New Efficient Algorithms for LCS and Constrained LCS Problem

Algorithm 1 AlgLCSNew1: Construct the set M using Algorithm 1 of [24]. Let M i = {(i, j) | (i, j) ∈ M, 1 ≤j ≤ n}.2: globalLCS.Instance = ɛ3: globalLCS.Value = ɛ4: H 0 = ɛ5: for i = 1 to n do6: H i = H i−17: for each (i, j) ∈ M i do8: maxresult = BoundedMax(H i−1 , j)9: T [i, j].Value = maxresult.Value + 110: T [i, j].Prev = maxresult.Instance11: if globalLCS.Value < T [i, j].Value then12: globalLCS.Value = T [i, j].Value13: globalLCS.Instance = (i, j)14: end if15: IncreaseV alue(H i , j, T [i, j].Value, (i, j)).16: end for17: Delete H i−1 .18: end for19: return globalLCSwhere⎧1 + T [i − 1, j − 1, k − 1] if(k = 1 or⎪⎨T ′ (k > 1 and T [i − 1, j − 1, k − 1] > 0))[i, j, k] =and X[i] = Y [j] = Z[k].⎪⎩0 otherwise.(3)and⎧⎪⎨ 1 + T [i − 1, j − 1, k] if (k = 0 or T [i − 1, j − 1, k] > 0)T ′′ [i, j, k] =and X[i] = Y [j].⎪⎩0 otherwise.(4)The following boundary conditions are assumed in Equations 2 to 4:T [i, 0, k] = T [0, j, k] = 0, 0 ≤ i, j ≤ n, 0 ≤ k ≤ p.It is straightforward to give a O(pn 2 ) algorithm realizing the dynamic programmingformulation presented in Equations 2 to 4. Our goal is to present a new efficient algorithmusing the parameter R. In line of Equation 1, we can reformulate Equations 3and 4 as follows: