題目：

LCS for input Sequences “ABCDGH” and “AEDFHR” is “ADH” of length 3.
LCS for input Sequences “AGGTAB” and “GXTXAYB” is “GTAB” of length 4.

實現起來比較簡單，狀態方程：

LCS F[i,v] = (max{F[i-1,v],F[i,v-1]},F[i-1,v-1] +1)[X[i] == Y[v]]

程式碼實現：G為標記函式

 import numpy as np
def LCS(X,Y,N,V):
#    L = [[None]*(n+1) for i in xrange(m+1)] 
 

    list = np.zeros((N+1,V+1),dtype = int)
    list[1:,1:] =-1
    G = np.full((N+1,V+1),'0')
    
    for i in range(1,N+1):
        for v in range(1,V+1):
            if X[i-1] == Y[v-1]:
                list[i][v] = list[i-1,v-1] +1
                G[i][v] = '7'
            else:
                if
 
 list[i-1,v] > list[i,v-1]:
                    list[i][v] = list[i-1,v]
                    G[i][v] = '^'
                else:
                    list[i][v] =list[i,v-1]
                    G[i][v] = '<'
    return list,G

主要介紹標記函式和追蹤解，雙針模型實現：

# 使用遞迴的方式
def print_LCS_Rec(X,Y,G,N,V):

    if
 
 N == 0 or V == 0:
        return 
    elif G[N,V] == '7':
        print_LCS_Rec(X,Y,G,N-1,V-1) 
        print X[N-1],
    elif G[N,V] == '^':
        print_LCS_Rec(X,Y,G,N,V-1)
    else :
        print_LCS_Rec(X,Y,G,N-1,V)
    return 
 # 使用非遞迴方式
def print_LCS_(X,Y,G,N,V):
    
    i = N
    v = V
    result = []
    
    while i > 0 and v > 0:
        if G[i,v] == '7':
            result += X[i-1]
            i -=1
            v -=1
            
        elif G[i,v] == '^':
            i -=1
        else:
            v -=1           
    return result[::-1]      
 # 直接使用DP表來追蹤也可以實現
def print_LCS(X,Y,list,N,V):
    
    i = N
    v = V
    result = []
    
    while i > 0 and v > 0:
        if X[i-1] == Y[v-1]:
            result += X[i-1]
            i -=1
            v -=1
            
        elif list[i-1,v] > list[i,v-1]:
            i -=1
        else:
            v -=1           
    return result[::-1]

執行結果

#%%
X = 'breather'
Y = 'conservatives'

N = len(X)
V = len(Y)

value,path = LCS(X,Y,N,V)
print value
print path

print_LCS_Rec(X,Y,path,N,V)                

print print_LCS(X,Y,value,N,V)
print print_LCS_(X,Y,path,N,V)


[[0 0 0 0 0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 1 1 1 1 1 1 1 1]
 [0 0 0 0 0 1 1 1 1 1 1 1 2 2]
 [0 0 0 0 0 1 1 1 2 2 2 2 2 2]
 [0 0 0 0 0 1 1 1 2 3 3 3 3 3]
 [0 0 0 0 0 1 1 1 2 3 3 3 3 3]
 [0 0 0 0 0 1 1 1 2 3 3 3 4 4]
 [0 0 0 0 0 1 2 2 2 3 3 3 4 4]]
[['0' '0' '0' '0' '0' '0' '0' '0' '0' '0' '0' '0' '0' '0']
 ['0' '<' '<' '<' '<' '<' '<' '<' '<' '<' '<' '<' '<' '<']
 ['0' '<' '<' '<' '<' '<' '7' '<' '<' '<' '<' '<' '<' '<']
 ['0' '<' '<' '<' '<' '7' '<' '<' '<' '<' '<' '<' '7' '<']
 ['0' '<' '<' '<' '<' '^' '<' '<' '7' '<' '<' '<' '<' '<']
 ['0' '<' '<' '<' '<' '^' '<' '<' '^' '7' '<' '<' '<' '<']
 ['0' '<' '<' '<' '<' '^' '<' '<' '^' '^' '<' '<' '<' '<']
 ['0' '<' '<' '<' '<' '7' '<' '<' '^' '^' '<' '<' '7' '<']
 ['0' '<' '<' '<' '<' '^' '7' '<' '<' '^' '<' '<' '^' '<']]
e
a
t
e
['e', 'a', 't', 'e']
['e', 'a', 't', 'e']