Mobile phones,POJ,1195

题目描述

Farmer John has gone into the cell phone business, and he has a rather unique business model. Instead of simply offering a few fixed plans for his users to choose from, he allows them to create their own plans by selecting a list of features, and then he charges them a certain amount per feature. He needs your help in determining the price he should quote for each plan.

The features that FJ offers are of two types: “Basic” features, and “Advanced” features. There are ( B ) basic features, which each user can either choose or decline. There are ( A ) advanced features, of which each user must choose exactly ( K ) (or else they get no service at all).

For each basic feature ( i ), FJ charges ( b_i ) dollars per month. For each combination of ( K ) advanced features, FJ charges a certain amount which is different for each combination. You job is to write a program that reads in the values for ( A ), ( B ), ( K ), and all of the associated costs, and determines the least expensive plan for each of FJ’s potential customers.

输入输出格式

输入格式:

Line 1: Three space-separated integers ( A ), ( B ), and ( K ) Lines 2..( A+1 ): Line ( i+1 ) contains the cost of the ( i )th advanced feature Lines ( A+2 )..( A+B+1 ): Line ( i+A+1 ) contains the cost of the ( i )th basic feature

输出格式:

Line 1: The cost of the cheapest plan

输入输出样例

输入样例#1:

3 2 2 100 200 300 10 20

输出样例#1:

220

题目分析

这道题目是一个组合优化的经典问题。我们需要为每个用户选择一组基本功能和一组高级功能，使得总费用最低。基本功能可以选择不选，而高级功能必须恰好选择 ( K ) 个。

解决思路

1. 处理基本功能费用：
   • 基本功能可以选择不选，因此每个基本功能的费用可以视为0或其本身的费用。
2. 处理高级功能费用：
   • 高级功能必须选择恰好 ( K ) 个，我们需要从 ( A ) 个高级功能中选择 ( K ) 个的组合，并计算其费用。
3. 组合费用计算：
   • 使用组合数学的方法，枚举所有可能的 ( K ) 个高级功能的组合，并计算其费用。
   • 对于每种组合，加上所有基本功能的费用（选或不选的最小值）。
4. 优化：
   • 使用动态规划或贪心算法优化组合费用的计算。

具体实现步骤

1. 输入读取：
   • 读取 ( A ), ( B ), ( K )。
   • 读取 ( A ) 个高级功能的费用。
   • 读取 ( B ) 个基本功能的费用。
2. 计算高级功能组合费用：
   • 使用组合枚举的方法，计算每种 ( K ) 个高级功能组合的费用。
3. 计算总费用：
   • 对于每种高级功能组合，加上基本功能的最小费用。
4. 输出最小费用。

代码实现（Python示例）

from itertools import combinations

A = int(data[0])
B = int(data[1])
K = int(data[2])

advanced_costs = list(map(int, data[3:3+A]))
basic_costs = list(map(int, data[3+A:3+A+B]))

# 计算所有基本功能的最小费用（选或不选）
min_basic_cost = sum(min(0, cost) for cost in basic_costs)

# 计算所有高级功能组合的费用
min_cost = float('inf')
for combo in combinations(advanced_costs, K):
    combo_cost = sum(combo)
    total_cost = combo_cost + min_basic_cost
    if total_cost < min_cost:
        min_cost = total_cost

print(min_cost)

if name == "main": main()

解释

1. 输入处理：
   • 使用 sys.stdin.read 快速读取所有输入，然后分割成列表处理。
2. 组合枚举：
   • 使用 itertools.combinations 枚举所有 ( K ) 个高级功能的组合。
3. 费用计算：
   • 对于每种组合，计算其费用并加上基本功能的最小费用。
4. 输出最小费用。

这个实现简洁明了，利用了Python标准库中的组合枚举功能，适合处理题目中给定的数据规模。对于更大规模的数据，可以考虑进一步优化算法，例如使用动态规划预处理组合费用等。