I know that, T(n,m) = T(n-1,m) + T(n,m-1) + c it's the recurrence equation of Longest Common Subsequence algorithm. And the time complexity of the LCS in case of recursive method is O(2^n+m).

The base condition is: when m or n = 0, T(m,n) = 1 i.e., T(0,n)=1 and T(m,0)=1.

But, how do I derive the time complexity from the above recurrence relation?

  • $\begingroup$ Please only post to one site at a time. Cross-posted here: math.stackexchange.com/questions/4678728 $\endgroup$
    – Matt Groff
    Apr 14, 2023 at 3:42
  • $\begingroup$ The recurrence is symmetric in $n,m$, but the solution that you show is not. This cannot be. $\endgroup$
    – user16034
    Apr 14, 2023 at 8:05

1 Answer 1



A solution of the homogeneous recurrence is well-known to be


by Pascal's identity.

And a particular solution of the non-homogeneous solution is simply


  • 1
    $\begingroup$ and 2^{m+n} is another homogeneous solution $\endgroup$
    – 1001
    Apr 14, 2023 at 21:11
  • $\begingroup$ @1001: that does not meet the limit conditions. $\endgroup$
    – user16034
    Apr 15, 2023 at 9:48

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