longest sub-sequence in both directions

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Given a character array A of size n, compute the length of a longest sub-sequence S of A such that, S read backward is also a sub-sequence of A.



Example: A = cabca



the sub-sequence S = abc is the longest of such sub-sequences.



reading S forward (abc): cabca



reading S reverse (cba): cabca



How to compute the length of such an S ?






algorithms dynamic-programming strings subsequences







share|cite|improve this question

























    up vote
    2
    down vote

    favorite












    Given a character array A of size n, compute the length of a longest sub-sequence S of A such that, S read backward is also a sub-sequence of A.



    Example: A = cabca



    the sub-sequence S = abc is the longest of such sub-sequences.



    reading S forward (abc): cabca



    reading S reverse (cba): cabca



    How to compute the length of such an S ?






    algorithms dynamic-programming strings subsequences







    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Given a character array A of size n, compute the length of a longest sub-sequence S of A such that, S read backward is also a sub-sequence of A.



      Example: A = cabca



      the sub-sequence S = abc is the longest of such sub-sequences.



      reading S forward (abc): cabca



      reading S reverse (cba): cabca



      How to compute the length of such an S ?






      algorithms dynamic-programming strings subsequences







      share|cite|improve this question













      Given a character array A of size n, compute the length of a longest sub-sequence S of A such that, S read backward is also a sub-sequence of A.



      Example: A = cabca



      the sub-sequence S = abc is the longest of such sub-sequences.



      reading S forward (abc): cabca



      reading S reverse (cba): cabca



      How to compute the length of such an S ?






      algorithms dynamic-programming strings subsequences




      algorithms dynamic-programming strings subsequences






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 5 hours ago









      user3222

      707




      707




















          1 Answer
          1






          active

          oldest

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          up vote
          2
          down vote













          One option is to reduce your problem to Longest Common Subsequence. I'll let you figure out how.






          share|cite|improve this answer




















          • Thanks for this. Could you give a bigger hint please ?
            – user3222
            2 hours ago










          • Unfortunately it would be difficult for me to give further hints without giving away the solution.
            – Yuval Filmus
            1 hour ago










          • I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
            – user3222
            45 mins ago











          • Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
            – user3222
            33 mins ago










          • Could you please tell me what is wrong with my recurrence ?
            – user3222
            12 mins ago










          Your Answer




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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote













          One option is to reduce your problem to Longest Common Subsequence. I'll let you figure out how.






          share|cite|improve this answer




















          • Thanks for this. Could you give a bigger hint please ?
            – user3222
            2 hours ago










          • Unfortunately it would be difficult for me to give further hints without giving away the solution.
            – Yuval Filmus
            1 hour ago










          • I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
            – user3222
            45 mins ago











          • Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
            – user3222
            33 mins ago










          • Could you please tell me what is wrong with my recurrence ?
            – user3222
            12 mins ago














          up vote
          2
          down vote













          One option is to reduce your problem to Longest Common Subsequence. I'll let you figure out how.






          share|cite|improve this answer




















          • Thanks for this. Could you give a bigger hint please ?
            – user3222
            2 hours ago










          • Unfortunately it would be difficult for me to give further hints without giving away the solution.
            – Yuval Filmus
            1 hour ago










          • I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
            – user3222
            45 mins ago











          • Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
            – user3222
            33 mins ago










          • Could you please tell me what is wrong with my recurrence ?
            – user3222
            12 mins ago












          up vote
          2
          down vote










          up vote
          2
          down vote









          One option is to reduce your problem to Longest Common Subsequence. I'll let you figure out how.






          share|cite|improve this answer












          One option is to reduce your problem to Longest Common Subsequence. I'll let you figure out how.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 4 hours ago









          Yuval Filmus

          185k12176336




          185k12176336











          • Thanks for this. Could you give a bigger hint please ?
            – user3222
            2 hours ago










          • Unfortunately it would be difficult for me to give further hints without giving away the solution.
            – Yuval Filmus
            1 hour ago










          • I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
            – user3222
            45 mins ago











          • Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
            – user3222
            33 mins ago










          • Could you please tell me what is wrong with my recurrence ?
            – user3222
            12 mins ago
















          • Thanks for this. Could you give a bigger hint please ?
            – user3222
            2 hours ago










          • Unfortunately it would be difficult for me to give further hints without giving away the solution.
            – Yuval Filmus
            1 hour ago










          • I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
            – user3222
            45 mins ago











          • Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
            – user3222
            33 mins ago










          • Could you please tell me what is wrong with my recurrence ?
            – user3222
            12 mins ago















          Thanks for this. Could you give a bigger hint please ?
          – user3222
          2 hours ago




          Thanks for this. Could you give a bigger hint please ?
          – user3222
          2 hours ago












          Unfortunately it would be difficult for me to give further hints without giving away the solution.
          – Yuval Filmus
          1 hour ago




          Unfortunately it would be difficult for me to give further hints without giving away the solution.
          – Yuval Filmus
          1 hour ago












          I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
          – user3222
          45 mins ago





          I call it longest 2 sided sub-sequence, the following is my recurrence. | 0 if i>j l2sss(i,j) = | 1 if i = j | 2 + l2sss(i+1, j-1) if s[i]==s[j] //include both | max (2 + l2sss(i+1, find_last_of(s[i])-1), l2sss(i+1, j) two cases within max 1> include s[i], so the problem size become i+1 (because s[i] was included) to find_last_of(s[i]-1) and 2> I exclude s[i]. This does not seem to work (when I implemented this, it does not evaluate to the right answer) and seems to force the string S to be a palindrome (it wasn't a requirement).
          – user3222
          45 mins ago













          Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
          – user3222
          33 mins ago




          Sorry, I tried to pretty print that recurrence equation, it did not work out as I expected.
          – user3222
          33 mins ago












          Could you please tell me what is wrong with my recurrence ?
          – user3222
          12 mins ago




          Could you please tell me what is wrong with my recurrence ?
          – user3222
          12 mins ago

















           

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