Algorithm Design

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270 Chapter 6 Dynamic Programming 2 1 0 0 I 0 0 0 0 ,.....-- 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 Figure 6.11 The two-dimensional table of OPT values. The leftmost column and bottom row is always 0. The entry for OPT(i, w) is computed from the two other entries OPT(i -- I, w) and OPT(i -- 1, w -- wi), as indicated by the arrows. Using (6.8) one can immediately prove by induction that the returned n and available weight W. ~ Analyzing the <strong>Algorithm</strong> Recall the tabular picture we considered in Figure 6.5, associated with weighted interval scheduling, where we also showed the way in which the array M for that algorithm was iteratively filled in. For the algorithm we’ve iust designed, we can use a similar representation, but we need a twodimensional table, reflecting the two-dimensional array of subproblems that, is being built up. Figure 6.11 shows the building up of subproblems in this case: the value M[i, w] is computed from the two other values M[i - 1, w] and M[i - 1, u? -- wi]. As an example of this algorithm executing, consider an instance with weight limit W = 6, and n = 3 items of sizes w 1 = w2 = 2 and w 3 = 3. We find that the optimal value OPT(3, 6) = 5 (which we get by using the third item and one of the first two items). Figure 6.12 illustrates the way the algorithm fills in the two-dimensional table of OPT values row by row. Next we w/l! worry about the running time of this algorithm. As before .in the case of weighted interval scheduling, we are building up a table of solutions M, and we compute each of the values M[i, w] in O(1) time using the previous values. Thus the running time is proportional to the number of entries in the table. 3 2 1 0 6.4 Subset Sums and Knapsacks: Adding a Variable Knapsack size W - 6, items w 1 = 2, w 2 = 2, w 3 = 3 3 2 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 o 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Initial values Filling in values for i = 1 0 0 2 2 4 4 4 0 0 2 2 2 2 2 0 0 0 0 0 o 0 0 1 2 3 4 5 6 Filling in values for i = 2 1 0 0 0 2 3 4 5 5 0 o 2 2 4 4 4 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 1 2 3 4 5 6 Filling in values for i = 3 Figure 6.12 The iterations of the algorithm on a sample instance of the Subset Sum Problem. (6.9) The Subset-Sum(n, W) <strong>Algorithm</strong> correctly computes the optimal value of the problem, and runs in O(nW) time. Note that this method is not as efficient as our dynamic program for the Weighted Interval Scheduling Problem. Indeed, its running time is not a polynomial function of n; rather, it is a polynomial function of n and W, the largest integer involved in defining the problem. We cal! such algorithms pseudo-polynomial. Pseudo-polynomial algorithms can be reasonably efficient when the numbers {uai} involved in the input are reasonably small; however, they become less practical as these numbers grow large. To recover an optimal set S of items, we can trace back through the array M by a procedure similar to those we developed in the previous sections. (6.10) Given a table M of the optimal values of thesubproblems, the optimal set S can be found in O(n) time. Extension: The Knapsack Problem The Knapsack Problem is a bit more complex than the scheduling problem we discussed earlier. Consider a situation in which each item i has a normegative weight w i as before, and also a distinct value vi. Our goal is now to find a 271
272 Chapter 6 Dynamic Programming subset S of maximum value ~],i~s vi, subject to the restriction that the total weight of the set should not exceed W: ~i~s wi
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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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10 Chapter 1 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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Page 141 and 142: 256 Chapter 6 Dynamic Programming 6
Page 143 and 144: 260 Index 1 2 3 4 5 6 L~ I = 2 Chap
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Page 155 and 156: 284 Figure 6.18 The OPT values for
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Page 183 and 184: 54O Chapter 7 Network Flow define f
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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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440 Chapter 7 Network Flow going to
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Chapter 7 Network Flow 44. 45. Give
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448 Chapter 7 Network Flow 51. The
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452 Chapter 8 NP and Computational
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472 Chapter 8 NP and Computation!!
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5OO Chapter 8 NP and Computational
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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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58O 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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636 Chapter !! Approximation Algori
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640 Chapter 11 Approximation Mgofit
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656 Chapter 11 Approximation Mgorit
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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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696 Chapter 12 Local Search Of cour
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7OO Chapter 12 Loca! Search and for
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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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