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Sec. 7.2 Three Θ(n 2 ) Sorting Algorithms 227of comparisons will be n − 1, which is the number of times the outer for loopexecutes. Thus, the cost for Insertion Sort in the best case is Θ(n).While the best case is significantly faster than the worst case, the worst caseis usually a more reliable indication of the “typical” running time. However, thereare situations where we can expect the input to be in sorted or nearly sorted order.One example is when an already sorted list is slightly disordered by a small numberof additions to the list; restoring sorted order using Insertion Sort might be a goodidea if we know that the disordering is slight. Examples of algorithms that take advantageof Insertion Sort’s near-best-case running time are the Shellsort algorithmof Section 7.3 and the Quicksort algorithm of Section 7.5.What is the average-case cost of Insertion Sort? When record i is processed,the number of times through the inner for loop depends on how far “out of order”the record is. In particular, the inner for loop is executed once for each key greaterthan the key of record i that appears in array positions 0 through i−1. For example,in the leftmost column of Figure 7.1 the value 15 is preceded by five values greaterthan 15. Each such occurrence is called an inversion. The number of inversions(i.e., the number of values greater than a given value that occur prior to it in thearray) will determine the number of comparisons and swaps that must take place.We need to determine what the average number of inversions will be for the recordin position i. We expect on average that half of the keys in the first i − 1 arraypositions will have a value greater than that of the key at position i. Thus, theaverage case should be about half the cost of the worst case, or around n 2 /4, whichis still Θ(n 2 ). So, the average case is no better than the worst case in asymptoticcomplexity.Counting comparisons or swaps yields similar results. Each time through theinner for loop yields both a comparison and a swap, except the last (i.e., thecomparison that fails the inner for loop’s test), which has no swap. Thus, thenumber of swaps for the entire sort operation is n − 1 less than the number ofcomparisons. This is 0 in the best case, and Θ(n 2 ) in the average and worst cases.7.2.2 Bubble SortOur next sorting algorithm is called Bubble Sort. Bubble Sort is often taught tonovice programmers in introductory computer science courses. This is unfortunate,because Bubble Sort has no redeeming features whatsoever. It is a relatively slowsort, it is no easier to understand than Insertion Sort, it does not correspond to anyintuitive counterpart in “everyday” use, and it has a poor best-case running time.However, Bubble Sort can serve as the inspiration for a better sorting algorithm thatwill be presented in Section 7.2.3.Bubble Sort consists of a simple double for loop. The first iteration of theinner for loop moves through the record array from bottom to top, comparingadjacent keys. If the lower-indexed key’s value is greater than its higher-indexed
228 Chap. 7 Internal Sorting4220171328142315i=0 1 2 3 4 5 61342201714281513144220171528231314154220172328131415174220232813141517204223281314151720234223 281314151720232842Figure 7.2 An illustration of Bubble Sort. Each column shows the array afterthe iteration with the indicated value of i in the outer for loop. Values above theline in each column have been sorted. Arrows indicate the swaps that take placeduring a given iteration.neighbor, then the two values are swapped. Once the smallest value is encountered,this process will cause it to “bubble” up to the top of the array. The second passthrough the array repeats this process. However, because we know that the smallestvalue reached the top of the array on the first pass, there is no need to comparethe top two elements on the second pass. Likewise, each succeeding pass throughthe array compares adjacent elements, looking at one less value than the precedingpass. Figure 7.2 illustrates Bubble Sort. A Java implementation is as follows:static
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Data Structures and AlgorithmAnalys
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viContents6.1 General Tree Definiti
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viiiContents10.5.2 B-Tree Analysis
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xContents15.7 Optimal Sorting 50115
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PrefaceWe study data structures so
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PrefacexvA sophomore-level class wh
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Prefacexviithe form ofcalls to meth
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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. 1.3 Design Patterns 13that tra
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Sec. 1.3 Design Patterns 15will cal
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Sec. 1.4 Problems, Algorithms, and
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Sec. 1.5 Further Reading 19vides po
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Sec. 1.6 Exercises 21than 10,000 of
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2Mathematical PreliminariesThis cha
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Sec. 2.1 Sets and Relations 25A seq
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Sec. 2.2 Miscellaneous Notation 27E
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Sec. 2.3 Logarithms 29Unfortunately
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Sec. 2.8 Further Reading 45one inch
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Sec. 2.9 Exercises 512.38 How many
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3Algorithm AnalysisHow long will it
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Sec. 3.1 Introduction 55efficient.
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Sec. 3.1 Introduction 57n! 2 n2n 25
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Sec. 3.13 Exercises 853.13 Exercise
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4Lists, Stacks, and QueuesIf your p
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Sec. 4.1 Lists 95list ADT. Interfac
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Sec. 4.1 Lists 97for (L.moveToStart
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Sec. 4.1 Lists 99public void moveTo
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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 111Array-based lists
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Sec. 4.1 Lists 115/** Insert "it" a
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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 127frontfront20 512
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Sec. 4.3 Queues 129/** Array-based
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Sec. 4.4 Dictionaries 1314.3.3 Comp
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Sec. 4.4 Dictionaries 133not get at
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Sec. 4.4 Dictionaries 135/** Contai
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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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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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302 Chap. 9 Searchingintroduced in
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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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322 Chap. 9 Searching01HashTable100
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324 Chap. 9 Searching/** Insert rec
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332 Chap. 9 SearchingThe cost for a
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10IndexingMany large-scale computin
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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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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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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. 16.3 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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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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