CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1....

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Instructor: Lilian de Greef Quarter: Summer 2017 CSE 373: Data Structures and Algorithms Lecture 11: Finish AVL Trees; Priority Queues; Binary Heaps

Transcript of CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1....

Page 1: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Instructor:LiliandeGreefQuarter:Summer2017

CSE373:DataStructuresandAlgorithmsLecture11:FinishAVLTrees;PriorityQueues;BinaryHeaps

Page 2: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Today

• Announcements• FinishAVLTrees• PriorityQueues• BinaryHeaps

Page 3: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Announcements

• ChangestoOfficeHours:fromnowon…• NomoreWednesdaymorning&shorterWednesdayafternoonofficehours• New Thursdayofficehours!• Kyle’sWednesdayhourisnow1:30-2:30pm• Ben’sofficehoursisnowThursday1:00-3:00pm

• AVLTreeHeight• Insection,computedthatminimum#nodesinAVLtreeofacertainheightisS(h)=1+S(h-1)+S(h-2)whereh=heightoftree• Postedaproofnexttotheselectureslidesonlineforyourperusal

Page 4: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Announcements

• Midterm• NextFriday!(atusualclasstime&location)• Everythingwecoverinclassuntilexamdateisfairgame(minusclearly-marked“bonusmaterial”).Thatincludesnextweek’smaterial!

• Today’shw3duedatedesignedtogiveyoutimetostudy.

• CourseFeedback• Headsup:officialUW-mediatedcoursefeedbacksessionforpartofWednesday• Alsowanttobetterunderstandananonymousconcernoncoursepacing→ Poll

Page 5: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

BacktoAVLTreesFinishinguplastcouplecasesforinsert,thenwrappingupBSTs

Page 6: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

TheAVLTreeDataStructure

AnAVLtreeisaself-balancingbinarysearchtree.

Structuralproperties1. Binarytreeproperty(sameasBST)2. Order property(sameasforBST)

3. Balancecondition:balanceofeverynodeisbetween-1and1

wherebalance(node)=height(node.left)– height(node.right)

Result:Worst-case depthisO(logn)

Page 7: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

SingleRotations

(FiguresbyMelissaO’Neill,reprintedwithherpermissiontoLilian)

Page 8: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Case#3:

(FiguresbyMelissaO’Neill,reprintedwithherpermissiontoLilian)

Page 9: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

ABetterLookatCase#3:

(FiguresbyMelissaO’Neill,reprintedwithherpermissiontoLilian)

Page 10: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Case#3:Right-LeftCase(afteronerotation)

(FiguresbyMelissaO’Neill,reprintedwithherpermissiontoLilian)

rightrotate

Page 11: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Case#3:Right-LeftCase(aftertworotations)

(FiguresbyMelissaO’Neill,reprintedwithherpermissiontoLilian)

rightrotate

leftrotate

Awaytorememberit:Movedtograndparent’sposition.PuteverythingelseintheironlylegalpositionsforaBST.

Page 12: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Practicetime!ExampleofCase#4

A)

5010

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42

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8042

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8042

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42

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8060

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B) C) D)

WhichofthefollowingistheupdatedAVLtreeafterinserting42?

StartingwiththisAVLtree:

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Practicetime!ExampleofCase#4

5010

8030

60 WhichofthefollowingistheupdatedAVLtreeafterinserting42?

StartingwiththisAVLtree:

Whatrotationsdidwedo?What’sthenameofthiscase?

Page 14: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

Insert,summarized

• InsertasinourgenericBST• Checkbackuppathforimbalance,whichwillbe1of4cases:

• Node’sleft-left grandchildistootall• Node’s left-rightgrandchildistootall• Node’sright-left grandchildistootall• Node’sright-rightgrandchildistootall

• Onlyoccursbecause• Aftertheappropriatesingleordoublerotation,thesmallest-unbalancedsubtreehasthesameheightasbeforetheinsertion• Soallancestorsarenowbalanced

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one case tree was balanced before insert

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AVLTreeEfficiency

• Worst-casecomplexityoffind:

• Worst-casecomplexityofinsert:

• Worst-casecomplexityofbuildTree:

Takessomemorerotationactiontohandledelete…

Page 16: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

ProsandConsofAVLTreesArgumentsforAVLtrees:

1. Alloperationslogarithmicworst-casebecausetreesarealways balanced2. Heightbalancingaddsnomorethanaconstantfactortothespeedof

insert anddelete

ArgumentsagainstAVLtrees:

1. Difficulttoprogram&debug[butdoneonceinalibrary!]2. Morespaceforheightfield3. Asymptoticallyfasterbutrebalancingtakesalittletime4. Ifamortized logarithmictimeisenough,usesplaytrees(alsointhetext,

notcoveredinthisclass)

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LotsofcoolSelf-BalancingBSTsoutthere!

Popularself-balancingBSTsinclude:• AVLtree• Splaytree• 2-3tree• AAtree• Red-blacktree• Scapegoattree• Treap(Fromhttps://en.wikipedia.org/wiki/Self-balancing_binary_search_tree#Implementations)

(Notcoveredinthisclass,butseveralareinthetextbookandallofthemareonline!)

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WrappingUp:HashTablevsBSTDictionary

HashTableadvantages:

BSTadvantages:• Cangetkeysinsortedorderwithoutmuchextraeffort• Samefororderingstatistics,findingclosestelements,rangequeries• Easiertoimplementifdon’thavehashfunction(whicharehard!).• CanguaranteeO(logn)worst-casetime,whereashashtablesareO(1)average time.

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PriorityQueueADTLikeaQueue,butwithprioritiesforeachelement.

Page 20: CSE 373: Data Structures and Algorithms · Pros and Cons of AVL Trees Arguments for AVL trees: 1. All operations logarithmic worst-case because trees are alwaysbalanced 2. Height

AnIntroductoryExample…

GillBates,theCEOofthesoftwarecompanyMillisoft,builtarobotsecretarytomanageherhundredsofemails.Duringtheday,Batesonlywantstolookatafewemailseverynowandthensoshecanstayfocused.Therobotsecretarysendsherthemostimportantemailseachtime.Todoso,heassignseachemailapriority whenhegetsthemandonlysendsBatesthehighestpriority emailsuponrequest.

Allofyourcomputerserversareonfire!Priority:

Here’sareminderforourmeetingin2months.Priority:

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PriorityQueueADTApriorityqueue holdscompare-abledata

• Likedictionaries,weneedtocompareitems• Givenx andy,isx lessthan,equalto,orgreaterthany• Meaningoftheorderingcandependonyourdata

• Thedatatypicallyhastwofields:• We’llnowuseintegersforexamples,butcanuseothertypes/objectsforprioritiestoo!

newemail highestpriorityemail

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Inourintroductoryexample:

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PriorityQueueADTMeaning:• Apriorityqueue holdscompare-abledata• Keyproperty:

Operations:

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PriorityQueue:Example

insert x1 withpriority5insert x2 withpriority3a = deleteMininsert x3 withpriority2insert x4 withpriority6c = deleteMind = deleteMin

Analogy:insert islikeenqueue,deleteMin islikedequeueButthewholepointistouseprioritiesinsteadofFIFO

insert deleteMin


Part-F2 AVL Trees

Part-F2 AVL Trees

Augmenting AVL trees

Augmenting AVL trees

AVL Trees II

AVL Trees II

Lec-14 AVL Trees Implementation

Lec-14 AVL Trees Implementation

Balanced BST and AVL Trees - University of Waterloocmoreno/ece250/2012-02-15--balanced_bst--AVL.pdfBalanced BST and AVL Trees Last time on this topic: Introduced AVL trees Discussed

Balanced BST and AVL Trees - University of Waterloocmoreno/ece250/2012-02-15--balanced_bst--AVL.pdfBalanced BST and AVL Trees Last time on this topic: Introduced AVL trees Discussed

AVL TREES

AVL TREES

1 Lecture 12 AVL Trees. 2 trees static dynamic game treessearch trees priority queues and heaps graphs binary search trees AVL trees 2-3 treestries Huffman.

1 Lecture 12 AVL Trees. 2 trees static dynamic game treessearch trees priority queues and heaps graphs binary search trees AVL trees 2-3 treestries Huffman.

AVL-Trees: Motivation

AVL-Trees: Motivation

AVL Search Trees

AVL Search Trees

Lec-13 AVL Trees

Lec-13 AVL Trees

Concurrent AVL Trees

Concurrent AVL Trees

Avl trees final

Avl trees final

s12 AVL trees - York University · AVL Trees (11.2, 11.3) EECS 2011 25 February 2020 2 AVL Trees • AVL trees are balanced. • An AVL Tree is a binary search tree such that for

s12 AVL trees - York University · AVL Trees (11.2, 11.3) EECS 2011 25 February 2020 2 AVL Trees • AVL trees are balanced. • An AVL Tree is a binary search tree such that for

Binary Trees, Binary Search Trees and AVL Trees

Binary Trees, Binary Search Trees and AVL Trees

Balanced Trees AVL Trees Red-Black Trees 2-3 Trees 2-3-4 Trees.

Balanced Trees AVL Trees Red-Black Trees 2-3 Trees 2-3-4 Trees.

Balanced Search Trees AVL Trees 2-3 Trees 2-4 Trees.

Balanced Search Trees AVL Trees 2-3 Trees 2-4 Trees.

Balanced Trees (AVL and RedBlack)

Balanced Trees (AVL and RedBlack)

TREES General trees Binary trees Binary search trees AVL trees Balanced and Threaded trees.

TREES General trees Binary trees Binary search trees AVL trees Balanced and Threaded trees.

Balanced Search Trees. 2-3 Trees 2-3-4 Trees Red-Black Trees AVL Trees.

Balanced Search Trees. 2-3 Trees 2-3-4 Trees Red-Black Trees AVL Trees.

Tree Balancing: AVL Trees

Tree Balancing: AVL Trees