高级算法：动态规划（一）

一、暴力枚举

1、实现代码

def fib(n):
    f = [1,1]
    for i in range(2,n+1):
        f.append(f[-1]+f[-2])
    print(f)
    return f(n)

fib(5)

2、输出

"C:\Program Files\Python35\python.exe" E:/工作目录/python/test/DP.py
[1, 1, 2, 3, 5, 8]

Process finished with exit code 1

二、动态规划定义

1、什么是动态规划？

动态规划的英文名，是一种分阶段求解决策略的数学思想，它不止用于编程领域，也用于管理学，经济学、生物学

2、初始为1

实现代码

def LTS(x):
    F = [0 for _ in range(len(x))]
    F[0] = 1
    for k in range(1,len(F)):
        max_loc = None
        max_num = 0
        for i in range(1,k):
            if x[i] < x[k]:
                if F[i] > max_num:
                    max_loc = i
                    max_num = F[i]
        F[k] = max_num + 1
    return F

print(LTS([1,7,2,8,3,5,2]))

输出

"C:\Program Files\Python35\python.exe" E:/工作目录/python/test/DP.py
[1, 1, 1, 2, 2, 3, 1]

Process finished with exit code 0

2、初始为0

1、实现代码

def LIS(x):
    F = [0 for _ in range(len(x))]
    #初始化
    F[0] = 1
    for k in range(1,len(F)):
        max_loc = None
        max_num = 0
        #内层循环表示[0:R] 里所有小于x[k]的对应位置的F[i]最大值
        for i in range(0,k):
            if x[i] < x[k]:
                if F[i] > max_num:
                    max_loc = i
                    max_num = F[i]
        F[k] = max_num + 1
    return F

print(LIS([1,7,2,8,3,5,2]))

2、输出

"C:\Program Files\Python35\python.exe" E:/工作目录/python/test/DP.py
[1, 2, 2, 3, 3, 4, 2]

Process finished with exit code 0

三、动态规划最长上升子序列

1、实现代码

def fib(n):
    f = [1,1]
    for i in range(2,n+1):
        f.append(f[-1]+f[-2])
    print(f)
    return f(n)

# fib(5)

def LIS(x):
    F = [0 for _ in range(len(x))]
    p = [-1 for _ in range(len(x))]
    #初始化
    F[0] = 1
    p[0] = -1
    for k in range(1,len(F)):
        max_loc = -1
        max_num = 0
        #内层循环表示[0:R] 里所有小于x[k]的对应位置的F[i]最大值
        for i in range(0,k):
            if x[i] < x[k]:
                if F[i] > max_num:
                    max_loc = i
                    max_num = F[i]
        F[k] = max_num + 1
        p[k] = max_loc

    max_i = 0
    for i in range(1,len(F)):
        if F[i] > F[max_i]:
            max_i = i
    lis = []
    i = max_i
    while i >= 0:
        lis.append(x[i])
        i = p[i]
    lis.reverse()
    return lis

print(LIS([1,7,2,8,3,5,2]))

2、输出结果

"C:\Program Files\Python35\python.exe" E:/工作目录/python/test/DP.py
[1, 2, 3, 5]

Process finished with exit code 0

最长公共子序列2

 最长公共子序列1

 

动态规划最优子结构