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Sec. 3.13 Exercises 853.13 Exercises3.1 For each of the six expressions of Figure 3.1, give the range of values of nfor which that expression is most efficient.3.2 Graph the following expressions. For each expression, state the range ofvalues of n for which that expression is the most efficient.4n 2 log 3 n 3 n 20n 2 log 2 n n 2/33.3 Arrange the following expressions by growth rate from slowest to fastest.4n 2 log 3 n n! 3 n 20n 2 log 2 n n 2/3See Stirling’s approximation in Section 2.2 for help in classifying n!.3.4 (a) Suppose that a particular algorithm has time complexity T(n) = 3 ×2 n , and that executing an implementation of it on a particular machinetakes t seconds for n inputs. Now suppose that we are presented with amachine that is 64 times as fast. How many inputs could we process onthe new machine in t seconds?(b) Suppose that another algorithm has time complexity T(n) = n 2 , andthat executing an implementation of it on a particular machine takest seconds for n inputs. Now suppose that we are presented with a machinethat is 64 times as fast. How many inputs could we process onthe new machine in t seconds?(c) A third algorithm has time complexity T(n) = 8n. Executing an implementationof it on a particular machine takes t seconds for n inputs.Given a new machine that is 64 times as fast, how many inputs couldwe process in t seconds?3.5 Hardware vendor XYZ Corp. claims that their latest computer will run 100times faster than that of their competitor, Prunes, Inc. If the Prunes, Inc.computer can execute a program on input of size n in one hour, what sizeinput can XYZ’s computer execute in one hour for each algorithm with thefollowing growth rate equations?n n 2 n 3 2 n3.6 (a) Find a growth rate that squares the run time when we double the inputsize. That is, if T(n) = X, then T(2n) = x 2(b) Find a growth rate that cubes the run time when we double the inputsize. That is, if T(n) = X, then T(2n) = x 33.7 Using the definition of big-Oh, show that 1 is in O(1) and that 1 is in O(n).3.8 Using the definitions of big-Oh and Ω, find the upper and lower bounds forthe following expressions. Be sure to state appropriate values for c and n 0 .
86 Chap. 3 Algorithm Analysis(a) c 1 n(b) c 2 n 3 + c 3(c) c 4 n log n + c 5 n(d) c 6 2 n + c 7 n 63.9 (a) What is the smallest integer k such that √ n = O(n k )?(b) What is the smallest integer k such that n log n = O(n k )?3.10 (a) Is 2n = Θ(3n)? Explain why or why not.(b) Is 2 n = Θ(3 n )? Explain why or why not.3.11 For each of the following pairs of functions, either f(n) is in O(g(n)), f(n)is in Ω(g(n)), or f(n) = Θ(g(n)). For each pair, determine which relationshipis correct. Justify your answer, using the method of limits discussed inSection 3.4.5.(a) f(n) = log n 2 ; g(n) = log n + 5.(b) f(n) = √ n; g(n) = log n 2 .(c) f(n) = log 2 n; g(n) = log n.(d) f(n) = n; g(n) = log 2 n.(e) f(n) = n log n + n; g(n) = log n.(f) f(n) = log n 2 ; g(n) = (log n) 2 .(g) f(n) = 10; g(n) = log 10.(h) f(n) = 2 n ; g(n) = 10n 2 .(i) f(n) = 2 n ; g(n) = n log n.(j) f(n) = 2 n ; g(n) = 3 n .(k) f(n) = 2 n ; g(n) = n n .3.12 Determine Θ for the following code fragments in the average case. Assumethat all variables are of type int.(a) a = b + c;d = a + e;(b) sum = 0;for (i=0; i
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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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PrefacexixFor the second edition, I
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1Data Structures and AlgorithmsHow
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Sec. 1.1 A Philosophy of Data Struc
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Sec. 1.1 A Philosophy of Data Struc
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Sec. 1.2 Abstract Data Types and Da
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Sec. 1.5 Further Reading 19vides po
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2Mathematical PreliminariesThis cha
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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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204 Chap. 6 Non-Binary Treesditiona
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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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7Internal SortingWe sort many thing
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Sec. 7.3 Shellsort 231Insertion Bub
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302 Chap. 9 Searchingintroduced in
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308 Chap. 9 Searchingrecords by exp
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10IndexingMany large-scale computin
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Sec. 10.1 Linear Indexing 343Linear
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Sec. 10.4 2-3 Trees 351/** 2-3 tree
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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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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. 12.3 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. 14.4 Further Reading 479compar
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Sec. 14.5 Exercises 48114.14 For th
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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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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. 16.2 Randomized Algorithms 515
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Sec. 16.3 Numerical Algorithms 523
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Sec. 16.6 Projects 533value depth5
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536 Chap. 17 Limits to ComputationT
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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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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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