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CSC212 Data Structure

- Section FG

Lecture17B-TreesandtheSetClass

Instructor:FengHUDepartmentofComputerScience

CityCollegeofNewYork

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Topics

• WhyB-Tree• Theproblemofanunbalancedtree

• TheB-TreeRules• TheSetClassADTwithB-Trees• SearchforanIteminaB-Tree• InsertanIteminaB-Tree (*)• RemoveaItemfromaB-Tree (*)

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TheproblemofanunbalancedBST

• MaximumdepthofaBSTwithnentires:n-1

❐An Example: Insert 1, 2, 3, 4, 5 in that order into a bag using a BST

❐Run BagTest!

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Worst-CaseTimesforBSTs

• Adding,deletingorsearchingforanentryinaBSTwithnentries isO(d)intheworstcase,wheredisthedepthoftheBST

• Sincedisnomorethann-1,theoperationsintheworstcaseis(n-1).• Conclusion:theworstcasetimefortheadd,deleteorsearchoperationofaBSTisO(n)

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Solutionstotheproblem

• Solution1• Periodicallybalancethesearchtree• Project10.9,page516

• Solution2• Aparticularkindoftree:B-Tree• proposed byBayer&McCreightin1972

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TheB-TreeBasics

• Similartoabinarysearchtree(BST)• wheretheimplementationrequirestheabilitytocomparetwoentriesviaaless-thanoperator(<)

•ButaB-treeisNOTaBST– infactitisnotevenabinarytree• B-treenodeshavemany(morethantwo)children

•Anotherimportantproperty• eachnodecontainsmorethanjustasingleentry

•Advantages:• Easytosearch,andnottoodeep

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Applications: bag andset

• TheDifference• twoormoreequalentriescanoccurmanytimesinabag, butnotinaset• C++STL:setandmultiset(=bag)

• TheB-TreeRulesforaSet• Wewilllookata“set formulation”oftheB-Treerules,butkeepinmindthata“bag formulation” isalsopossible

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TheB-TreeRules

• TheentriesinaB-treenode• B-tree Rule 1:Therootmayhaveasfewasoneentry(or0entryifnochildren);everyothernodehasatleastMINIMUMentries

• B-tree Rule 2:Themaximumnumber ofentriesinanode is2*MINIMUM.• B-tree Rule 3:TheentriesofeachB-treenodearestoredinapartiallyfilledarray,sortedfromthesmallesttothelargest.

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TheB-TreeRules(cont.)

• ThesubtreesbelowaB-treenode• B-tree Rule 4:Thenumber ofthesubtreesbelowanon-leafnodewithnentriesisalwaysn+1

• B-tree Rule 5:Foranynon-leafnode:• (a).Anentryatindex iisgreater thanalltheentries insubtreenumberiofthenode• (b)Anentryatindex iislessthanalltheentries insubtreenumberi+1ofthenode

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AnExampleofB-Tree

93 and 107

subtree number 0 subtree

number 1

subtree number 2

[0] [1]

each entry < 93 each entry

Î (93,107)each entry > 107

What kind traversal can print a sorted list?

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TheB-TreeRules(cont.)

• AB-treeisbalanced• B-tree Rule 6:EveryleafinaB-treehasthesamedepth

• ThisruleensuresthataB-tree isbalanced

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AnotherExample,MINIMUM=1

Can you verify that all 6 rules are satisfied?

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10531

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Theset ADTwithaB-Tree

• Combine fixedsizearraywithlinkednodes

• data[]• *subset[]

• numberofentriesvary

• data_count• upto200!

• numberofchildrenvary

• child_count• =data_count+1?

set.h (p 528-529) template <class Item>class set{public:

... ...bool insert(const Item& entry);std::size_t erase(const Item& target);std::size_t count(const Item& target) const;

private:// MEMBER CONSTANTSstatic const std::size_t MINIMUM = 200;static const std::size_t MAXIMUM = 2 * MINIMUM;// MEMBER VARIABLESstd::size_t data_count;Item data[MAXIMUM+1]; // why +1? -for insert/erasestd::size_t child_count;set *subset[MAXIMUM+2]; // why +2? - one more

};

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Invariantfortheset Class

• TheentriesofasetisstoredinaB-tree,satisfyingthesixB-treerules.

• Thenumberofentriesinanodeisstoredindata_count,andtheentriesarestoredindata[0]throughdata[data_count-1]

• Thenumberofsubtreesofanodeisstoredinchild_count,andthesubtreesarepointedbysetpointerssubset[0]throughsubset[child_count-1]

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SearchforaIteminaB-Tree

• Prototype:• std::size_tcount(const Item&target)const;

• Post-condition:• Returnsthenumber ofitemsequaltothetarget• (either0or1foraset).

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SearchingforanItem:count

Startattheroot.1) locateiso

that!(data[i]<target)2) If(data[i]istarget)

return1;elseif(nochildren)

return0;else

returnsubset[i]->count(target);

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search for 10: cout << count (10);

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SearchingforanItem:count

Startattheroot.1) locateiso

that!(data[i]<target)2) If(data[i]istarget)

return1;elseif(nochildren)

return0;else

returnsubset[i]->count(target);

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search for 10: cout << count (10);i = 1

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SearchingforanItem:count

Startattheroot.1) locateiso

that!(data[i]<target)2) If(data[i]istarget)

return1;elseif(nochildren)

return0;else

returnsubset[i]->count(target);

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search for 10: cout << count (10);

i = 1

subset[1]

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SearchingforanItem:count

Startattheroot.1) locateiso

that!(data[i]<target)2) If(data[i]istarget)

return1;elseif(nochildren)

return0;else

returnsubset[i]->count(target);

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search for 10: cout << count (10);

i = 0

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SearchingforanItem:count

Startattheroot.1) locateiso

that!(data[i]<target)2) If(data[i]istarget)

return1;elseif(nochildren)

return0;else

returnsubset[i]->count(target);

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search for 10: cout << count (10);

i = 0

subset[0]

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SearchingforanItem:count

Startattheroot.1) locateiso

that!(data[i]<target)2) If(data[i]istarget)

return1;elseif(nochildren)

return0;else

returnsubset[i]->count(target);

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search for 10: cout << count (10);

i = 0data[i] is target !

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InsertaItemintoaB-Tree

• Prototype:• bool insert(const Item&entry);

• Post-condition:• Ifanequalentrywasalreadyintheset,thesetisunchangedandthereturnvalueisfalse.

• Otherwise,entrywasaddedtothesetandthereturnvalueistrue.

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InsertanIteminaB-Tree

Startattheroot.1) locateiso

that!(data[i]<entry)2) If(data[i]isentry)

returnfalse;//nowork!elseif(nochildren)

insertentryati;returntrue;

elsereturnsubset[i]->insert(entry);

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insert (11);

i = 0data[i] is target !

i = 0

i = 1

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InsertanIteminaB-Tree

Startattheroot.1) locateiso

that!(data[i]<entry)2) If(data[i]isentry)

returnfalse;//nowork!elseif(nochildren)

insertentryati;returntrue;

elsereturnsubset[i]->insert(entry);

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insert (11); // MIN = 1 -> MAX = 2

i = 1data[0] < entry !

i = 0

i = 1

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InsertanIteminaB-Tree

Startattheroot.1) locateiso

that!(data[i]<entry)2) If(data[i]isentry)

returnfalse;//nowork!elseif(nochildren)

insertentryati;returntrue;

elsereturnsubset[i]->insert(entry);

insert (11); // MIN = 1 -> MAX = 2

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i = 1put entry in data[1]

i = 0

i = 1

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InsertanIteminaB-Tree

Startattheroot.1) locateiso

that!(data[i]<entry)2) If(data[i]isentry)

returnfalse;//nowork!elseif(nochildren)

insertentryati;returntrue;

elsereturnsubset[i]->insert(entry);

insert (1); // MIN = 1 -> MAX = 2

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i = 0=> put entry in data[0]

i = 0

i = 0

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InsertanIteminaB-Tree

Startattheroot.1) locateiso

that!(data[i]<entry)2) If(data[i]isentry)

returnfalse;//nowork!elseif(nochildren)

insertentryati;returntrue;

elsereturnsubset[i]->insert(entry);

insert (1); // MIN = 1 -> MAX = 2

a node has MAX+1 = 3 entries!

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i = 0

i = 0

i = 0

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InsertanIteminaB-Tree

FixthenodewithMAX+1entries

❶ splitthenodeintotwofromthemiddle

❶ movethemiddleentryup

insert (1); // MIN = 1 -> MAX = 2

a node has MAX+1 = 3 entries!

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InsertanIteminaB-Tree

FixthenodewithMAX+1entries

❶ splitthenodeintotwofromthemiddle

❶ movethemiddleentryup

insert (1); // MIN = 1 -> MAX = 2

Note: This shall be done recursively... the recursive function returns the middle entry to the root of the subset.

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InsertinganItemintoaB-Tree

• WhatifthenodealreadyhaveMAXIMUMnumberofitems?• Solution– loose insertion (p551– 557)

• AlooseinsertmayresultsinMAX+1entriesintherootofasubset• Twostepstofixtheproblem:

• fixit– buttheproblemmaymovetotherootoftheset• fixtherootoftheset

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ErasinganItemfromaB-Tree

• Prototype:• std::size_terase(constItem&target);

• Post-Condition:• Iftargetwasintheset,thenithasbeenremovedfromthesetandthereturnvalueis1.

• Otherwisethesetisunchangedandthereturnvalueiszero.

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ErasinganItemfromaB-Tree

• Similarly,after“looseerase”,therootofasubsetmayjusthaveMINIMUM–1entries

• Solution:(p557– 562)• Fixtheshortage ofthesubsetroot– butthismaymovetheproblemtotherootoftheentireset

• Fixtheroot oftheentireset(tree)

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Summary

•AB-treeisatreeforsortingentriesfollowingthesixrules

•B-Treeisbalanced- everyleafinaB-treehasthesamedepth

•Adding,erasingandsearchinganiteminaB-treehaveworst-casetimeO(logn),wherenisthenumberofentries

•HowevertheimplementationofaddinganderasinganiteminaB-treeisnotatrivialtask.