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List of boxes Bases and logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Two’s complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Is your social security number a prime? . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Hey, that was group theory! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Carmichael numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Randomized <strong>algorithms</strong>: a virtual chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 37 An application of number theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Binary search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 An n log n lower bound for sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 The Unix sort command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Why multiply polynomials? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 The slow spread of a fast algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 How big is your graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Crawling fast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Which heap is best? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 A randomized algorithm for minimum cut . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Recursion? No, thanks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Programming? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Common subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Of mice and men . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Memoization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 On time and memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 A magic trick called duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Matrix-vector notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Visualizing duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Linear programming in polynomial time . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 The story of Sissa and Moore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7
Page 1: Algorithms Copyright c○2006 S. Da
Page 4 and 5: 4 Paths in graphs 109 4.1 Distances
Page 8 and 9: Why P and NP? . . . . . . . . . . .
Page 10 and 11: and excursions for the mathematical
Page 12 and 13: Since then, this decimal positional
Page 14 and 15: £¢¡ §¦ ¥¤ Figure 0.1 The pro
Page 16 and 17: Figure 0.2 Which running time is be
Page 18 and 19: Exercises 0.1. In each of the follo
Page 21 and 22: Chapter 1 Algorithms with numbers O
Page 23 and 24: of this sum gets computed in a fixe
Page 25 and 26: Figure 1.2 Division. function divid
Page 27 and 28: Taken together with the substitutio
Page 29 and 30: Figure 1.5 Euclid’s algorithm for
Page 31 and 32: Proof. The first thing to confirm i
Page 33 and 34: 1.3 Primality testing Is there some
Page 35 and 36: The set {1, 2, . . . , N − 1} b a
Page 37 and 38: long. What makes this task quite ea
Page 39 and 40: Alice and Bob are worried that the
Page 41 and 42: The security of AES has not been ri
Page 43 and 44: entries, the vast majority of them
Page 45 and 46: Let us now prove the preceding prop
Page 47 and 48: 1.15. Determine necessary and suffi
Page 49 and 50: (d) Unlike Fermat’s Little theore
Page 51 and 52: Chapter 2 Divide-and-conquer algori
Page 53 and 54: Figure 2.1 A divide-and-conquer alg
Page 55 and 56: Figure 2.3 Each problem of size n i
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Figure 2.4 The sequence of merge op
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2.4 Medians The median of a list of
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This follows by taking expected val
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2.6 The fast Fourier transform We h
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2.6.1 An alternative representation
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The original problem of size n is i
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Figure 2.7 The fast Fourier transfo
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Figure 2.8 The FFT takes points in
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Figure 2.9 The fast Fourier transfo
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¡ £¢ ¥¤ §¦ ©¨ #" %
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Exercises 2.1. Use the divide-and-c
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2.12. How many lines, as a function
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2.24. On page 62 there is a high-le
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(a) Show that the following rule is
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Figure 3.1 (a) A map and (b) its gr
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Figure 3.2 Exploring a graph is rat
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Figure 3.4 The result of explore(A)
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Figure 3.7 DFS on a directed graph.
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Figure 3.8 A directed acyclic graph
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Figure 3.9 (a) A directed graph and
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to successively output the second s
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(i) A (ii) B E C F D A B C D E F G
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3.17. Infinite paths. Let G = (V, E
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3.28. In the 2SAT problem, you are
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108
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Figure 4.2 A physical model of a gr
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Figure 4.5 Edge lengths often matte
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More generally, at any given moment
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Figure 4.9 A complete run of Dijkst
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Initialize dist(s) to 0, other dist
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4.5 Priority queue implementations
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Figure 4.12 Dijkstra’s algorithm
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Figure 4.14 The Bellman-Ford algori
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Exercises 4.1. Suppose Dijkstra’s
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Figure 4.16 Operations on a binary
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4.19. Generalized shortest-paths pr
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132
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Output: A tree T = (V, E ′ ), wit
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£¢ ¡ ©¨ Figure 5.2 T ∪ {e}
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Figure 5.4 Kruskal’s minimum span
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Figure 5.6 A sequence of disjoint-s
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compression) only touch the insides
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£¢ ¡ ©¨ Figure 5.8 Prim’s
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5.2 Huffman encoding In the MP3 aud
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f 1 + f 2 f 5 f 4 f 1 f 2 f 3 The l
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Entropy The annual county horse rac
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while there is an implication that
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e −x 1 − x 0 1 x Thus n t ≤ n
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5.7. Show how to find the maximum s
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(b) Now consider arbitrary distribu
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5.28. A prefix-free encoding of a f
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left-to-right order of Figure 6.1,
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same bookkeeping device we used for
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Figure 6.3 The subproblem E(7, 5).
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Figure 6.5 The underlying dag, and
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Of mice and men Our bodies are extr
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K(0) = 0 for w = 1 to W : K(w) = ma
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Figure 6.6 A × B × C × D = (A ×
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Figure 6.8 We want a path from s to
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Figure 6.9 The optimal traveling sa
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Figure 6.10 The largest independent
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Give an efficient algorithm to comp
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maximum sum of selling prices. You
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6.22. Give an O(nt) algorithm for t
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6.30. Reconstructing evolutionary t
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Figure 7.1 (a) The feasible region
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Figure 7.2 The feasible polyhedron
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How can we handle the fluctuations
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andwidth is exceeded and that each
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We will now show that these various
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Figure 7.4 (a) A network with edge
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c f : { cuv − f uv if (u, v) ∈
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Figure 7.6 The max-flow algorithm a
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Figure 7.7 An edge between two peop
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Figure 7.9 By design, dual feasible
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When the primal is the LP that expr
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to announce his or her strategy in
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Figure 7.12 A polyhedron defined by
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u y 2 y 1 x Specifically, if one of
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7.6.3 Loose ends There are several
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220
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Linear programming in polynomial ti
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Another parting thought: by what ot
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(b) Graph the feasible region, give
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ingredient energy protein fat carbo
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1 A 2 S 1 T 3 B 1 (a) Write the pro
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(b) Show that the fattest s − t p
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The story of Sissa and Moore Accord
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Figure 8.1 The optimal traveling sa
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This graph has multiple edges betwe
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further cut would then be needed to
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special case in which E consists of
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Why P and NP? Okay, P must stand fo
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Figure 8.6 The space NP of all sear
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RUDRATA (s, t)-PATH Instance: G = (
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It is usually cleaner to prove the
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p 1 b 0 g 0 p 0 p 2 g 1 b 1 p 3 Sup
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ZOE−→SUBSET SUM This is a reduc
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Figure 8.11 Reducing ZOE to RUDRATA
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Figure 8.12 A gadget for enforcing
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Figure 8.13 An instance of CIRCUIT
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Unsolvable problems At least an NP-
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(a) Show that if each literal appea
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EXPERIMENTAL CUISINE Input: n, the
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268
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Unfortunately, this approach does n
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Figure 9.1 Backtracking reveals tha
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Figure 9.2 (a) A graph and its opti
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Figure 9.3 A graph whose optimal ve
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Figure 9.5 Some data points and the
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By the triangle inequality, these b
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• Finally, there is another class
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Figure 9.7 (a) Nine American cities
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Figure 9.9 An instance of GRAPH PAR
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epeat randomly choose a solution s
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Figure 9.11 Simulated annealing. Th
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9.6. In the MINIMUM STEINER TREE pr
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294
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Figure 10.1 An electron can be in a
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Entanglement ∣ 〉 ∣ 〉 Suppos
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the hidden pattern. All this probab
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Input: A superposition of m = log M
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Let’s make this more precise. Lem
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FFT M (input: α 0 , . . . , α M
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Setting up a periodic superposition
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O(n 3 ) steps (as we saw in Section
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(d) Conclude therefore that for any
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Chapters 8 and 9 The notion of NP-c
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shortest path, 120, 157 disjoint se
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RSA cryptosystem, 39-40, 262 Rudrat
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