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I am trying to find the maximum palindrome sub-sequence and after going through some tutorials, I came up with a memoized version.But I am not sure about the runtime.I want to know if the following algorithm will work.Could also someone explain what the runtime will be?

Memoized-Palindrome(A,n)
initialize longest [i][j] =0 for all i and j
then return Memoized-Palindrome1(A,1,n,longest)

Memoized-Palindrome1(A,i,j,longest)
if longest[i][j]>0 return longest [i][j]
if (j-i) <=1 return j-i
if A[i]==A[j] 
      then longest[i][j] = 2 + Memoized-Palindrome1(A,i+1,j-1,longest)
    else 
      longest[i][j]= max(Memoized-Palindrome1(A,i+1,j,longest),Memoized-Palindrome1(A,i,j+1,longest)
return longest[i][j]
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Replace j+1 with j-1, also in the base case you should make a case analysis: if i == j and if j == i + 1 then you have to test whether A[i] == A[j]. –  jmad Nov 12 '12 at 19:26
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1 Answer 1

up vote 1 down vote accepted

This kind of approach is more often called dynamic programming than memoization but they are, indeed, related.

In dynamic programming, however, you usually write the order in which your function is called. (This is one key difference between them.) In this case, you will clearly see that you will compute $n$ longest[i][i] first, then $n-1$ longest[i][i+1], then $n-2$ longest[i][i+2] etc. each computation being in constant time. Giving a complexity of $n+(n-1)+\dots+2+1$=$n(n+1)/2$ i.e. $O(n^2)$.

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but One can say that the memoization belongs to dynamic programming, right? I think that is a top-down case of dp. –  d555 Nov 12 '12 at 20:50
    
@d555: not really: if dynamic programming could be seen as an example of memoization, the contrary is awkward. But even then, the techniques used are so different that dynamic programming algorithms are usually not qualified memoized. –  jmad Nov 12 '12 at 21:43
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