Download pdf - Daa Summary

20/11/2013

1

DAASummaryCSE3rd YearGCET

Syllabus

UnitI:Introduction:Algorithms,Analyzingalgorithms Complexity of algorithms Growth ofalgorithms,Complexityofalgorithms,Growthoffunctions,Performancemeasurements,SortingandorderStatistics Shellsort,Quicksort,Mergesort,Heapsort,Comparisonofsortingalgorithms,Sortinginlineartime.

UnitII:AdvancedDataStructures:RedBlacktrees,B trees,BinomialHeaps,FibonacciHeaps.

11Nov2013 2DAA(ECS502)

20/11/2013

2

Syllabus Unit III:DivideandConquerwithexamplessuchasSorting,MatrixMultiplication,ConvexhullandSearching Greedy methods with examples such asSearching.GreedymethodswithexamplessuchasOptimalReliabilityAllocation,Knapsack,MinimumSpanningtrees PrimsandKruskals algorithms,Singlesourceshortestpaths Dijkstras andBellmanFordalgorithms.

Unit IV:DynamicprogrammingwithexamplessuchK k All i h t t th W h l dasKanpsack,Allpairshortestpaths Warshals and

Floydsalgorithms,Resourceallocationproblem.Backtracking,BranchandBoundwithexamplessuchasTravellingSalesmanProblem, GraphColoring, nQueenProblem, HamiltonianCyclesandSumofsubsets.


Syllabus

UnitV:SelectedTopics:AlgebraicComputation, FastFourierTransform, StringMatching, TheoryofNPcompleteness,ApproximationalgorithmsandRandomizedalgorithms.

References:1. ThomasH.Coreman,CharlesE.Leiserson andRonaldL.Rivest,

IntroductiontoAlgorithms,Printice HallofIndia.2. RCT Lee, SS Tseng, RC Chang and YT Tsai, Introduction to the Design2. RCTLee,SSTseng,RCChangandYTTsai, IntroductiontotheDesign

andAnalysisofAlgorithms,McGraw Hill,2005.3. E.Horowitz&SSahni,"FundamentalsofComputerAlgorithms",4. Aho,Hopcraft,Ullman,TheDesignandAnalysisofComputer

AlgorithmsPearson


20/11/2013

3

INSERTIONSORTAlg.: INSERTIONSORT(A)forj 2 ton a8a7a6a5a4a3a2a1

1 2 3 4 5 6 7 8

dokeyA[ j ]InsertA[ j ] intothesortedsequenceA[1 . . j -1]i j - 1whilei > 0 andA[i] > key

d A[i 1] A[i]

key

5

doA[i + 1] A[i]i i 1

A[i + 1] key Insertionsort sortstheelementsinplace

MergeSortAlg.: MERGESORT(A, p, r)

1 2 3 4 5 6 7 8

62317425

p rq

ifp < r Checkforbasecasethenq (p + r)/2 Divide

MERGESORT(A, p, q) ConquerMERGESORT(A, q + 1, r) Conquer

(A )

6

MERGE(A, p, q, r) Combine

Initialcall: MERGESORT(A, 1, n)

20/11/2013

4

Merge PseudocodeAlg.: MERGE(A,p,q,r)1. Computen1 and n2

1 2 3 4 5 6 7 8

63217542

p rq

2. Copythefirstn1 elementsinto L[1 . . n1 + 1] andthenextn2 elementsintoR[1 . . n2 + 1]

3. L[n1 + 1] ; R[n2 + 1] 4. i 1; j 15. fork p tor6 do if L[ i ] R[ j ]

p q

7542

6321rq+1

L

R

n1 n2

7

6. doifL[ i ] R[ j ]7. thenA[k] L[ i ]8. i i + 19. elseA[k] R[ j ]10. j j + 1

RandomizedQuicksortAlg. : RANDOMIZEDQUICKSORT(A, p, r)

ifp < r

thenq RANDOMIZEDPARTITION(A, p, r)

(A )

8

RANDOMIZEDQUICKSORT(A, p, q)

RANDOMIZEDQUICKSORT(A, q + 1, r)

20/11/2013

5

RandomizedPARTITION

Alg.: RANDOMIZEDPARTITION(A, p, r)

i RANDOM(p, r)

exchange A[p] A[i]

9

exchangeA[p] A[i]

returnPARTITION(A, p, r)

PARTITIONAlg.:PARTITION(A,p,r)

x A[r]i p - 1

A[pi] x A[i+1j-1] > xp i i+1 rj1 ji p 1

forj p tor - 1doifA[ j ] x

theni i + 1exchangeA[i]A[j]

exchangeA[i + 1]A[r]

unknown

pivot

10

returni + 1ChoosesthelastelementofthearrayasapivotGrowsasubarray [p..i]ofelementsxGrowsasubarray [i+1..j1]ofelements>xRunningTime:(n),wheren=rp+1

20/11/2013

6

Alg: HEAPSORT(A)

1. BUILDMAXHEAP(A) O(n)2. fori length[A] downto23. doexchangeA[1] A[i]4. MAXHEAPIFY(A, 1, i - 1)

( )

O(lgn)

n-1 times

11

Runningtime:O(nlgn) --- Can be shown to be (nlgn)

BuildingaHeap ConvertanarrayA[1 n] intoamaxheap(n = length[A]) TheelementsinthesubarrayA[(n/2+1) .. n] areleaves

Alg: BUILDMAXHEAP(A)1. n =length[A]2. for i n/2 downto 1

ApplyMAXHEAPIFYonelementsbetween1 andn/2

1

4

3

1

2 3

12

3. doMAXHEAPIFY(A, i, n) 214 8

16

7

9 104 5 6 7

8 9 10

4 1 3 2 16 9 10 14 8 7A:

20/11/2013

7

MaintainingtheHeapProperty Assumptions: LeftandRight

bt f i

Alg: MAXHEAPIFY(A, i, n)1. lLEFT(i)2 RIGHT(i)subtrees ofi

aremaxheaps A[i] maybe

smallerthanitschildren

2. rRIGHT(i)3. if l n andA[l] > A[i]4. then largest l5. else largest i6. if r n andA[r] > A[largest]7 then largest r

13

7. then largest r8. if largest i9. then exchangeA[i] A[largest]10. MAXHEAPIFY(A, largest, n)

AnalysisofCountingSortAlg.: COUNTINGSORT(A,B,n,k)1. fori 0 tor2 do C[ i ] 0 (r)2. doC[ i ] 03. forj 1 ton4. doC[A[ j ]] C[A[ j ]] + 15. C[i] containsthenumberofelementsequaltoi6. fori 1 tor7. doC[ i ] C[ i ] + C[i -1]

(n)

(r)

14

[ ] [ ] [ ]8. C[i] containsthenumberofelements i9. forj n downto 110. doB[C[A[ j ]]] A[ j ]11. C[A[ j ]] C[A[ j ]] - 1

(n)

Overalltime:(n + r)

20/11/2013

8

AnalysisofBucketSortAlg.: BUCKETSORT(A,n)

for i 1 to nfori 1 tondoinsertA[i]intolistB[nA[i]]

fori 0 ton - 1dosortlistB[i] withquicksortsort

B[0] B[1] B[ 1]

O(n)

(n)

15

concatenatelistsB[0], B[1], . . . , B[n -1]togetherinorderreturntheconcatenatedlists

O(n)

(n)

RBINSERT(T, z)1. y NIL Initializenodesxandy

Th h t th l ith i t

26

17 41

30 47

38 50

2. x root[T]3. whilex NIL4. doy x5. ifkey[z] < key[x]

Throughoutthealgorithmypointstotheparentofx

Godownthetreeuntilreachingaleaf Atthatpointyisthe

16

6. thenx left[x]7. elsex right[x]8. p[z] y

p yparentofthenodetobeinserted

Setstheparentofztobey

20/11/2013

9

RBINSERT(T, z)9. ify = NIL10. thenroot[T] z Thetreewasempty:setthenewnodetobetheroot

26

17 41

30 47

38 50

11. elseifkey[z] < key[y]12. thenleft[y] z13. elseright[y] z14. left[z] NIL

Otherwise,setztobetheleftorrightchildofy,dependingonwhethertheinsertednodeissmallerorlargerthanyskey

17

15. right[z] NIL16. color[z] RED17. RBINSERTFIXUP(T, z)

Setthefieldsofthenewlyaddednode

Fixanyinconsistenciesthatcouldhavebeenintroducedbyaddingthisnewrednode

RBINSERTFIXUP(T, z)1. whilecolor[p[z]] =RED2. doifp[z] = left[p[p[z]]]

i h [ [ [ ]]]

Thewhilelooprepeatsonlywhencase1isexecuted:O(lgn) times

Set the value of xs uncle3. theny right[p[p[z]]]4. ifcolor[y] = RED5. thenCase16. elseifz = right[p[z]]7. thenCase2

Setthevalueofx s uncle

18

8. Case39. else(sameasthenclausewithright

andleftexchanged)10. color[root[T]] BLACK Wejustinsertedtheroot,or

Theredviolationreachedtheroot

20/11/2013

10

LEFTROTATE(T,x)1. y right[x] Sety2. right[x] left[y] ysleftsubtreebecomesxsrightsubtree3 if left[y] NIL3. ifleft[y] NIL4. thenp[left[y]] x Settheparentrelationfromleft[y]tox5. p[y] p[x] Theparentofxbecomestheparentofy6. ifp[x] = NIL7. thenroot[T] y8 else if x = left[p[x]]

19

8. elseifx = left[p[x]]9. thenleft[p[x]] y10. elseright[p[x]] y11. left[y] x Putxonysleft12. p[x] y ybecomesxsparent

PRIM(V, E, w, r)1. Q 2. foreachu V3. dokey[u] O(V) ifQisimplementedas

Totaltime:O(VlgV + ElgV) = O(ElgV)

y[ ]4. [u]NIL5. INSERT(Q, u)6. DECREASEKEY(Q, r, 0) key[r] 07. whileQ 8. douEXTRACTMIN(Q)

aminheap

Executed|V|timesTakesO(lgV)

Minheapoperations:O(VlgV)

O(lgV)

20

9. foreachv Adj[u]10. doifv Q andw(u, v) < key[v]11. then[v] u12. DECREASEKEY(Q, v, w(u, v))

ExecutedO(E)timestotalConstant

TakesO(lgV)O(ElgV)

20/11/2013

11

UsingFibonacciHeaps Dependingontheheapimplementation,runningtimecould be improved!couldbeimproved!

21

1. A2. foreachvertexv V3. doMAKESET(v)

KRUSKAL(V, E, w)

O(V)3. doMAKE SET(v)4. sortEintonondecreasingorderbyw5. foreach(u, v) takenfromthesortedlist6. doifFINDSET(u) FINDSET(v)7. thenAA {(u, v)}8. UNION(u v)

O(ElgE)

O(E)

O(lgV)

22

8. UNION(u, v)9. returnARunningtime:O(V+ElgE+ElgV)=O(ElgE) dependentonthe

implementationofthedisjointsetdatastructure Runningtime:O(V+ElgE+ElgV)=O(ElgE) SinceE=O(V2),wehavelgE=O(2lgV)=O(lgV)

O(lgV)

O(ElgV)

20/11/2013

12

BELLMANFORD(V, E, w, s)1. INITIALIZESINGLESOURCE(V,s)2. fori1to|V| 1

(V)O(V)

O(VE)3. doforeachedge(u,v) E4. doRELAX(u,v,w)5. foreachedge(u,v) E6. doifd[v]>d[u]+w(u,v)7. then return FALSE

O(E)

O(E)

O(VE)

23

7. thenreturnFALSE8. returnTRUE

Runningtime:O(V+VE+E)=O(VE)

RelaxationStep Relaxinganedge(u,v)=testingwhetherwecan

improvetheshortestpathtovfoundsofarbygoingthroughu

d[ ] d[ ] ( ) Ifd[v] > d[u] + w(u, v) wecanimprovetheshortestpathtov d[v]=d[u]+w(u,v)[v] u

2u v

2u v

Afterrelaxation:d[v] d[u] + w(u, v)s s

24

5 9

5 72

u v

RELAX(u,v,w)

5 6

5 62

u v

RELAX(u,v,w)

nochange

20/11/2013

13

Dijkstra(G,w,s)1. INITIALIZESINGLESOURCE(V, s)2. S3 Q V[G]

(V)

O(V) build min heap3. QV[G]4. whileQ 5. do uEXTRACTMIN(Q)6. SS {u}7. foreachvertexv Adj[u]

O(V)buildminheapExecutedO(V)times

O(lgV)

O(E)times

O(VlgV)

25

j8. doRELAX(u, v, w)9. UpdateQ(DECREASE_KEY)

Runningtime:O(VlgV + ElgV) = O(ElgV)

( )(total)

O(lgV)

O(ElgV)

Floyd'sAlgorithm:Using2Dmatrices

Floyd1.DW//initializeD arraytoW[]2.P 0//initializeParrayto[0]3.fork 1ton

//ComputingDfromD4.dofori 1ton5.doforj 1ton6. if(D[i,j ]>D[i,k ]+ D[k,j ])7 th D[ i j ] D[ i k ] D[ k j ]

FloydsAlgorithm26

7. thenD[i,j ] D[i,k ]+ D[k,j ]8. P[i,j ] k;9. elseD[i,j ] D[i,j ]10.MoveDtoD.

20/11/2013

14

Printingintermediatenodesonshortestpathfromqtor

path(index q,r)if (P[q,r]0)

h( P[ ]) 30 0

1 2 3

1path(q,P[q,r])println(v+P[q,r])path(P[q,r],r)return;

//nointermediatenodeselse return

BeforecallingpathcheckD[q,r]

20/11/2013

15

rightDiagonal[row-col]=!available;if (row< squares-1)

tQ ( +1)putQueen(row+1);else

print(" solution found);column[col]=available;leftDiagonal[row+col]=available;rightDiagonal[row-col]= available;

}}}

Navestringmatching

Runningtime:O((nm+1)m).

20/11/2013

16

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

20/11/2013

17

11Nov2013 DAA(ECS502) 33

11Nov2013 DAA(ECS502) 34