Problem: Longest Common Subsequence

 c[i-1][j-1]+1 & \quad \text{i,j>0,且$x_{i}=y_{j}$}

 max({c[i][j-1],c[i-1][j]}) & \quad \text{i,j>0,且$x_{i} \ne y_{j}$}   \end{array} \right. $$

def lcs0(i,j):
    #string starts at index 0, not 1
    if i<0 or j<0: return 0 #attention to this!!!
    if x[i]==y[j]:  return lcs0(i-1,j-1)+1
    return max(lcs0(i-1,j),lcs0(i,j-1))

x,y='abcde','oaob'
lenx,leny=len(x),len(y)
print(lcs0(lenx-1,leny-1)) #2

from functools import wraps

def memo(func):
    cache={}
    @wraps(func)
    def wrap(*args):
        if args not in cache:
            cache[args]=func(*args)
        return cache[args]
    return wrap

@memo
def lcs1(i,j):
    #string starts at index 0, not 1
    if i<0 or j<0: return 0 #attention to this!!!
    if x[i]==y[j]:  return lcs1(i-1,j-1)+1
    return max(lcs1(i-1,j),lcs1(i,j-1))

x,y='abcde','oaob'
lenx,leny=len(x),len(y)
print(lcs1(lenx-1,leny-1)) #2

def lcs2(x,y):
    lenx,leny=len(x),len(y)
    minlen,maxlen=0,0
    if lenx<leny: minlen,maxlen=lenx,leny; x,y=y,x
    else: minlen,maxlen=leny,lenx;
    #s is maxlen * minlen
    s=[[0 for j in range(minlen)] for i in range(maxlen)]
    for i in range(maxlen): #so, let x be the longer string!!!
        for j in range(minlen):
            if x[i]==y[j]: s[i][j]=s[i-1][j-1]+1
            else: s[i][j]=max(s[i-1][j],s[i][j-1])
    return s

x,y='abcde','oaob'
s=lcs2(x,y)
print(s) #[[0, 1, 1, 1], [0, 1, 1, 2], [0, 1, 1, 2], [0, 1, 1, 2], [0, 1, 1, 2]]

def lcs3(x,y):
    lenx,leny=len(x),len(y)
    minlen,maxlen=0,0
    if lenx<leny: minlen,maxlen=lenx,leny; x,y=y,x
    else: minlen,maxlen=leny,lenx;
    #s is maxlen * minlen
    pre=[0 for j in range(minlen)]
    cur=[0 for j in range(minlen)]
    for i in range(maxlen): #so, let x be the longer string!!!
        for j in range(minlen):
            if x[i]==y[j]: cur[j]=pre[j-1]+1
            else: cur[j]=max(pre[j],cur[j-1])
        pre[:]=cur[:]
    return cur

x,y='abcde','oaob'
s=lcs3(x,y)
print(s) #[2, 2, 2, 2]

def lcs4(x,y):
    lenx,leny=len(x),len(y)
    minlen,maxlen=0,0
    if lenx<leny: minlen,maxlen=lenx,leny; x,y=y,x
    else: minlen,maxlen=leny,lenx;
    #s is maxlen * minlen
    s=[0 for j in range(minlen)]
    t=0
    for i in range(maxlen): #so, let x be the longer string!!!
        for j in range(minlen):
            if x[i]==y[j]: s[j]=t+1
            else: s[j]=max(s[j],s[j-1])
            t=s[j]
    return s

x,y='abcde','oaobce'
s=lcs4(x,y)
print(s) #[3, 3, 3, 3, 4]