Algorithm Design

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542 Chapter 9 PSPACE: A Class of Problems beyond NP Does this really reduce the space usage to a polynomial amount ~. We first write down the procedure carefully, and then analyze it. We will think of L as a power of 2, which it is for our purposes. Path(~1, ~2, L) If L= 1 then If there is an operator 0 converting ~i to ~2 then return ’yes’ ’ Else return echO,, Endif Else (L > l) Enumerate al! configurations (Y using an n-bit counter For each ~’ do the following: Compute x = Path(~l, ~’, [L/2]) Delete all intermediate work, saving only the return value x Compute y = Path(~’, (~2, [L/Z]) Delete all intermediate work, saving only the return value y If both x and y are equal to eyes’’, then return ~yes’’ Endfor If ~~yes’’ was not returned for any ~’ then Return ~no’’ Endif Endif Again, note that this procedure solves a generalization of our original question, which simply asked for Path(~0, ~*, 2n) ¯ This does mean, however, that we should remember to view L as an exponentially large parameter: log L = n. . ~ Analyzing the <strong>Algorithm</strong> The following claim therefore implies that Planning can be solved in polynomial space. i (9’8) Path(el, e2, L) returns ,’yes" i[ and only i]: there is a sequence o[ operators of length at most L leading from el to e 2. Its space usage is polynomial in n, k, and log L. Proof. The correctness follows by induction on L. It clearly holds when L = 1, since all operators are considered explicitly. Now consider a larger value of L. If there is a sequence of operators from e~ to e2, of length L’
544 Chapter 9 PSPACE: A Class of Problems beyond NP This basic PSPACE-complete problem can then serve as a good "root" from which to discover other PSPACE-complete problems. By strict analogy with the case of 3g~P, it’s easy to see from the definition that if a problem Y is PSPACEcomplete, and a problem X in PSPACE has the property that Y
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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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6 Chapter 1 Introduction: Some Repr
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10 Chapter 1 Introduction: Some Rep
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14 Chapter ! Introduction: Some Rep
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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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74 Chapter 3 Graphs 3.1 Basic Defin
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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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Before swapping: After swapping: d
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134 Chapter 4 Greedy Algorithms wit
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138 Chapter 4 Greedy Algorithms 4.4
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142 Chapter 4 Greedy Algorithms 4.5
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146 Chapter 4 Greedy Algorithms (e
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150 Chapter 4 Greedy Algorithms her
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154 Chapter 4 Greedy Algorithms we
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158 Chapter 4 Greedy Algorithms spa
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162 Chapter 4 Greedy Algorithms ~ T
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166 Chapter 4 Greedy Algorithms g2(
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170 Chapter 4 Greedy Algorithms Sha
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174 Chapter 4 Greedy Algorithms ...
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178 Chapter 4 Greedy Algorithms Fig
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182 Chapter 4 Greedy Algorithms The
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186 Chapter 4 Greedy Algorithms Sol
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192 Chapter 4 Greedy Algorithms 8.
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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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220 Chapter 5 Divide and Conquer mo
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224 Chapter 5 Divide and Conquer El
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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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244 Chapter 5 Divide and Conquer Th
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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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268 Chapter 6 Dynamic Programming t
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272 Chapter 6 Dynamic Programming s
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280 Chapter 6 Dynamic Programming t
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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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300 Chapter 6 Dynamic Programming p
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304 Chapter 6 Dynamic Programming e
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308 Chapter 6 Dynamic Programming o
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312 Chapter 6 Dynamic Programming E
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316 Chapter 6 Dynamic Programming T
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320 Chapter 6 Dynamic Programming (
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328 Chapter 6 Dynamic Programming T
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336 Chapter 6 Dynamic Programming h
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54O Chapter 7 Network Flow define f
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344 Chapter 7 Network Flow This aug
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348 Chapter 7 Network Flow Proof. ~
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352 Chapter 7 Network Flow In the n
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356 Chapter 7 Network Flow (7.19) T
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360 Chapter 7 Network Flow preflow
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364 Chapter 7 Network Flow Heights
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368 Chapter 7 Network Flow ~ The Pr
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372 Chapter 7 Network Flow that are
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376 Chapter 7 Network Flow I~low ar
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380 Chapter 7 Network Flow Figure 7
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384 Chapter 7 Network Flow Lower bo
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388 BOS 6 DCA 7 DCA 8 Chapter 7 Net
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392 Chapter 7 Network Flow natural
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396 Chapter 7 Network Flow 7.11 Pro
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400 Chapter 7 Network Flow 7.12 Bas
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404 Chapter 7 Network Flow to cross
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4O8 Chapter 7 Network Flow Why are
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412 Chapter 7 Network Flow ProoL Th
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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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644 Chapter 11 Approximation Mgofit
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648 Chapter 11 Approximation Algori
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652 Chapter 11 Approximation Algori
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656 Chapter 11 Approximation Mgorit
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660 Chapter 11 Approximation Mgofit
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664 Chapter 12 Local Search basic p
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668 Chapter 12 Local Search moves b
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672 Chapter 12 Local Search of all
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676 Chapter 12 Local Search 12.4 Ma
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680 Chapter 12 Local Search saw ear
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684 Chapter 12 Local Search utting
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688 Chapter 12 Local Search this di
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692 Chapter 12 Local Search sides a
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696 Chapter 12 Local Search Of cour
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7OO Chapter 12 Loca! Search and for
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7O4 Chapter 12 Local Search A previ
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708 Chapter 13 Randomized Algorithm
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724 Chapter 13 Randomized Mgorithms
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736 Chapter 13 Randomized Mgofithms
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792 Chapter !3 Randomized Algorithm
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796 Epilogue: Algorithms That Run F
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800 Epilogue: Algorithms That Run F
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8O4 Epilogue: Algorithms That Run F
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808 References L. R. Ford. Network
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812 References M. Mitzenmacher and
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816 Index Approximation algorithms
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820 Cushions in packet switching, 8
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824 Index Graphs (cont.) queues and
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828 Index Mapping genomes, 279, 521
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832 Index Index 833 Preflow-Push Ma
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836 Index Stale items in randomized
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Algorithm Design

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