A Practical Introduction to Data Structures and Algorithm Analysis

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Sec. 2.9 Exercises 532.17 Prove by contradiction that the number of primes is infinite.2.18 (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.2.19 Prove that √ 2 is irrational.2.20 Explain whyn∑ n∑n−1∑i = (n − i + 1) = (n − i).i=1i=12.21 Prove Equation 2.2 using mathematical induction.2.22 Prove Equation 2.6 using mathematical induction.2.23 Prove Equation 2.7 using mathematical induction.2.24 Find a closed-form solution and prove (using induction) that your solution iscorrect for the summationn∑3 i .i=1i=02.25 Prove that the sum of the first n even numbers is n 2 + n(a) indirectly by assuming that the sum of the first n odd numbers is n 2 .(b) directly by mathematical induction.2.26 Give a closed-form formula for the summation ∑ nbetween 1 and n.2.27 Prove that Fib(n) < ( 5 3 )n .2.28 Prove, for n ≥ 1, thatn∑i 3 = n2 (n + 1) 2.42.29 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.(a) Prove the Pigeonhole Principle using proof by contradiction.(b) Prove the Pigeonhole Principle using mathematical induction.2.30 For this problem, you will consider arrangements of infinite lines in the planesuch that three or more lines never intersect at a single point.(a) Give a recurrence relation that expresses the number of regions formedby n lines, and explain why your recurrence is correct.
54 Chap. 2 Mathematical Preliminaries(b) Give the summation that results from expanding your recurrence.(c) Give a closed-form solution for the summation.2.31 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)/2.2.32 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) = 0.2.33 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) = 1.2.34 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.2.35 What is the total volume of your body in liters (or, if you prefer, gallons)?2.36 An art historian has a database of 20,000 full-screen color images.(a) About how much space will this require? How many CD-ROMs wouldbe required to store the database? (A CD-ROM holds about 600MBof data). Be sure to explain all assumptions you made to derive youranswer.(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 CD-ROM if the images are compressed?2.37 How many cubic miles of water flow out of the mouth of the MississippiRiver each day? DO NOT look up the answer or any supplemental facts. Besure to describe all assumptions made in arriving at your answer.
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xviPrefacemake the examples as clea
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xviiiPrefaceOne of the most importa
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xxPrefaceOthers at Prentice Hall wh
Page 22 and 23: 1Data Structures and AlgorithmsHow
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Sec. 4.1 Lists 103mentations, asser
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Sec. 4.1 Lists 105/** Array-based l
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Sec. 4.1 Lists 107Insert 23:1312 20
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Sec. 4.1 Lists 111// Linked list im
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Sec. 4.1 Lists 113curr... 23 12 ...
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Sec. 4.1 Lists 115// Singly linked
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Sec. 4.1 Lists 117determined size.
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Sec. 4.1 Lists 119pointer to each e
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Sec. 4.1 Lists 121// Insert "it" at
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Sec. 4.1 Lists 123curr... 20curr23
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Sec. 4.2 Stacks 125/** Stack ADT */
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Sec. 4.2 Stacks 127// Linked stack
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Sec. 4.2 Stacks 129Currptr β 1Curr
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Sec. 4.2 Stacks 131Example 4.3 The
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Sec. 4.3 Queues 133/** Queue ADT */
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Sec. 4.3 Queues 135frontfront20 512
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Sec. 4.3 Queues 137// Array-based q
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Sec. 4.4 Dictionaries 139enqueue pl
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Sec. 4.4 Dictionaries 141Method cle
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Sec. 4.4 Dictionaries 143// Contain
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Sec. 4.4 Dictionaries 145/** Dictio
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Sec. 4.5 Further Reading 1474.5 Fur
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Sec. 4.6 Exercises 149(a) The data
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Sec. 4.7 Projects 1514.3 Use singly
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5Binary TreesThe list representatio
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Sec. 5.1 Definitions and Properties
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Sec. 5.1 Definitions and Properties
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Sec. 5.2 Binary Tree Traversals 159
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Sec. 5.2 Binary Tree Traversals 161
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Sec. 5.3 Binary Tree Node Implement
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Sec. 5.4 Binary Search Trees 171Thi
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Sec. 5.4 Binary Search Trees 173120
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Sec. 5.4 Binary Search Trees 175/**
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Sec. 5.4 Binary Search Trees 177min
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Sec. 5.5 Heaps and Priority Queues
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Sec. 5.6 Huffman Coding Trees 189Le
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Sec. 5.6 Huffman Coding Trees 191St
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Sec. 5.6 Huffman Coding Trees 193/*
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Sec. 5.6 Huffman Coding Trees 195//
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Sec. 5.6 Huffman Coding Trees 197A
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Sec. 5.8 Exercises 199Many techniqu
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Sec. 5.8 Exercises 2015.19 Write a
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Sec. 5.9 Projects 203(d) The record
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6Non-Binary TreesMany organizations
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Sec. 6.1 General Tree Definitions a
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Sec. 6.3 General Tree Implementatio
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Sec. 6.4 K-ary Trees 221RRABABCDEFC
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Sec. 6.5 Sequential Tree Implementa
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Sec. 6.7 Exercises 2276.2 Write an
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Sec. 6.7 Exercises 229CAFBEHGDIFigu
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Sec. 6.8 Projects 231proved perform
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7Internal SortingWe sort many thing
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Sec. 7.2 Three Θ(n 2 ) Sorting Alg
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Sec. 7.3 Shellsort 24559 20 17 13 2
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Sec. 7.4 Mergesort 24736 20 17 13 2
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Sec. 7.5 Quicksort 249static
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Sec. 7.5 Quicksort 251static
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Sec. 7.5 Quicksort 253InitialPass 1
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Sec. 7.5 Quicksort 255The initial c
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Sec. 7.6 Heapsort 257and fast. The
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Sec. 7.7 Binsort and Radix Sort 259
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Sec. 7.8 An Empirical Comparison of
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Sec. 7.9 Lower Bounds for Sorting 2
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Sec. 7.10 Further Reading 271• A
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Sec. 7.11 Exercises 2737.7 Recall t
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Sec. 7.12 Projects 2757.16 (a) Devi
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Sec. 7.12 Projects 277classmates? W
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280 Chap. 8 File Processing and Ext
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318 Chap. 9 SearchingThis and the f
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320 Chap. 9 Searchingelement in L,
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322 Chap. 9 SearchingThis is equal
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324 Chap. 9 Searching9.2 Self-Organ
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326 Chap. 9 SearchingWhen a frequen
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328 Chap. 9 Searchingand the total
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330 Chap. 9 SearchingThe set differ
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334 Chap. 9 SearchingExample 9.6 A
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352 Chap. 9 SearchingBentley et al.
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354 Chap. 9 Searching(c) How many s
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356 Chap. 9 Searching9.5 Implement
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358 Chap. 10 Indexingfile is create
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PART IVAdvanced Data Structures387
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390 Chap. 11 Graphs04123147123(a)(b
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392 Chap. 11 Graphs0 1 2 3 40201 11
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394 Chap. 11 GraphsThe adjacency ma
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396 Chap. 11 Graphsedge list. Funct
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398 Chap. 11 Graphspublic int weigh
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400 Chap. 11 Graphs}// Store edge w
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402 Chap. 11 GraphsABABCCDDFFEE(a)(
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404 Chap. 11 Graphs// Breadth first
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406 Chap. 11 GraphsAInitial call to
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408 Chap. 11 Graphsstatic void tops
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410 Chap. 11 Graphs// Compute short
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412 Chap. 11 Graphs// Dijkstra’s
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414 Chap. 11 Graphs// Compute a min
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416 Chap. 11 GraphsMarkedVertices v
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418 Chap. 11 GraphsInitialA B C D E
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420 Chap. 11 Graphs2 511032041032 6
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422 Chap. 11 Graphstriangular matri
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424 Chap. 12 Lists and Arrays Revis
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PART VTheory of Algorithms479
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482 Chap. 14 Analysis Techniquesthi
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488 Chap. 14 Analysis Techniquesof
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15Lower BoundsHow do I know if I ha
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Sec. 15.2 Lower Bounds on Searching
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Sec. 15.3 Finding the Maximum Value
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Sec. 15.4 Adversarial Lower Bounds
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Sec. 15.4 Adversarial Lower Bounds
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Sec. 15.5 State Space Lower Bounds
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Sec. 15.5 State Space Lower Bounds
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Sec. 15.6 Finding the ith Best Elem
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Sec. 15.7 Optimal Sorting 525search
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Sec. 15.8 Further Reading 527Merge
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Sec. 15.9 Exercises 52915.10 Show t
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16Patterns of AlgorithmsThis chapte
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Sec. 16.2 Randomized Algorithms 537
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Sec. 16.3 Numerical Algorithms 545e
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Sec. 16.3 Numerical Algorithms 547d
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Sec. 16.3 Numerical Algorithms 549W
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Sec. 16.3 Numerical Algorithms 5511
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Sec. 16.3 Numerical Algorithms 553i
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Sec. 16.3 Numerical Algorithms 555b
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Sec. 16.6 Projects 55716.8 What is
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560 Chap. 17 Limits to Computations
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562 Chap. 17 Limits to Computationf
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592 Chap. 17 Limits to Computationt
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594 BIBLIOGRAPHY[BG00] Sara Baase a
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596 BIBLIOGRAPHY[LLKS85] E.L. Lawle
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598 BIBLIOGRAPHY[WMB99][Zei07]I.H.
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600 INDEXbest fit, see memory manag
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602 INDEXrelation, 27, 50estimating
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604 INDEXinteger representation, 5,
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606 INDEXpath compression, 215-216,
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608 INDEXterminology, 236-237spatia
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A Practical Introduction to Data Structures and Algorithm Analysis

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