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CourseCode: CSCI2100AFinalExamination Page

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CourseExaminationsSecondTerm,2014-2015

CSCI2100AFinalExamination

CourseCode

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Title

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1. Hashing(20)

Giventheinput{84,12,48,16,38,108,36,60,72,63} andahashfunctionh x) = x mod 10, show

whathappenswheneachelementinthelistisinsertedintothehashtableof size10 usingthefollowing

collisionpolicy.

(a) (4)Open hashtable.Explainwhatmethod

of

insertionyouareusing.

(b) (4)Closedhashtableusinglinearprobing(assumingthattheincrementis

1 .

(c) (4)Closedhashtablewithsecondhashfunctionh2 X)

=

3- x mod 3.

(d) (4)

What

aretheadvantagesanddisadvantages

of

thevariouscollisionstrategiesinthefirsttwo

questions?Justifyyouranswer.

(e) (4)Whichhashfunctionisabetterhashfunctionfortheabovesequence :

1) i x) = x

mod

12

or

(2)

h2 X)

=

x mod11? Justify youranswer. Note thatthe sizeof hashtablewillchange

accordingtohashfunction.

2. Graph (27)

(a) (3)Drawthecorrespondingadjacencymatrixforthegraph.

(b) (3)Pleaselistthedepth-firstsearchsequencestartedfromvertex

D

(c) (3)Pleaselistthebreath-firstsearchsequencestartedfromvertex

A.

(d) (6)Findtheshortestweighedpath fromvertex  to

G

Illustrateintermediatestepswithatable

orfigures.

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Course

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CSCI 2100A Final Examination Page 2 o 2

(e) (8)UseKruskal's algorithm to find the minimumcostspanning treeof thegraph. Show the

intermediatesteps

of

the algorithmand the final resultof theminimumcostspanningtreethat

youhaveobtained.

(f) (4)Findthemaximumflow in thenetworkaboveassumingthat the sourceisnodeA andthe

sinkisnodeG.

No

intermediatestepsarerequired.

3. Sorting (27)

(a) (3)Showall theinversionpairsinthe list  4,1,5,2, 3)? Listthemoutin an organizedmanner,

i.e.,sorttheminanascendingorderbythefirstelementandthenbythesecondelement.

(b) (5)Giventhesequence

  4,6,1,8,3,7,2,5),

sortthesequenceusingtheInsertionSortalgorithm.

Illustrate the resultafter eachpass. Howmany positions movedhave youperformed at each

pass?Justifyyouranswerbyusingatablelikeinthelecturenotes.

(c) (5)Giventhe sequence 5,2,8,4,1,6,3,7), sort the sequence usingMergesort. Illustratethe

resultaftereachpass.Howmanycomparisonshaveyouperformed?

(d) (6)

You

havealist

of

elementswithonlythreedifferentelements,e.g. 0,1.2. Forexample, the

listcanbe{0,1,1,0,2,},0,0,2}.

a)(3)Pleasegiveanalgorithmwhosetimecomplexityis

O n).

b)(3)Pleasegiveanalgorithmwhosetimecomplexityis O n) whilespacecomplexity is 0 1).

(e) (4)Given an arraywith

n

elements

to

sort, what

is

the

best

and

worst

timecomplexitywhen

usingBubblesort,Selectionsort,InsertionsortandMergesortinterms of Big-Onotation.

(f) (4)Quicksortwiththemedian

of

the first, middle,and lastelement

of

thearray

as

pivotanda

cutoffof 3,demonstrateONLY theprocessof partitionthesequence,  7,6,1,4,8,2,3,5), into

twopartsof theQuicksort(donotshowthewholeQuicksortalgorithm!). Notethatassumethe

size of array is

n,

themiddleelementindex is

nj2

where index starts from°o

n

1 and

the first stepinthePartitionalgorithmistoswaptheselectedpivot(medianof 3)withthefirst

element,

Le

afterthisstep,itisthesame

as

thecodeintroducedinthelecturenoteswhereusing

the first element as thepivot,afterpartition,thelefthalfelementsislessor equaltothepivot,

therighthalfisgreater thanthepivot. Illustrateintermediatestepsclearly.

4. Short Answers (26)(Pleasegiveonlyconciseandshortanswers!)

(a) (6) How many different stack-permutable final sequences are there if the input sequence is

{1,2,3,4,5}?E.g., {5,4,3,2,1}and {1,3,2,5,4}arebothvalidfinalsequences.

(b) (6)

The

followings arethepartialresultsof abinarytree'straversalsin pre-order, in-orderand

post-order(XrepresentsUNKNOWN).Pleasedrawthecorrespondingtree. Notethat the tree

has9nodes in total,whicharefromA to1.

Pre-order: XAXIXXBXC In-order:XDXAHXFXC Post-order:XEDXBXGFH

(c) (14)What

is

theAverage- andWorst-caseruntimesof asingleDeleteMin operationonthe

following types

of

datastructures?Theanswersshouldbeexpressedin thebig-Ohnotationin

termsof,

n

thenumber

of

elementsstoredinthedatastructure.

DataStructure

 

Average-Case

 

Worst-Case

 

Descending-SortedDoubleLinkedList

UnsortedArray

Ascending-SortedArray

BinarySearchTree

AVLTree

Max-Heap

Min-Heap

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