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Sec. 2.9 Exercises 492.16 Write a recursive algorithm to print all of the subsets for the set of the first npositive integers.2.17 The Largest Common Factor (LCF) for two positive integers n and m isthe largest integer that divides both n and m evenly. LCF(n, m) is at leastone, and at most m, assuming that n ≥ m. Over two thousand years ago,Euclid provided an efficient algorithm based on the observation that, whenn mod m ≠ 0, LCF(n, m) = LCF(m, n mod m). Use this fact to write twoalgorithms to find the LCF for two positive integers. The first version shouldcompute the value iteratively. The second version should compute the valueusing recursion.2.18 Prove by contradiction that the number of primes is infinite.2.19 (a) Use induction to show that n 2 − n is always even.(b) Give a direct proof in one or two sentences that n 2 − n is always even.(c) Show that n 3 − n is always divisible by three.(d) Is n 5 − n aways divisible by 5? Explain your answer.2.20 Prove that √ 2 is irrational.2.21 Explain whyn∑ n∑n−1∑i = (n − i + 1) = (n − i).i=1i=12.22 Prove Equation 2.2 using mathematical induction.2.23 Prove Equation 2.6 using mathematical induction.2.24 Prove Equation 2.7 using mathematical induction.2.25 Find a closed-form solution and prove (using induction) that your solution iscorrect for the summationn∑3 i .i=1i=02.26 Prove that the sum of the first n even numbers is n 2 + n(a) by assuming that the sum of the first n odd numbers is n 2 .(b) by mathematical induction.2.27 Give a closed-form formula for the summation ∑ nbetween 1 and n.2.28 Prove that Fib(n) < ( 5 3 )n .2.29 Prove, for n ≥ 1, thatn∑i 3 = n2 (n + 1) 2.42.30 The following theorem is called the Pigeonhole Principle.i=1i=ai where a is an integerTheorem 2.10 When n + 1 pigeons roost in n holes, there must be somehole containing at least two pigeons.
50 Chap. 2 Mathematical Preliminaries(a) Prove the Pigeonhole Principle using proof by contradiction.(b) Prove the Pigeonhole Principle using mathematical induction.2.31 For this problem, you will consider arrangements of infinite lines in the planesuch that three or more lines never intersect at a single point and no two linesare parallel.(a) Give a recurrence relation that expresses the number of regions formedby n lines, and explain why your recurrence is correct.(b) Give the summation that results from expanding your recurrence.(c) Give a closed-form solution for the summation.2.32 Prove (using induction) that the recurrence T(n) = T(n − 1) + n; T(1) = 1has as its closed-form solution T(n) = n(n + 1)/2.2.33 Expand the following recurrence to help you find a closed-form solution, andthen use induction to prove your answer is correct.T(n) = 2T(n − 1) + 1 for n > 0; T(0) = 0.2.34 Expand the following recurrence to help you find a closed-form solution, andthen use induction to prove your answer is correct.T(n) = T(n − 1) + 3n + 1 for n > 0; T(0) = 1.2.35 Assume that an n-bit integer (represented by standard binary notation) takesany value in the range 0 to 2 n − 1 with equal probability.(a) For each bit position, what is the probability of its value being 1 andwhat is the probability of its value being 0?(b) What is the average number of “1” bits for an n-bit random number?(c) What is the expected value for the position of the leftmost “1” bit? Inother words, how many positions on average must we examine whenmoving from left to right before encountering a “1” bit? Show theappropriate summation.2.36 What is the total volume of your body in liters (or, if you prefer, gallons)?2.37 An art historian has a database of 20,000 full-screen color images.(a) About how much space will this require? How many CDs would berequired to store the database? (A CD holds about 600MB of data). Besure to explain all assumptions you made to derive your answer.(b) Now, assume that you have access to a good image compression techniquethat can store the images in only 1/10 of the space required foran uncompressed image. Will the entire database fit onto a single CDif the images are compressed?
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Data Structures and AlgorithmAnalys
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PrefaceWe study data structures so
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PrefacexvA sophomore-level class wh
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Sec. 4.1 Lists 101/** Singly linked
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Sec. 4.1 Lists 105/** Set curr at l
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Sec. 4.1 Lists 109/** Singly linked
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Sec. 4.1 Lists 111Array-based lists
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Sec. 4.1 Lists 115/** Insert "it" a
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Sec. 4.2 Stacks 117The principle be
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Sec. 4.2 Stacks 119/** Array-based
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Sec. 4.2 Stacks 121top1top2Figure 4
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Sec. 4.2 Stacks 123you might want t
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Sec. 4.3 Queues 125/** Queue ADT */
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Sec. 4.3 Queues 129/** Array-based
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Sec. 4.4 Dictionaries 137}/** @retu
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146 Chap. 5 Binary TreesABCDEFGHIFi
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148 Chap. 5 Binary TreesAny number
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150 Chap. 5 Binary Trees/** ADT for
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152 Chap. 5 Binary Treestends to be
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154 Chap. 5 Binary Trees5.3 Binary
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156 Chap. 5 Binary TreesABCDEFGHIFi
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158 Chap. 5 Binary Treesto its call
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160 Chap. 5 Binary Trees5.3.2 Space
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162 Chap. 5 Binary Trees012345 678
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164 Chap. 5 Binary Trees12037422442
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168 Chap. 5 Binary Treessubroot105
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170 Chap. 5 Binary Treesin the case
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172 Chap. 5 Binary Treessynonymous
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174 Chap. 5 Binary Trees/** Heapify
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176 Chap. 5 Binary TreesRH1H2Figure
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178 Chap. 5 Binary Trees5.6 Huffman
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180 Chap. 5 Binary TreesLetter C D
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182 Chap. 5 Binary Trees/** Huffman
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184 Chap. 5 Binary Trees/** Build a
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186 Chap. 5 Binary Treesa reverse p
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188 Chap. 5 Binary Trees18 bits, mo
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190 Chap. 5 Binary Trees5.12 Write
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192 Chap. 5 Binary Trees(c) What is
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194 Chap. 5 Binary Trees5.7 Complet
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196 Chap. 6 Non-Binary TreesRootRAn
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198 Chap. 6 Non-Binary TreesRABCD E
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200 Chap. 6 Non-Binary Trees/** Gen
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202 Chap. 6 Non-Binary Treesspond t
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204 Chap. 6 Non-Binary Treesditiona
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206 Chap. 6 Non-Binary Treeslence o
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208 Chap. 6 Non-Binary TreesR’RA
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210 Chap. 6 Non-Binary TreesRRABABC
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212 Chap. 6 Non-Binary Trees(a)(b)F
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214 Chap. 6 Non-Binary Treestwo chi
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216 Chap. 6 Non-Binary Treeschild v
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218 Chap. 6 Non-Binary TreesCAFBEHG
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7Internal SortingWe sort many thing
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Sec. 7.3 Shellsort 231Insertion Bub
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Sec. 7.10 Further Reading 257• Eq
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Sec. 7.11 Exercises 259algorithm to
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266 Chap. 8 File Processing and Ext
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302 Chap. 9 Searchingintroduced in
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304 Chap. 9 Searchingvalue in L gre
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306 Chap. 9 SearchingWe now show th
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308 Chap. 9 Searchingrecords by exp
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310 Chap. 9 SearchingThis is potent
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312 Chap. 9 Searchingand the total
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322 Chap. 9 Searching01HashTable100
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324 Chap. 9 Searching/** Insert rec
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328 Chap. 9 SearchingExample 9.9 Co
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330 Chap. 9 Searching/** Dictionary
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332 Chap. 9 SearchingThe cost for a
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334 Chap. 9 Searchingthat is not in
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10IndexingMany large-scale computin
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Sec. 10.1 Linear Indexing 343Linear
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PART IVAdvanced Data Structures369
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372 Chap. 11 Graphs04123147123(a)(b
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374 Chap. 11 Graphs0 1 2 3 40201 11
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376 Chap. 11 Graphstime. This is a
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378 Chap. 11 GraphsIt is reasonably
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380 Chap. 11 Graphs}/** @return an
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382 Chap. 11 Graphs}/** Set the wei
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384 Chap. 11 GraphsABABCCDDFFEE(a)(
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386 Chap. 11 Graphs/** Breadth firs
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388 Chap. 11 Graphs/** Recursive to
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390 Chap. 11 GraphsA103B2205D11C15E
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392 Chap. 11 Graphs/** Dijkstra’s
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394 Chap. 11 GraphsA7 5CB91 26ED 21
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396 Chap. 11 GraphsPrim’s algorit
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398 Chap. 11 GraphsInitialA B C D E
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400 Chap. 11 Graphs(a) IF an undire
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402 Chap. 11 Graphs11.8 Projects11.
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12Lists and Arrays RevisitedSimple
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Sec. 12.1 Multilists 407L1L2L3bcdL4
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Sec. 12.3 Memory Management 413/**
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Sec. 12.3 Memory Management 419tend
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13Advanced Tree StructuresThis chap
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14Analysis TechniquesOften it is ea
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Sec. 14.1 Summation Techniques 463w
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Sec. 14.3 Amortized Analysis 477(ch
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Sec. 14.4 Further Reading 479compar
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Sec. 14.5 Exercises 48114.14 For th
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Sec. 14.6 Projects 48314.24 Use mat
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486 Chap. 15 Lower Boundsthe concep
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488 Chap. 15 Lower Boundsat least o
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490 Chap. 15 Lower BoundsABCGDEFFig
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492 Chap. 15 Lower BoundsProof 1: T
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494 Chap. 15 Lower BoundsFigure 15.
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496 Chap. 15 Lower BoundsTo minimiz
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498 Chap. 15 Lower BoundsThis recur
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500 Chap. 15 Lower BoundsFigure 15.
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502 Chap. 15 Lower Boundsnot only t
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504 Chap. 15 Lower Bounds1234Figure
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506 Chap. 15 Lower Bounds(a) Assume
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16Patterns of AlgorithmsThis chapte
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Sec. 16.1 Dynamic Programming 513on
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Sec. 16.2 Randomized Algorithms 515
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Sec. 16.3 Numerical Algorithms 523
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Sec. 16.3 Numerical Algorithms 527B
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Sec. 16.3 Numerical Algorithms 529o
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Sec. 16.3 Numerical Algorithms 531o
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Sec. 16.6 Projects 533value depth5
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536 Chap. 17 Limits to ComputationT
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540 Chap. 17 Limits to ComputationS
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542 Chap. 17 Limits to Computationa
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544 Chap. 17 Limits to Computation3
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548 Chap. 17 Limits to Computationp
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550 Chap. 17 Limits to ComputationE
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552 Chap. 17 Limits to Computation1
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554 Chap. 17 Limits to ComputationM
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556 Chap. 17 Limits to Computationw
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558 Chap. 17 Limits to Computationx
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560 Chap. 17 Limits to Computationt
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562 Chap. 17 Limits to Computationm
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564 Chap. 17 Limits to Computation(
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Bibliography[AG06] Ken Arnold and J
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BIBLIOGRAPHY 569[GI91] Zvi Galil an
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BIBLIOGRAPHY 571[SJH93] Clifford A.
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Index80/20 rule, 309, 333abstract d
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INDEX 575image space, 430key space,
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INDEX 577building, 172-177for memor
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INDEX 579octree, 451Ω notation, 65
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INDEX 581comparing algorithms, 224-
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