计算机科学外文翻译.docx

计算机科学外文翻译.docx

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计算机科学外文翻译

Binomialheap

Incomputerscience,abinomialheapisaheapsimilartoabinaryheapbutalsosupportsquickmergingoftwoheaps.Thisisachievedbyusingaspecialtreestructure.Itisimportantasanimplementationofthemergeableheapabstractdatatype（alsocalledmeldableheap）,whichisapriorityqueuesupportingmergeoperation.

Binomialtree

Abinomialheapisimplementedasacollectionofbinomialtrees（comparewithabinaryheap,whichhasashapeofasinglebinarytree）.Abinomialtreeisdefinedrecursively:

∙Abinomialtreeoforder0isasinglenode

∙Abinomialtreeoforderkhasarootnodewhosechildrenarerootsofbinomialtreesofordersk−1,k−2,...,2,1,0（inthisorder）.

Binomialtreesoforder0to3:

Eachtreehasarootnodewithsubtreesofalllowerorderedbinomialtrees,whichhavebeenhighlighted.Forexample,theorder3binomialtreeisconnectedtoanorder2,1,and0（highlightedasblue,greenandredrespectively）binomialtree.

Abinomialtreeoforderkhas2knodes,heightk.

Becauseofitsuniquestructure,abinomialtreeoforderkcanbeconstructedfromtwotreesoforderk−1triviallybyattachingoneofthemastheleftmostchildofrootoftheotherone.Thisfeatureiscentraltothemergeoperationofabinomialheap,whichisitsmajoradvantageoverotherconventionalheaps.

Thenamecomesfromtheshape:

abinomialtreeoforder

has

nodesatdepth

.

Structureofabinomialheap

Abinomialheapisimplementedasasetofbinomialtreesthatsatisfythebinomialheapproperties:

∙Eachbinomialtreeinaheapobeystheminimum-heapproperty:

thekeyofanodeisgreaterthanorequaltothekeyofitsparent.

∙Therecanonlybeeitheroneorzerobinomialtreesforeachorder,includingzeroorder.

Thefirstpropertyensuresthattherootofeachbinomialtreecontainsthesmallestkeyinthetree,whichappliestotheentireheap.

Thesecondpropertyimpliesthatabinomialheapwithnnodesconsistsofatmostlogn+1binomialtrees.Infact,thenumberandordersofthesetreesareuniquelydeterminedbythenumberofnodesn:

eachbinomialtreecorrespondstoonedigitinthebinaryrepresentationofnumbern.Forexamplenumber13is1101inbinary,

andthusabinomialheapwith13nodeswillconsistofthreebinomialtreesoforders3,2,and0（seefigurebelow）.

Exampleofabinomialheapcontaining13nodeswithdistinctkeys.

Theheapconsistsofthreebinomialtreeswithorders0,2,and3.

Implementation

Becausenooperationrequiresrandomaccesstotherootnodesofthebinomialtrees,therootsofthebinomialtreescanbestoredinalinkedlist,orderedbyincreasingorderofthetree.

Merge

Asmentionedabove,thesimplestandmostimportantoperationisthemergingoftwobinomialtreesofthesameorderwithintwobinomialheaps.Duetothestructureofbinomialtrees,theycanbemergedtrivially.Astheirrootnodeisthesmallestelementwithinthetree,bycomparingthetwokeys,thesmallerofthemistheminimumkey,andbecomesthenewrootnode.Thentheothertreebecomeasubtreeofthecombinedtree.Thisoperationisbasictothecompletemergingoftwobinomialheaps.

functionmergeTree（p,q）

ifp.root.key<=q.root.key

returnp.addSubTree（q）

else

returnq.addSubTree（p）

Tomergetwobinomialtreesofthesameorder,firstcomparetherootkey.Since7>3,theblacktreeontheleft（withrootnode7）isattachedtothegreytreeontheright（withrootnode3）asasubtree.Theresultisatreeoforder3.

Theoperationofmergingtwoheapsisperhapsthemostinterestingandcanbeusedasasubroutineinmostotheroperations.Thelistsofrootsofbothheapsaretraversedsimultaneously,similarlyasinthemergealgorithm

Ifonlyoneoftheheapscontainsatreeoforderj,thistreeismovedtothemergedheap.Ifbothheapscontainatreeoforderj,thetwotreesaremergedtoonetreeoforderj+1sothattheminimum-heappropertyissatisfied.Notethatitmaylaterbenecessarytomergethistreewithsomeothertreeoforderj+1presentinoneoftheheaps.Inthecourseofthealgorithm,weneedtoexamineatmostthreetreesofanyorder（twofromthetwoheapswemergeandonecomposedoftwosmallertrees）.

Becauseeachbinomialtreeinabinomialheapcorrespondstoabitinthebinaryrepresentationofitssize,thereisananalogybetweenthemergingoftwoheapsandthebinaryadditionofthesizesofthetwoheaps,fromright-to-left.Wheneveracarryoccursduringaddition,thiscorrespondstoamergingoftwobinomialtreesduringthemerge.

EachtreehasorderatmostlognandthereforetherunningtimeisO（logn）.

functionmerge（p,q）

whilenot（p.end（）andq.end（））

tree=mergeTree（p.currentTree（）,q.currentTree（））

ifnotheap.currentTree（）.empty（）

tree=mergeTree（tree,heap.currentTree（））

heap.