Longest Common Subsequence(LCS)

研究生 鍾聖彥 指導老師 許慶昇

Dynamic Programming

12014/05/07

最長共同子序列

Dynamic Programming

Optimal substructure(當一個問題存在著最佳解，則表示其所有子問題也必存在著最佳解)

Overlapping subproblems(子問題重複出現)

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Longest Common Subsequence???

Biological applications often need to compare the DNA of tow(or more) different organisms.

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Subsequence

A subsequence of a given sequence is just the given sequence with zero or more elements left out.

Ex: app、le、ple and so on are subsequences of “apple”.

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Common Subsequence

X = (A, B, C, B, D, A, B)Y = (B, D, C, A, B, A)

Two sequences:

Sequence Z is a common subsequence of X and Y if Z is a subsequence of both X and Y

Z = (B, C, A) — length 3 Z = (B, C, A, B) - length 4 Z = (B, D, A, B) — length 4

Z= — length 5 ???

longest

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What is longest Common Subsequence problem?

X = (x1, x2,……., xm) Y = (y1, y2,……., yn)

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Find a maximum-length common subsequence of X and Y

How to do?Dynamic Programming!!!

Brute Force!!!

Step 1: Characterize optimality

Sequence X = (x1, x2,……., xm)

Define the ith prefix of X, for i = 0, 1,…, m as Xi = (x1, x2, ..., xi)

with X0 representing the empty sequence.

EX: if X = (A, B, C, A, D, A, B) then X4 = (A, B, C, A)X0 = ( ) empty sequence

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Theorem (Optimal substructure of LCS)

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1. If Xm = Yn, then Zk = Xm = Yn and Zk-1 is a LCS of Xm-1 and Yn-1

2. If Xm ≠ Yn, then Zk ≠ Xm implies that Z is a LCS of Xm-1 and Y

3. If Xm ≠ Yn, then Zk ≠ Yn implies that Z is a LCS of X and Yn-1

X = (X1, X2,…, Xm) and Y = (Y1, Y2,…, Yn)

Sequences

Z = (Z1, Z2,…, Zk) be any LCS of X and Y

We assume:

Optimal substructure problem

The LCS of the original two sequences contains a LCS of prefixes of the two sequences.

(當一個問題存在著最佳解，則表示其所有子問題也必存在著最佳解)

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Step 2: A recursive solutionXi and Yj end with xi=yj

Zk is Zk -1 followed by Zk = Xi = Yj where Zk-1 is an LCS of Xi-1 and Yj -1

LenLCS(i, j) = LenLCS(i-1, j-1)+1

Xi x1 x2 … xi-1 xi

Yj y1 y2 … yj-1 yj=xi

Zk z1 z2…zk-1 zk =yj=xi

Case 1:

Step 2: A recursive solutionCase 2,3: Xi and Yj end with xi ≠ yj

Xi x1 x2 … xi-1 xi

Yj y1 y2 … yj-1 yj

Zk z1 z2…zk-1 zk ≠yj

Xi x1 x2 … xi-1 x i

Yj yj y1 y2 …yj-1 yj

Zk z1 z2…zk-1 zk ≠ xi

Zk is an LCS of Xi and Yj -1 Zk is an LCS of Xi-1 and Yj

LenLCS(i, j)=max{LenLCS(i, j-1), LenLCS(i-1, j)}

Step 2:A recursive solution

Let c[i,j] be the length of a LCS for Xi and Yj the recursion described by the above cases as

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Case 1 Reduces to the single subproblem of finding a LCS of

Xm-1, Yn-1 and adding Xm = Yn to the end of Z.

Cases 2 and 3 Reduces to two subproblems of finding a LCS of Xm-1, Y and X, Yn-1 and selecting the longer of the two.

Step 3: Compute the length of the LCS

LCS problem has only ɵ(mn) distinct subproblems.

So? Use Dynamic programming!!!

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Step 3: Compute the length of the LCSProcedure 1

LCS-length takes two Sequences X = (x1, x2,…, xm) and Y = (y1, y2,…, yn) as input.

Procedure 2It stores the c[i, j] values in a table c[0..m, 0..n] and

it computes the entries in row-major order.Procedure 3

Table b[1..m, 1..n] to construct an optimal solution. b[i, j] points to the table entry corresponding to the

optimal solution chosen when computing c[i, j]Procedure 4

Return the b and c tables; c[m, n] contains the length of an LCS X and Y

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LCS-Length(X, Y)1 m = X.length 2 n = Y.length 3 let b[1..m, 1..n] and c[0..m, 0..n] be new tables. 4 for i 1 to m do 5 c[i, 0] = 0 6 for j 1 to n do 7 c[0, j] = 0 8 for i 1 to m do 9 for j 1 to n do 10 if xi ==yj11 c[i, j] = c[i-1, j-1]+112 b[i, j] = “ ” 13 else if c[i-1, j] ≥ c[i, j-1]14 c[i, j] = c[i-1, j] 15 b[i, j] = “ ”16 else 17 c[i, j] = c[i, j-1]18 b[i, j] = “ ” 19 return c and b 15

The table produced by LCS-Length on the sequences X = (A, B, C, B, D, A, B) and Y = (B, D, C, A, B, A).

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The running time of the procedure is O(mn), since each table entry table O(1) time to compute

Step 4: Construct an optimal LCS

PRINT-LCS(b, X, i, j)PRINT-LCS(b, X, X.length, Y.length)

1 if i == 0 or j == 0 2 return 3 if b[i, j] == “ ” 4 PRINT-LCS(b,X,i-1, j-1) 5 print Xi 6 else if b[i, j] == “ ” 7 PRINT-LCS(b,X,i-1, j) 8 else PRINT-LCS(b,X,i, j-1)

This procedure prints BCBA. The procedure takes time O(m+n)

ExampleX = <A, B, C, B, A> Y = <B, D, C, A, B>

We will fill in the table in row-major order starting in the upper left corner using the following formulas:

ExampleX = <A, B, C, B, A> Y = <B, D, C, A, B>

We will fill in the table in row-major order starting in the upper left corner using the following formulas:

Answer

Thus the optimal LCS length is c[m,n] = 3.Optimal LCS starting at c[5,5] we get Z = <B, C, B>

Alternatively start at c[5,4] we would produce Z = <B, C, A>.

*Note that the LCS is not unique but the optimal length of the LCS is.

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ReferenceLecture 13: Dynamic Programming - Longest Common Subsequence http://faculty.ycp.edu/~dbabcock/cs360/lectures/lecture13.html

http://www.csie.ntnu.edu.tw/~u91029/LongestCommonSubsequence.html

Longest common subsequence (Cormen et al., Sec. 15.4)

https://www.youtube.com/watch?v=Wv1y45iqsbk

https://www.youtube.com/watch?v=wJ-rP9hJXO0