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thatproducesthesameresultatlowercost.Ifacharacterisreplacedbyacharactera
andthenreplacedagainbyacharacterb,thenbothoperationscanbereplacedby“re
placeb”.Thismeansthatallinsertedandreplacedcharactersareneverchangedafter
performingtheinsertionorreplacement.Noticethatafterremovingtheseoperations
thatintroducedependencies,anysequenceofinsert,delete,andreplaceoperationscan
bereorderedsothattheyoccurfromlefttorightwithoutaffectingtheoutcomeofthe
transformation.
（c）Weshowthatcomputingeditdistanceforstringsxandycanbedonebyfindingthe
editdistanceofsubproblems.Defineacostfunction
（i,j）=d（x,y[1..i]||x[j+1..m]）
（1）
Thatis
（i,j）istheminimumcostoftransformingthefirstjcharactersofxinto
thefirsticharactersofy.Thend（x,y）=
（n,m）.Nowconsiderasequenceof
operationsS=（o1,o2,...,ok）thattransformsxtoywithcostC（S）=d（x,y）.
LetSibethesubsequenceofScontainingthefirstioperationsofS.Letzibethe
auxilliarystringafterperformingoperationsSi,wherez0=xandzk=y.
（d）Wecancalculatetheeditdistanced（x,y）usingthedefinitiono𝐜
𝐱
𝐲
（i,j）fromEqua
tion1.Recallthatd（x,y）=
（m,n）.Sinceweshowedinpart（a）thatthereexists
asequenceofoperationsthatachievesd（x,y）withoutusingthe“left”operation,we
onlyneedtoconsiderthefouroperations“right”,“replace”,“delete”,and“insert”.
Wecancalculate
（m,n）recursively.Thebasecaseiswhennotransformationop
erationshavebeenperformed,so𝐜
（0,0）=0.
（e）ConstructatableTwhereeachentryT[i,j]=
（i,j）.Sinceeachvalue
of
（i,j）onlydependson𝐜
（i1,j2）wherei1<
=iandj1<
=j,wecancomputethe
entriesofTrowbyrowusingEquation2:
EDIT-DISTANCE（x[1..m],y[1..n]）
1T[0,0]←0
2fori←1ton
3forj←1tom
4doT[i,j]←min𝐓
𝐢
−𝟏
𝐣
𝐢
𝐟
>
𝟎
𝐚
𝐧
𝐝
𝐣
[𝐢
]=𝐲
[𝐣
]𝐓
+𝟒
𝐢
𝐓
+𝟐
𝐢
+𝟑
5returnT[n,m]
Therunningtimeofthisalgorithmis
（mn）.Thisalgorithmrequires
（mn）space.
（f）
x:
electricalengineering
y:
computerscience
EditDistance:
54
Oper|c|Total|z
initial|0|0|*electricalengineering
delete|2|2|*lectricalengineering
delete|2|4|*ectricalengineering
delete|2|6|*ctricalengineering
right|0|6|c*tricalengineering
insert|3|9|co*tricalengineering
insert|3|12|com*tricalengineering
insert|3|15|comp*tricalengineering
insert|3|18|compu*tricalengineering
right|0|18|comput*ricalengineering
insert|3|21|compute*ricalengineering
right|0|21|computer*icalengineering
delete|2|23|computer*calengineering
delete|2|25|computer*alengineering
delete|2|27|computer*lengineering
delete|2|29|computer*engineering
right|0|29|computer*engineering
delete|2|31|computer*ngineering
replace|4|35|computers*gineering
replace|4|39|computersc*ineering
right|0|39|computersci*neering
delete|2|41|computersci*eering
delete|2|43|computersci*ering
right|0|43|computerscie*ring
delete|2|45|computerscie*ing
delete|2|47|computerscie*ng
right|0|47|computerscien*g
insert|3|50|computerscienc*g
replace|4|54|computerscience*
（g）InputFiled（x,y）
Input11816
Input21824
Input31829
（h）AllthetransformationoperationscanbeimplementedinO
（1）timeusing
twostacks,LandR.Initially,LisemptyandRcontainsallthecharactersofxin
order.
OperationImplementation
leftIfnotEMPTY（L）,thenPUSH（R,POP（L））
rightIfnotEMPTY（R）,thenPUSH（L,POP（R））
replacebycIfnotEMPTY（R）,thenPOP（R）,PUSH（L,c）
deleteIfnotEMPTY（R）,thenPOP（R）
insertcPUSH（L,c）
EachstackoperationrequiresO
（1）time,andeachtransformationoperationrequires
onlyO
（1）stackoperations.ThereforeeachoperationrequiresO
（1）time.
代码部分：
#include<
assert.h>
stdio.h>
stdlib.h>
string.h>
//OperationCosts
#defineMOVE_COST0
#defineREPLACE_COST4
#defineINSERT_COST3
#defineDELETE_COST2
//OperationTypes
typedefenum{initial,right,insert,delete,replace,final}opType;
char*opStrings[]={"
initial"
"
right"
insert"
delete"
replace"
final"
};
//Recoversandprintsouttheeditsnecessarytochangexintoy.
staticvoidrecoverEdits（char*x,intm,
char*y,intn,
int**d,opType**opPerformed）;
//Computestheeditdistancebetweenxandy
staticvoidcomputeEditDistance（char*x,intm,
char*y,intn）{
//Inthisdynamicprogram,d[i][j]storestheeditdistancebetween
//thestringx[i..m-1]andy[j..n-1].
//NOTE!
Carraysstartindexingat0.
int**d;
opType**opPerformed;
//storeoperationsforreconstructingtheanswer
opTypeopDone;
inti,j;
d=（int**）malloc（（m+1）*sizeof（int*））;
assert（d!
=NULL）;
opPerformed=（opType**）malloc（（m+1）*sizeof（opType*））;
assert（opPerformed!
for（i=0;
i<
=m;
i++）{
d[i]=（int*）malloc（（n+1）*sizeof（int））;
assert（d[i]!
opPerformed[i]=（opType*）malloc（（n+1）*sizeof（opType））;
assert（opPerformed[i]!
}
//Initializethebasecases.
//x[m]andy[n]arebothnullstrings.Editdistancebetweenthemis0.
d[m][n]=0;
opPerformed[m][n]=final;
m;
//Toconvertx[i..m-1]tothenullstringy[n],deleteall（m-i）
//remainingcharactersinx.
d[i][n]=DELETE_COST*（m-i）;
opPerformed[i][n]=delete;
for（j=0;
j<
n;
j++）{
//Toconvertx[m]（thenullstring）intoy[j..n-1],insert
//themissingn-jcharacters.
d[m][j]=INSERT_COST*（n-j）;
opPerformed[m][j]=insert;
//Startatd[m][n]andloopbackwards
for（i=m-1;
i>
=0;
i--）{
for（j=n-1;
j>
=0;
j--）{
intcostForReplaceOrMove;
intcostForInsert;
intcostForDelete;
intminValue;
//Computed[i][j]astheminimumof4terms:
//Ifx[i]==y[j],wecouldmoveright.
//Otherwise,wecanreplacex[i]withy[j]and
//incrementiandj.
costForReplaceOrMove=d[i+1][j+1]+REPLACE_COST*（x[i]!
=y[j]）+MOVE_COST*（x[i]==y[j]）;
//Ifweinsertacharacterintoxtomatchy[j],then
//weincrementjby1
costForInsert=d[i][j+1]+INSERT_COST;
//Ifwedeleteacharacterinx,thenwe
//incrementiby1.
costForDelete=d[i+1][j]+DELETE_COST;
//Oftheaboveoperations,findonethatgivesus
//aminimumcost.
minValue=costForReplaceOrMove;
if（x[i]!
=y[j]）{
opDone=replace;
else{
opDone=right;
if（minValue>
costForInsert）{
minValue=costForInsert;
opDone=insert;
costForDelete）{
minValue=costForDelete;
opDone=delete;
d[i][j]=minValue;
opPerformed[i][j]=opDone;
//Finalanswer
printf（"
EditDistance=%d\n"
d[0][0]）;
//Recoveroperations/printoutansweronlyifthestringsaren'
ttoolong.
if（（m<
75）&
&
（n<
75））{
recoverEdits（x,m,y,n,d,opPerformed）;
free（d[i]）;
free（opPerformed[i]）;
free（d）;
free（opPerformed）;
}
int**d,opType**opPerformed）{
inti=0;
intj=0;
intk=0;
inta;
intopDone;
intcostSoFar=0;
intstepCost=0;
char*newString;
newString=（char*）malloc（sizeof（char）*（m+n））;
//Ratherthanimplementingastringobjectforxthatallowsus
//constant-timeinsertsanddeletesatthecursor,
//wearejustgoingtohavetwocopiesofthestring.
//newString[0..k]storeseverythingbeforethecursorthatwehave
//changed,andx[i..m-1]willstoreallthepartsafterthecursorthat
//arestillthesame.
strncpy（newString,x,m）;
\n"
）;
（i,j）:
%8s|c|Total|z\n"
Oper"
---------------------------------------------------------------------\n"
（%2d,%2d）:
%8s|%4d|%4d|"
i,j,"
costSoFar,stepCost）;
*%s\n"
x）;
while（opPerformed[i][j]!
=final）{
opDone=opPerformed[i][j];
switch（opDone）{
caseright:
newString[k]=x[i];
i++;
j++;
stepCost=MOVE_COST;
break;
casereplace:
newString[k]=y[j];
stepCost=REPLACE_COST;
caseinsert:
stepCost=INSERT_COST;
casedelete:
stepCost=DELETE_COST;
default:
//Weshouldneverafollowapointertoinitialorfinal.
ERROR.\n"
exit
（1）;
if（opDone!
=delete）{
k++;
costSoFar+=stepCost;
%8s|
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