Problem:
Longest Common Subsequence For given two sequences $X$ and $Y$, a sequence $Z$ is a common subsequence of $X$ and $Y$ if $Z$ is a subsequence of both $X$ and $Y$. For example, if $X = \{a,b,c,b,d,a,b\}$ and $Y = \{b,d,c,a,b,a\}$, the sequence $\{b,c,a\}$ is a common subsequence of both $X$ and $Y$. On the other hand, the sequence $\{b,c,a\}$ is not a longest common subsequence (LCS) of $X$ and $Y$, since it has length 3 and the sequence $\{b,c,b,a\}$, which is also common to both $X$ and $Y$, has length 4. The sequence $\{b,c,b,a\}$ is an LCS of $X$ and $Y$, since there is no common subsequence of length 5 or greater. Write a program which finds the length of LCS of given two sequences $X$ and $Y$. The sequence consists of alphabetical characters. Input The input consists of multiple datasets. In the first line, an integer $q$ which is the number of datasets is given. In the following $2 \times q$ lines, each dataset which consists of the two sequences $X$ and $Y$ are given. Output For each dataset, print the length of LCS of $X$ and $Y$ in a line. Constraints  $1 \leq q \leq 150$  $1 \leq$ length of $X$ and $Y$ $\leq 1,000$  $q \leq 20$ if the dataset includes a sequence whose length is more than 100  Sample Input 1 3
abcbdab
bdcaba
abc
abc
abc
bc Sample Output 1 4
3
2 Reference Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution (Python):
def lcs(X, Y):
    dp = []
    for y in Y:
        bgn_idx = 0
        for i, cur_idx in enumerate(dp):
            chr_idx = X.find(y, bgn_idx) + 1
            
            if not chr_idx:
                break
            dp[i] = min(cur_idx, chr_idx)
            bgn_idx = cur_idx
        else:
            chr_idx = X.find(y, bgn_idx) + 1
            if chr_idx:
                dp.append(chr_idx)
    return len(dp)

q = int(input())
for i in range(q):
    X = input()
    Y = input()
    print(lcs(X, Y))

Solution (Python):
from typing import List


def compute_lcs(x_str: str, y_str: str) -> int:
    if len(x_str) <= len(y_str):
        shorter_str, longer_str = x_str, y_str
    else:
        shorter_str, longer_str = y_str, x_str

    lcs_idx: List[int] = []
    for s_ch in shorter_str:
        l_bgn_idx = 0
        for i, l_cur_idx in enumerate(lcs_idx):
            l_idx = longer_str.find(s_ch, l_bgn_idx)
            if -1 == l_idx:
                break
            lcs_idx[i] = min(l_cur_idx, l_idx)
            l_bgn_idx = l_cur_idx + 1  
        l_idx = longer_str.find(s_ch, l_bgn_idx)
        if -1 != l_idx:
            lcs_idx.append(l_idx)
    return len(lcs_idx)


if __name__ == "__main__":
    num_pairs = int(input())

    for _ in range(num_pairs):
        x_str = input()
        y_str = input()
        lcs = compute_lcs(x_str, y_str)
        print(f"{lcs}")

Solution (Python):
def longest_common_subsequence(X, Y):
    X = " " + X
    Y = " " + Y

    length_X = len(X)
    length_Y = len(Y)

    c = [0] * length_Y

    for i in range(1, length_X):
        
        c_ = c[:]

        for j in range(1, length_Y):
            if X[i] == Y[j]:
                c[j] = c_[j - 1] + 1
            elif c_[j] < c[j - 1]:
                c[j] = c[j - 1]

    return c[-1]


q = int(input())


for _ in range(q):

    X = input()
    Y = input()

    print(longest_common_subsequence(X, Y))

Solution (Python):
def lcs(X, Y):
    C = []
    for y in Y:
        bgn_idx = 0
        for i, cur_idx in enumerate(C):
            chr_idx = X.find(y, bgn_idx) + 1
            if chr_idx == 0:
                break
            C[i] = min(cur_idx, chr_idx)
            bgn_idx = cur_idx
        chr_idx = X.find(y, bgn_idx) + 1
        if chr_idx != 0:
            C.append(chr_idx)
    return len(C)

q = int(input())
for _ in range(q):
    X = input()
    Y = input()
    print(lcs(X, Y))

Solution (Python):
def lcs(s1_raw, s2_raw):
    dp = []
    for s2 in s2_raw:
        start = 0
        for i, cnt in enumerate(dp):
            index = s1_raw.find(s2, start) + 1
            if not index:
                break
            else:
                dp[i]  = min(cnt , index)
                start = cnt
        index = s1_raw.find(s2, start) + 1
        if index:
            dp.append(index)
    return len(dp)


if __name__ == "__main__":
    q = int(input())
    for _ in range(q):
        x = input()
        y = input()
        print(lcs(x, y))

Solution (Python):
def main():
    N = int(input())

    string_data = [input() for _ in range(N * 2)]

    for _ in range(N):
        s = string_data.pop(0)
        t = string_data.pop(0)

        dp = [0] * (len(t) + 1)

        for i, sc in enumerate(s):
            tmp = dp[:]
            for j, tc in enumerate(t):
                if sc == tc:
                    dp[j + 1] = tmp[j] + 1
                elif dp[j + 1] < dp[j]:
                    dp[j + 1] = dp[j]

        print(dp[-1])


if __name__ == '__main__':
    main()

Solution (Python):

"""
http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ALDS1_10_C

最長共通部分列
"""


c = []
def prep_array(X, Y):
    global c
    """ 値が0の2次元配列を用意する。サイズは入力データの最大値を想定して縦横(1000+1)以上必要。 """
    
    
    c = [[0]*(len(Y)+2) for _ in range(len(X)+2)] 
    return c


def check_max(array):
    """ 2次元配列の中の最大の値を返す """
    max_lcs = 0
    for row in array:
        temp_max = max(row)
        if temp_max > max_lcs:
            max_lcs = temp_max
    return max_lcs


def calc_lcs4(X, Y, c):
    """
    Y[j]のアクセス方法を変更
    リストの値を一旦変数に保存するように変更
    最長共通部分長を関数内で計算するように変更
    4/5 のテストデータを使用してプロファイリングすると実効時間は 5.6秒
    """
    X = ' ' + X
    Y = ' ' + Y
    m = len(X)
    n = len(Y)
    max_lcs = 0

    pre_row = c[0]
    for i in range(1, m):
        row = c[i]
        XX = X[i]
        for j, YY in enumerate(Y):
            if XX == YY:
                pr_j1 = pre_row[j - 1]
                row[j] = pr_j1 + 1
                if pr_j1+1 > max_lcs:
                   max_lcs = pr_j1 + 1
            else:
                pr_j = pre_row[j]
                r_j1 = row[j-1]
                if pr_j >= r_j1:
                    row[j] = pr_j
                else:
                    row[j] = r_j1
        pre_row = row
    return max_lcs


if __name__ == '__main__':
    
    num = int(input())
    
    

    for _ in range(num):
        X = input().strip()
        Y = input().strip()

        
        result = calc_lcs4(X, Y, prep_array(X, Y))

        
        print(result)

Solution (Python):


def lcs(str1, str2):
    
    dp = [[0 for _ in range(len(str2)+10)] for _ in range(len(str1)+10)]
    
    for i in range(0,len(str1)):
        for j in range(0,len(str2)):
            v = max(dp[i+1][j],dp[i][j+1])
            if str1[i] == str2[j]:
                dp[i+1][j+1] = max(v,dp[i][j] + 1)
            else:
                dp[i+1][j+1] = max(v,dp[i][j])
    return dp[len(str1)][len(str2)]


def lcs_another(str1, str2):
    
    
    dp = [0 for _ in range(len(str2)+10)]
    for c1 in str1:
        tmp = dp[:]
        for i, c2 in enumerate(str2):
            if c1 == c2:
                dp[i+1] = tmp[i]+1
            elif dp[i+1] < dp[i]:
                dp[i+1] = dp[i]
    return dp[len(str2)]

p = int(input().rstrip())
for _ in range(p):
    line1 = input().rstrip()
    line2 = input().rstrip()
    print(lcs_another(line1,line2))

Solution (Python):
import bisect

INF = float('inf')
q = int(input())
for i in range(q):
    x = input()
    y = input()
    dic = {}
    for i,xx in enumerate(x):
        if xx in dic : dic[xx].append(i)
        else : dic[xx] = [i]
    inc = []
    size = -1
    for j,yy in enumerate(y):
        if not (yy in dic):continue
        inc.append([])
        size += 1
        for k in dic[yy]:
            inc[size].append(k)
    
    n = len(inc)
    dp =  [INF for i in range(n)]
    temp = [INF for i in range(n)]

    for ii in inc:
        for i in ii:
            p = bisect.bisect_left(dp, i)
            temp[p] = min(i,temp[p])
        dp = temp[:]


    print(bisect.bisect_left(dp, INF))

Solution (Python):
def lcs(s1, s2):
    dp = []
    for s2_k in s2:
        bgn_idx = 0
        for i, cur_idx in enumerate(dp):
            chr_idx = s1.find(s2_k, bgn_idx) + 1
            if not chr_idx:
                break
            dp[i] = min(cur_idx, chr_idx)
            bgn_idx = cur_idx
        else:
            chr_idx = s1.find(s2_k, bgn_idx) + 1
            if chr_idx:
                dp.append(chr_idx)
    return len(dp)


n = int(input())
for _ in range(n):
    x = input()
    y = input()
    print(lcs(x, y))

Solution (Python):
import sys
def longest_common_subsequence(line1, line2):
    n_rows = len(line2) + 1
    array = [0] * n_rows

    ix_ = [x for x in range(n_rows)]
    ix = [x+1 for x in range(n_rows)]

    for l1_letter in line1:
        prev_array = array[:]
        for i_row_, i_row, l2_letter in zip(ix_, ix, line2):
            if l1_letter == l2_letter:
                array[i_row] = prev_array[i_row_] + 1
            else:
                up = prev_array[i_row]
                left = array[i_row_]
                array[i_row] = up if up > left else left
    return array[-1]

calc_time = False
if calc_time: import time; start = time.time()
def main():
    n = int(input())
    lines = sys.stdin.readlines()
    for i in range(0, 2*n, 2):
        ans = longest_common_subsequence(lines[i].rstrip(), lines[i+1].rstrip())
        print(ans)
    return

main()
if calc_time:print("elapsed:",time.time()-start)

Solution (Python):
def LCS(X, Y):
    n = len(X)
    m = len(Y)
    T = [0]*(n+1)
    
    for j in range(m):
        T_old = T[:]
        c = Y[j]
        for i in range(n):
            if X[i] == c:
                T[i] = T_old[i-1] + 1
            elif T[i-1] > T[i]:
                T[i] = T[i-1]            
    return T[len(X)-1] 



q = int(input())
for cnt in range(q):
    X = input()
    Y = input()
    print(LCS(X, Y))

Solution (Python):



from sys import setrecursionlimit
from functools import lru_cache


def lcs(s1, s2):
    cost = [0]
    for c in s2:
        
        for i in reversed(range(len(cost))):
            index = s1.find(c, cost[i]) + 1
            if index:
                if i + 1 < len(cost):
                    cost[i + 1] = min(cost[i + 1], index)
                else:
                    cost.append(index)
            
    return len(cost) - 1


def main():
    setrecursionlimit(10000)

    q = int(input())

    for _ in range(q):
        x = input()
        y = input()
        print(lcs(x, y))


if __name__ == '__main__':
    
    
    
    main()

Solution (Python):


def lcs(x, y):
    m = len(x)
    n = len(y)
    c = [0] * (n + 1)
    for i in range(m):
        c2 = c[:]
        for j in range(n):
            if x[i] == y[j]:
                c[j + 1] = c2[j] + 1
            elif c[j + 1] < c[j]:
                c[j + 1] = c[j]
    return c[-1]


n = int(input())
Z = [0] * n
for i in range(n):
    X = input()
    Y = input()
    Z[i] = lcs(X, Y)
for z in Z:
    print(z)

Solution (Python):
def longest_common(x, y):
  cost = [0]
  for c in y:
    for i in range(len(cost)-1, -1, -1):
      index = x.find(c, cost[i]) + 1
      if index:
        if i + 1 < len(cost):
          cost[i + 1] = min(cost[i+1], index)
        else:
          cost.append(index)
  return len(cost) -1

n = int(input())
for _ in range(n):
  x = input()
  y = input()
  print(longest_common(x, y))

Solution (Python):
def common(x,y):
    n = len(x)
    m = len(y)
    c1 = [0for i in range(m+1)]
    for i in range(n):
        e1 = x[i]
        c2 = c1[:]
        for j in range(m):
            if (e1 == y[j]):
                c1[j+1] = c2[j] + 1
            elif (c1[j+1] < c1[j]):
                c1[j+1] = c1[j]
    return(c1[-1])


t = int(input())
for w in range(t):
    fseq = str(input()) 
    sseq = str(input()) 
    print(common(fseq,sseq))

Solution (Python):
def lcs(x, y):
    a = len(x)
    b = len(y)
    c1 = [0] * (b + 1)
    for i in range(a):
        e1 = x[i]
        c2 = c1[:]
        for j in range(b):
            if e1 == y[j]:
                c1[j+1] = c2[j] + 1
            elif c1[j+1] < c1[j]:
                c1[j+1] = c1[j]
    return c1[-1]
    
n = int(input().rstrip())
ret = []
for i in range(n):
    s1 = input().rstrip()
    s2 = input().rstrip()
    ret.append(lcs(s1, s2))
print(*ret, sep="\n")

Solution (Python):

def lcs_len(X,Y):
    m,n = len(X),len(Y)
    LCS_r = [0 for _ in range(m+1)]
    for y in Y:
        previous = LCS_r[:]
        for i,x in enumerate(X):
            if x == y:
                LCS_r[i+1] = previous[i] + 1
            elif LCS_r[i] > previous[i+1]:
                LCS_r[i+1] = LCS_r[i]
    return LCS_r[-1]

q = int(input())
for _ in range(q):
    X = input()
    Y = input()
    print(lcs_len(X,Y))