Edit Distance Algorithm and Knapsack Problem, Study notes of Algorithms and Programming

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Design and analysis of algorithmsLecture 25& 26

Edyta Szyma ´nska [email protected]

Edit distance

Application: Spell Checkers

Edit distance

Application: Spell Checkers ocurrance^? “Perhaps you mean

occurrence^ ?”

Edit distance

Application: Spell Checkers ocurrance^? “Perhaps you mean

occurrence^ ?” ocurrance^ and occurrence^ are^

close by

Edit distance

Application: Spell Checkers ocurrance^? “Perhaps you mean

occurrence^ ?” ocurrance^ and occurrence^ are^

close by How to define “closeness”? minimum number of “edits”-insertions, deletions and substitutions ofcharacters needed to transform string

x^ into string^ y. Example 1:^ d(to, fro

)= 2, to→^ fo (substitution)

→^ fro (insertion)

Edit distance

Solution? - dynamic programming?

Edit distance

Solution? - dynamic programming ?What are the subproblems? Example 2:^ d(exponential, polynomial

) =?^ CS3510 A, Fall 2005 – p. 3/

Edit distance

Solution? - dynamic programming ?What are the subproblems? Example 2:^ d(exponential, polynomial

possible subproblem: the smallest number of edits neededto transform some prefix of

x,^ say^ expo^ into some prefix of y,^ say^ pol.

Edit distance

Solution? - dynamic programming ?What are the subproblems? Example 2:^ d(exponential, polynomial

possible subproblem: the smallest number of edits neededto transform some prefix of

x,^ say^ expo^ into some prefix of y,^ say^ pol. How to transform

expo^ to^ pol^? left to right right to left middle-out?

Edit distance

INPUT:^ two words

x^ :^ |x|^ =^ m^ and

y^ :^ |y|^ =^ n insert y[j], substitute x[i]^ →

y[j]

Edit distance

INPUT:^ two words

x^ :^ |x|^ =^ m^ and

y^ :^ |y|^ =^ n OUTPUT:^ E(m, n)

,^ where^ E(i, j)^ is the edit distance between prefixes^ x[1^... i

]^ and^ y[1^... j] To figure out^ E(

i, j)^ check^ x[i] =

y[j]. If^ x[i]^6 =^ y[j]^ then we have one of the following operations:^ delete x[i],^ insert y[j],^ substitute x

[i]^ →^ y[j]

Edit distance

INPUT:^ two words

x^ :^ |x|^ =^ m^ and

y^ :^ |y|^ =^ n OUTPUT:^ E(m, n)

,^ where^ E(i, j)^ is the edit distance between prefixes^ x[1^... i

]^ and^ y[1^... j] To figure out^ E(

i, j)^ check^ x[i] =

y[j]. If^ x[i]^6 =^ y[j]^ then we have one of the following operations:^ delete x[i], E

(i, j) =^ E(i^ −^1 , j

insert y[j], E(i, j

) =^ E(i, j^ −^ 1) + 1 substitute x[i]^ →

y[j]^ E(i, j) =^ E

(i^ −^1 , j^ −^ 1) + 1^ CS3510 A, Fall 2005 – p. 4/

Edit distance

INPUT:^ two words

x^ :^ |x|^ =^ m^ and

y^ :^ |y|^ =^ n OUTPUT:^ E(m, n)

,^ where^ E(i, j)^ is the edit distance between prefixes^ x[1^... i

]^ and^ y[1^... j] To figure out^ E(

i, j)^ check^ x[i] =

y[j]. If^ x[i]^6 =^ y[j]^ then we have one of the following operations:^ delete x[i], E

(i, j) =^ E(i^ −^1 , j

insert y[j], E(i, j

) =^ E(i, j^ −^ 1) + 1 substitute x[i]^ →

y[j]^ E(i, j) =^ E

(i^ −^1 , j^ −^ 1) + 1 Otherwise (x[i] =

y[j]) we have^ E

(i, j) :=^ E(i^ −^1 , j

−^ 1)

Base case:^ E(

, j) =^ j^ and^ E(i,

1. =^ i.

CS3510 A, Fall 2005 – p. 4/

Edit distance

INPUT:^ two words

x^ :^ |x|^ =^ m^ and

y^ :^ |y|^ =^ n OUTPUT:^ E(m, n)

,^ where^ E(i, j)^ is the edit distance between prefixes^ x[1^... i

]^ and^ y[1^... j] To figure out^ E(

i, j)^ check^ x[i] =

y[j]. If^ x[i]^6 =^ y[j]^ then we have one of the following operations:^ delete x[i], E

(i, j) =^ E(i^ −^1 , j

insert y[j], E(i, j

) =^ E(i, j^ −^ 1) + 1 substitute x[i]^ →

y[j]^ E(i, j) =^ E

(i^ −^1 , j^ −^ 1) + 1 Otherwise (x[i] =

y[j]) we have^ E

(i, j) :=^ E(i^ −^1 , j

−^ 1)

Base case:^ E(

, j) =^ j^ and^ E(i,

1. =^ i.^ Let {^1 dif f (i, j) = if x[i]^6 =^ y[j]^

CS3510 A, Fall 2005 – p. 4/