# Module_6 - Dynamic programming techniques Topics Basics of...

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11/19/09 CSE 5311 Spring 2007 M Kumar 1 Dynamic programming techniques Topics Basics of DP Matrix-chain Multiplication Longest Common subsequence All-pairs Shortest paths Further Reading Chapter 6 Textbook

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11/19/09 CSE 5311 Spring 2007 M Kumar 2 Dynamic programming Solves problems by combining the solutions to subproblems DP is applicable when subproblems are not independent Subproblems share subsubproblems In such cases a simple Divide and Conquer strategy solves common subsubproblems . In DP every subproblem is solved just once and the solution is saved in a table for future reference (avoids re-computation). DP is typically applied to optimization problems A given problem may have many solutions, DP chooses the optimal solution.
11/19/09 CSE 5311 Spring 2007 M Kumar 3 Four stages of Dynamic Programming Characterize the structure of an optimal solution Recursively define the value of an optimal solution Compute the value of an optimal solution in a bottom-up fashion Construct an optimal solution from computed results

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11/19/09 CSE 5311 Spring 2007 M Kumar 4 Longest common subsequence A subsequence is formed from a list by deleting zero or more elements (the remaining elements are in order) A common subsequence of two lists is a subsequence of both. The longest common subsequence (LCS) of two lists is the longest among the common subsequences of the two lists. Example: abcabba and cbabac are two sequences baba is a subsequence of both
11/19/09 CSE 5311 Spring 2007 M Kumar 5 a b c a b b a b a b a c b a b a c

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11/19/09 CSE 5311 Spring 2007 M Kumar 6 To find the length of an LCS of lists x and y, we need to find the lengths of the LCSs of all pairs of prefixes. a prefix is an initial sublist of a list If x = ( a 1 ,a 2 ,a 3 , . . ., a m ) and y = ( b 1 ,b 2 ,b 3 , . . ., b n ) 0 i m and 0 j n Consider an LCS of the prefix ( a 1 ,a 2 ,a 3 , . . ., a i ) from x and of the prefix ( b 1 ,b 2 ,b 3 , . . ., b j ) from y. If i or j = 0 then one of the prefixes is ε and the only possible common subsequence between x and y is and the length of the LCS is zero.
11/19/09 CSE 5311 Spring 2007 M Kumar 7 L(i,j ) is the length of the LCS of ( a 1 ,a 2 ,a 3 , . . ., a i ) and ( b 1 ,b 2 ,b 3 , . . ., b j ). BASIS: If i+j = 0, then both i and j are zero and so the LCS is ε . INDUCTION: Consider i and j , and suppose we have already computed L( g,h ) for any g and h such that g+h < i+j . 1. If either i or j is 0 then L(i,j) = 0. 2. If i >0 and j >0, and a i b j then L( i,j ) = max(L( i,j-1 ),L( i-1,j )). 3. If i >0 and j > 0, and a i = b j then L( i,j ) = L( i-1,j-1 )+1.

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11/19/09 CSE 5311 Spring 2007 M Kumar 8 ε a b c a ε a c b 0 0 0 0 0 0 1 1 1 1 1.If either i or j is 0 then L(i,j) = 0. 2.If i >0 and j >0, and a i bj then L( i,j ) = max(L( i,j-1 ),L( i-1,j )). 3.If i >0 and j > 0, and a i = b j then L( i,j ) = L( i-1,j-1 )+1 . 0 1 1 2 2
11/19/09 CSE 5311 Spring 2007 M Kumar 9 Procedure LCS( x,y ) Input : The lists x and y Output : The longest common subsequence and its length 1. for j 0 to n do 2. L[0,j] 0; 3. for i 1 to m do 4. L[i,0] 0; 5. for j 1 to n do 6. if a[i] b[j] then 7. L[i,j] max {L[i-1,j],L[i,j-1]}; 8. else 9 L[i,j] 1+L[i-1,j-1];

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11/19/09 CSE 5311 Spring 2007 M Kumar 10 Example:
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Module_6 - Dynamic programming techniques Topics Basics of...

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