Algorithm Design

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742 Chapter 13 Randomized Algorithms separately. We have seen in Section 13.6 that dictionaries have a very efficient implementation using hashing, so abstracting out the dictionary operations allows us to treat the hashing as a "black box" and have the algorithm inherit an overall running time from whatever performance guarantee is satisfied by this hashing procedure. A concrete payoff of this is the following. It has been shown that with the right choice of hashing procedure (more powerful, and more complicated, than what we described in Section 13.6), one can make the underlying dictionary operations run in linear expected time as well, yielding an overall expected running time of O(n). Thus the randomized approach we describe here leads to an improvement over the running time of the divideand-conquer algorithm that we saw earlier. We will talk about the ideas that lead to this O(n) bound at the end of the section. It is worth remarking at the outset that randomization shows up for two independent reasons in this algorithm: the way in which the algorithm processes the input points will have a random component, regardless of how the dictionary data structure is implemented; and when the dictionary is implemented using hashing, this introduces an additional source of randomness as part of the hash-table operations. Expressing the running time via the number of dictionary operations allows us to cleanly separate the two uses of randomness. ~ The Problem Let us start by recalling the problem’s (very simple) statement. We are given n points in the plane, and we wish to find the pair that is closest together. As discussed in Chapter 5, this is one of the most basic geometric proximity problems, a topic with a wide range of applications. We wil! use the same notation as in our earlier discussion of the closestpair problem. We wil! denote the set of points by P = [Pl ..... Pn}, where Pi has coordinates (xi, y~); and for two points Pi, Pj E P, we use d(p~, pj) to denote the standard Euclidean distance between them. Our goal is to find the pair of points pi, pj that minimizes d(p~, pj). To simplify the discussion, we wi!l assume that the points are al! in the unit square: 0 < x~, y~ < 1 for all i = 1 ..... n. This is no loss of generality: in linear time, we can rescale all the x- and y-coordinates of the points so that they lie in a unit square, and .then we can translate them so that this unit square has its lower left corner: at the origin. L~ Designing the Algorithm The basic idea of the algorithm is very simple. We’ll consider the points in random order, and maintain a current value 3 for the closest pair as we process 13.7 Finding the Closest Pair of Points: A Randomized Approach the points in this order. When we get to a new point p, we look "in the vicinity" ofp to see if any of the previously considered points are at a distance less than 3 from p. If not, then the closest pair hasn’t changed, and we move on to the next point in the random order. If there is a point within a distance less than 3 from p, then the closest pair has changed, and we will need to update it. The challenge in turning this into an efficient algorithm is to figure out how to implement the task of looking for points in the vicinity ofp. It is here that the dictionary, data structure will come into play. We now begin making this more concrete. Let us assume for simplicity that the points in our random order are labeled Pl ..... Pn. The algorithm proceeds in stages; during each stage, the closest pair remains constant. The first stage starts by setting 3 = d(pl, P2), the distance of the first two points. The goal of a stage is to either verify that 3 is indeed the distance of the closest pair of points, or to find a pair of points Pi, Pj with d(Pi, pj) < 3. During a stage, we’ll gradually add points in the order Pl, P2 ..... Pn. The stage terminates when we reach a point Pi so that for some j < i, we have d(pi, pj) < 8. We then let 8 for the next stage be the closest distance found so far: 8 = min/:/
744 Chapter 13 Randomized Mgorithms (if p is involved in the clo~sest" i 5/2 I pair, then the other point l~s in a close subsquare. s_~ t5 T 8/2 Figure 13.2 Dividing the square into size $/2 subsquares. The point p lies in the subsquare Proof. If points p and q are in the same subsquare, then both coordinates Of the two. points differ by at most 3/2, and hence d(p, q)
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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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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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34 n= I0 n=30 n=50 = 100 = 1,000 10
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74 Chapter 3 Graphs 3.1 Basic Defin
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78 Chapter 3 Graphs Rooted trees ar
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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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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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we’ll just give a simple example
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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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284 Figure 6.18 The OPT values for
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54O Chapter 7 Network Flow define f
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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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Page 411 and 412: 796 Epilogue: Algorithms That Run F
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Page 417 and 418: 808 References L. R. Ford. Network
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Page 421 and 422: 816 Index Approximation algorithms
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Algorithm Design

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