PriorityQueue - - PDF document

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PriorityQueue

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Implementation

SortedSequence UnsortedSequence

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:k≤ ≤ ≤ ≤ k :ifk1≤ ≤ ≤ ≤ k2andk2≤ ≤ ≤ ≤ k1,thenk1 k2 :ifk1≤ ≤ ≤ ≤ k2andk2≤ ≤ ≤ ≤ k3,thenk1≤ ≤ ≤ ≤ k3

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Totalorderingexamples

• ≤ isatotalordering
• ≥ isalsoatotalordering
• Alphabeticalorder:wedefinea≤ bif‘a’ is

before‘b’ inalphabeticalorder

• Reversealphabeticalorder

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But…

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theyarenotreflexive

• isnotatotalorderingsincewecan’t

compareany2elementswith=. Givena,b,wedonotalwayshave

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Moreexamplesofordering

Wecanordertheco/ordinatepairs p=(x1,y1)andq=(x2,y2)by

• p≤ q ifx1≤ x2
• p≤ qif

y1 ≤ y2

• p≤ qifx1≤ x2and y1 ≤ y2

Thelastoneisonlyapartial ordering!

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• However,toimplementinsertItem(), wemustnowscanthroughthe

entiresequenceintheworstcase.ThusinsertItem() runsin

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Anobservation… Withanunsortedsequence… removeMin()always takesO(n) Butwithasortedsequence… insertItem()takesatmost O(n)

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AlgorithmPriorityQueueSort(S,P): Input:AsequenceSstoringnelements,onwhicha totalorderrelationisdefined,andaPriority QueuePthatcompareskeyswiththesamerelation Output:TheSequenceSsortedbythetotal

• rderrelation

while!S.isEmpty() do e← S.removeFirst() P.insertItem(e,e) whilePisnotemptydo e← P.removeMin() S.insertLast(e)

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VariationofPriorityQueueSort thatusesanunsortedsequence to implementthepriorityqueueP. – Phase1,theinsertionofanitemintoPtakesO(1)time – Phase2,removinganitemfromPtakestimeproportionaltothe currentnumberofelementsinP

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