动态规划小例子

1.最长增长子序列

# 最长递增子序列
def dp_1(lst):
    length = len(lst)
    dp = [1 for i in range(length)]
    ret = 1
    for i in range(1, length):
        for j in range(i):
            if lst[j] < lst[i]:
                dp[i] = max(dp[j] + 1, dp[i])
        ret = max(ret, dp[i])
    print(dp)
    return ret


if __name__ == '__main__':
    lst = list(map(lambda x: int(x), input('输入一列数字:').split(' ')))
    print(lst)
    print(dp_1(lst))

2.数字三角形

# 数字金字塔
def main():
    tower = []
    floor = int(input('请输入层数:'))
    for i in range(floor):
        tower.append(list(map(lambda x: int(x), input('输入第{}层的数字:'.format(i + 1)).split(' '))))
    print(tower)
    dp = [[0 for y in x] for x in tower]
    print(dp)
    for a in range(floor):
        dp[floor - 1][a] = tower[floor - 1][a]
    for b in range(floor - 2, -1, -1):
        for c in range(0, b + 1):
            dp[b][c] = max(dp[b + 1][c], dp[b + 1][c + 1]) + tower[b][c]
    print(dp[0][0])


if __name__ == '__main__':
    main()

3.斐波拉西数列

# 动态规划解决 fic

def main(len):
    status_list = [0 for i in range(len)]
    for i in range(len):
        if i == 0:
            status_list[i] = 1
        elif i == 1:
            status_list[i] = 1
        else:
            status_list[i] = status_list[i-1] + status_list[i-2]
    print(status_list)


if __name__ == '__main__':
    main(6)

4.求矩阵从左上到右下的最大路径和

res = 0
matrix = [
    [3, 4, 7],
    [2, 9, 8],
    [9, 5, 7],
    [8, 9, 9],
    [9, 3, 2]
]


def dp(x, y, status):
    global res
    if (x == y == 0):
        res = status + matrix[x][y]
    if (x - 1 >= 0 and y - 1 >= 0):
        # 状态转移方程
        dp(x, y - 1, status + matrix[x][y]) if matrix[x][y - 1] > matrix[x - 1][y] else dp(x - 1, y, status + matrix[x][y])
    if (x == 0 and y > 0):
        dp(x, y - 1, status + matrix[x][y])
    if (x > 0 and y == 0):
        dp(x - 1, y, status + matrix[x][y])


if __name__ == '__main__':
    dp(len(matrix) - 1, len(matrix[0]) - 1, 0)
    print(res)

解题思路：

• 1.找程序运行边界
• 2.大问题分解成子问题（递归或递推）
• 3.状态转移方程
• 4.保留每个子问题得到的状态值status