optimized version

sample.py

from typing import List
from collections import defaultdict

class Solution:
    def rob(self, nums: List[int]) -> int:
        N = len(nums)
        is_exists_cache = defaultdict(bool)
        cache_for_robbing = defaultdict(int)

        def rob_houses(index, sz):
            if index >= sz:
                return 0
            cache_key = str(sz) + str(index)
            if is_exists_cache[cache_key]:
                return cache_for_robbing[cache_key]

            amountFromFirst = rob_houses(index + 1, sz)
            amountFromSecond = rob_houses(index + 2, sz) + nums[index]
            is_exists_cache[cache_key] = True
            cache_for_robbing[cache_key] = max(amountFromFirst, amountFromSecond)
            return cache_for_robbing[cache_key]

        robingFromSecond = rob_houses(1, N)
        robbingFromFirst = rob_houses(2, N - 1) + nums[0]

        return max(robingFromSecond, robbingFromFirst)