Algorithm Design

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746 Chapter 13 Randomized Mgorithms Summary off the <strong>Algorithm</strong> We have now actually described the algorithm in full. To recap: Order the points in a random sequence Pl,Pz ..... Pn Let 8 denote the minimum distance found so far Initialize 8 : d(Pl, P2) Invoke MakeDictionarY for storing subsquares of side length For i=I, 2 ..... n: Determine the subsquare Sst containing Pi Look up the 25 subsquares close to Pi Compute the distance from Pi to any points found in these subsquares If there is a point p] (]
748 Chapter 13 Randomized Mgorithms Achieving Linear Expected Running Time up to this point, we have treated the dictionary data structure as a black box, and in 03.31) we bounded the running time of the algorithm in terms of computational time plus dictionary operations. We now want to give a bound on the actual expected running time, and so we need to analyze the work involved in performing these dictionary operations. To implement the dictionary, we’ll use a universal hashing scheme, like the one discussed in Section 13.6. Once the algorithm employs a hashing scheme, it is making use of randomness in two distinct ways: First, we randomly order the points to be added; and second, for each new minimum distance 8, we apply randomization to set up a new hash table using a universal hashing scheme. When inserting a new point Pi, the algorithm uses the hash-table Lookup operation to find all nodes in the 25 subsquares close to Pi. However, if the hash table has collisions, then these 25 Lookup operations can involve inspecting many more than 25 nodes. Statement (13.23) from Section ,13.6 shows that each such Lookup operation involves considering O(1) previously inserted points, in expectation. It seems intuitively clear that performing O(n) hash-table operations in expectation, each of which involves considering O(1) elements in expectation, will result in an expected running time of O(n) overall. To make this intuition precise, we need to be careful with how these two sources of randomness interact. (15.52) Assume we implement the randomized closest-pair algorithm using a universal hashing scheme. In expectation, the total number of points considered during the Lookup operations is bounded by O(n). Proof. From (13.31) we know that the expected number of Lookup operations is O(n), and from (13.23) we know that each of these Lookup operations involves considering only O(1) points in expectation. In order to conclude that this implies the expected number of points considered is O(n), we now consider the relationship between these two sources of randomness. Let X be a random variable denoting the number of Lookup operations performed by the algorithm. Now the random order a that the algorithm chooses for the points completely determines the sequence of minimumdistance values the algorithm will consider and the sequence of dictionary operations it will perform. As a result, the choice of a determines the value of X; we let X(a) denote this value, and we let ~a denote the event the algorithm chooses the random order or. Note that the conditional expectation E [X I ga] is equal to X(cr). Also, by (!3.31), we know that E [X] _< Co n, for some constant Co. 13.7 Finding the Closest Pair of Points: A Randomized Approach Now consider this sequence of Lookup operations for a fixed order or. For i = 1 ..... X(a), let Yi be the number of points that need to be inspected during the ith Lookup operations--namely, the number of previously inserted points that collide with the dictionary entry involved in this Lookup operation. We would like to bound the expected value V’x(~) of Yi, where expectation is over both the random choice of a and the random choice of hash function. By (13.23), we know that E [Y~ I g~] = O(1) for all cr and all values of i. It is useful to be able to refer to the constant in the expression O(1) here, so we will say that E [Yi I ~o] < cl for all cr and all values of i. Summing over all i, and using linearity of expectation, we get E [~i Yi I ~a] --< clX(cr)- NOW we have = q ~ E [X I g~] . Pr [g~] = qE [X] . Since we know that E [X] is at most con, the total expected number of points considered is at most Coqn = O(n), which proves the claim. [] Armed with this claim, we can use the universal hash functions from Section 13.6 in our closest-pair algorithm. In expectation, the algorithm will consider O(n) points during the Lookup operations. We have to set up multiple hash tables--a new one each time the minimum distance changes--and we have to compute O(n) hash-function values. All hash tables are set up for the same size, a prime p > n. We can select one prime and use the same table throughout the algorithm. Using this, we get the following bound on the runxfing time. (13.33) In expectation, the algorithm uses O(n) hash-function computations and O(n) additional time for finding the closest pair of points. Note the distinction between this statement and (13.31). There we counted each dictionary operation as a single, atomic step; here,, on the other hand, we’ve conceptually opened up the dictionary operations so as to account for the time incurred due to hash-table collisions and hash-function computations. Finally, consider the time needed for the O(n) hash-function computations. How fast is it to compute the value of a universal hash function h? The class of universal hash functions developed in Section 13.6 breaks numbers in our universe U into r ~ log N/log n smaller numbers of size O(log n) each, and 749
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Cornell University Boston San Franc
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Contents About the Authors Preface
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X Contents 9 10 Extending the Limit
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xiv Preface problems out in the wor
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Preface Chapters 2 and 3 also prese
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xxii Preface Preface xxiii Sue Whit
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2 Chapter 1 Introduction: Some Repr
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30 Chapter 2 Basics of Algorithm An
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34 n= I0 n=30 n=50 = 100 = 1,000 10
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78 Chapter 3 Graphs Rooted trees ar
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82 Chapter 3 Graphs Current compone
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86 Chapter 3 Graphs Proof. Suppose
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90 Chapter 3 Graphs Two of the simp
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94 Chapter 3 Graphs most ~u nv = 2m
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98 Chapter 3 Graphs It is important
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102 Chapter 3 Graphs ordering, that
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106 Chapter 3 Graphs We can already
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110 Chapter 3 Graphs Exercises 111
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Greedy A~gorithms In Wall Street, t
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118 Chapter 4 Greedy Mgofithms o In
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122 Chapter 4 Greedy Algorithms Our
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126 Chapter 4 Greedy Algorithms Len
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204 Chapter 4 Greedy Algorithms Not
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Chapter Divide artd Cortquer Divide
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212 Chapter 5 Divide and Conquer Th
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216 cn time plus recursive calls Ch
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228 Chapter 5 Divide and Conquer 5.
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232 Chapter 5 Divide and Conquer (a
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we’ll just give a simple example
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240 Chapter 5 Divide and Conquer po
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248 Chapter 5 Divide and Conquer It
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252 Chapter 6 Dynamic Programming b
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256 Chapter 6 Dynamic Programming 6
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260 Index 1 2 3 4 5 6 L~ I = 2 Chap
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264 Chapter 6 Dynamic Programming w
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284 Figure 6.18 The OPT values for
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288 Chapter 6 Dynamic Programming U
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292 (a) Figure 6.21 (a) With negati
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296 Chapter 6 Dynamic Programming i
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54O Chapter 7 Network Flow define f
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388 BOS 6 DCA 7 DCA 8 Chapter 7 Net
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416 Chapter 7 Network Flow Figure 7
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420 Chapter 7 Network Flow 11. Now
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424 Figure 7.29 An instance of Cove
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428 Chapter 7 Network Flow 22. This
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432 Chapter 7 Network Flow generall
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436 Chapter 7 Network Flow (a) (b)
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452 Chapter 8 NP and Computational
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472 Chapter 8 NP and Computation!!
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532 Chapter 9 PSPACE: A Class of Pr
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556 Chapter 10 Extending the Limits
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596 Figure 10.12 A triangulated cyc
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600 Chapter 11 Approximation Algori
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628 Chapter 11 Approximation Mgorit
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640 Chapter 11 Approximation Mgofit
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Page 411 and 412: 796 Epilogue: Algorithms That Run F
Page 413 and 414: 800 Epilogue: Algorithms That Run F
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Page 417 and 418: 808 References L. R. Ford. Network
Page 419 and 420: 812 References M. Mitzenmacher and
Page 421 and 422: 816 Index Approximation algorithms
Page 423 and 424: 820 Cushions in packet switching, 8
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Page 427 and 428: 828 Index Mapping genomes, 279, 521
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Algorithm Design

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