"""
动态规划的核心：
将大问题分解为多个小问题，通过解决一个个小问题，
并保存解决每个小问题时的状态最终解决大问题

动态规划实现找零：
1. 先看找零1元需要多少：1
2. 找零2元需要多少：两张1元
3. 找零3元需要多少：找零2元的结果+找零1元的结果
"""


# 动态规划找零
def dp_rec_mc1(changes: list[int], cash: int):
    min_changes = [0 for _ in range(cash + 1)]
    for i in range(1, cash + 1):
        min_amount = i
        for c in filter(lambda x: x <= i, changes):
            temp_min_amount = 1 + min_changes[i - c]
            if temp_min_amount < min_amount:
                min_amount = temp_min_amount

        min_changes[i] = min_amount

    return min_changes[cash]


# 动态规划找零，并保存每一种找零所需的纸币情况
def dp_rec_mc2(changes: list[int], cash: int):
    change_used: dict[int, dict[int, int]] = {}
    min_amounts = [0 for _ in range(cash + 1)]
    for i in range(1, cash + 1):
        min_amount = i
        changes_less_then_i = filter(lambda x: x <= i, changes)
        for c in changes_less_then_i:
            temp_min_amount = 1 + min_amounts[i - c]
            if temp_min_amount < min_amount:
                min_amount = temp_min_amount
                if i - c > 0:
                    change_used[i] = change_used[i - c].copy()
                    if change_used[i].get(c):
                        change_used[i][c] += 1
                    else:
                        change_used[i][c] = 1
                else:
                    change_used[i] = {c: 1}

        min_amounts[i] = min_amount
        if min_amount == i:
            change_used[i] = {1: i}
    return change_used[cash]