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
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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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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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