贪婪算法简介
贪婪算法（Greedy Algorithm）是一种在每一步选择中都采取当前状态下最好或最优（即最有利）的选择，从而希望导致结果是全局最好或最优的算法。贪婪算法通常用于解决优化问题，如最小化成本、最大化收益等。它并不保证得到最优解，但在很多问题中，它能产生很好的近似解或最优解。
贪婪算法的关键在于贪心选择性质：每一步所做的选择都是当前状态下的最优选择，并且这种选择不会依赖于子问题的解。
示例问题：找零钱问题
假设一个商店只销售价格为整数元的商品，并且只接受整数元的货币。给定一个商品的价格n元和一个无限数量的货币单位[1,2,5,10,20,50,100]（分别代表1元、2元、5元、10元、20元、50元和100元），编写一个程序计算找零给顾客的最少货币数量。
C# 实现
using System;
using System.Collections.Generic; class GreedyChangeMaker
{ static List<int> denominations = new List<int> { 100, 50, 20, 10, 5, 2, 1 }; static int MakeChange(int amount) { int count = 0; for (int i = 0; i < denominations.Count; i++) { while (amount >= denominations[i]) { amount -= denominations[i]; count++; } } return count; } static void Main() { int amount = 63; Console.WriteLine($"Minimum number of coins required: {MakeChange(amount)}"); }
}C++ 实现
#include <iostream>
#include <vector>
using namespace std; int makeChange(int amount) { vector<int> denominations = {100, 50, 20, 10, 5, 2, 1}; int count = 0; for (int coin : denominations) { while (amount >= coin) { amount -= coin; count++; } } return count;
} int main() { int amount = 63; cout << "Minimum number of coins required: " << makeChange(amount) << endl; return 0;
}Python 实现
def make_change(amount): denominations = [100, 50, 20, 10, 5, 2, 1] count = 0 for coin in denominations: count += amount // coin amount %= coin return count amount = 63
print(f"Minimum number of coins required: {make_change(amount)}")Java 实现
public class GreedyChangeMaker { private static final int[] denominations = {100, 50, 20, 10, 5, 2, 1}; public static int makeChange(int amount) { int count = 0; for (int denomination : denominations) { while (amount >= denomination) { amount -= denomination; count++; } } return count; } public static void main(String[] args) { int amount = 63; System.out.println("Minimum number of coins required: " + makeChange(amount)); }
}
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