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4.5 Priority queue implementations 4.5.1 Array The simplest implementation of a priority queue is as an unordered array of key values for all potential elements (the vertices of the graph, in the case of Dijkstra’s algorithm). Initially, these values are set to ∞. An insert or decreasekey is fast, because it just involves adjusting a key value, an O(1) operation. To deletemin, on the other hand, requires a linear-time scan of the list. 4.5.2 Binary heap Here elements are stored in a complete binary tree, namely, a binary tree in which each level is filled in from left to right, and must be full before the next level is started. In addition, a special ordering constraint is enforced: the key value of any node of the tree is less than or equal to that of its children. In particular, therefore, the root always contains the smallest element. See Figure 4.11(a) for an example. To insert, place the new element at the bottom of the tree (in the first available position), and let it “bubble up.” That is, if it is smaller than its parent, swap the two and repeat (Figure 4.11(b)–(d)). The number of swaps is at most the height of the tree, which is ⌊log 2 n⌋ when there are n elements. A decreasekey is similar, except that the element is already in the tree, so we let it bubble up from its current position. To deletemin, return the root value. To then remove this element from the heap, take the last node in the tree (in the rightmost position in the bottom row) and place it at the root. Let it “sift down”: if it is bigger than either child, swap it with the smaller child and repeat (Figure 4.11(e)–(g)). Again this takes O(log n) time. The regularity of a complete binary tree makes it easy to represent using an array. The tree nodes have a natural ordering: row by row, starting at the root and moving left to right within each row. If there are n nodes, this ordering specifies their positions 1, 2, . . . , n within the array. Moving up and down the tree is easily simulated on the array, using the fact that node number j has parent ⌊j/2⌋ and children 2j and 2j + 1 (Exercise 4.16). 4.5.3 d-ary heap A d-ary heap is identical to a binary heap, except that nodes have d children instead of just two. This reduces the height of a tree with n elements to Θ(log d n) = Θ((log n)/(log d)). Inserts are therefore speeded up by a factor of Θ(log d). Deletemin operations, however, take a little longer, namely O(d log d n) (do you see why?). The array representation of a binary heap is easily extended to the d-ary case. This time, node number j has parent ⌈(j − 1)/d⌉ and children {(j − 1)d + 2, . . . , min{n, (j − 1)d + d + 1}} (Exercise 4.16). 120
Figure 4.11 (a) A binary heap with 10 elements. Only the key values are shown. (b)–(d) The intermediate “bubble-up” steps in inserting an element with key 7. (e)–(g) The “sift-down” steps in a delete-min operation. (a) 3 (b) 3 10 5 10 5 11 12 6 8 11 12 6 8 15 20 13 15 20 13 7 (c) 3 (d) 3 10 5 7 5 11 7 6 8 11 10 6 8 15 20 13 12 15 20 13 12 (e) (f) 12 7 5 7 5 11 10 6 8 11 10 6 8 15 20 13 12 15 20 13 (g) 5 (h) 5 7 12 7 6 11 10 6 8 11 10 12 8 15 20 13 15 20 13 121
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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
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 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: Which heap is best? The running tim
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
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
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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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