$$\color{Red}{货币系统:完全背包求方案数（二维——》一维）}$$

我做的所有的题解

包括基础提高以及一些零散刷的各种各样的题

证明

板子题，可以看我之前的题解。
完全背包问题:三种语言+原版和滚动数组优化

java 二维朴素

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;

public class Main {
    static int N = 3010, n, m;
    static long[][] f = new long[20][N];
    static BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

    public static void main(String[] args) throws IOException {
        String[] s1 = br.readLine().split(" ");
        n = Integer.parseInt(s1[0]);
        m = Integer.parseInt(s1[1]);
        f[0][0] = 1;
        for (int i = 1; i <= n; i++) {
            int v = Integer.parseInt(br.readLine());
            for (int j = 0; j <= m; j++) 
                for (int k = 0; k * v <= j; k++) 
                    f[i][j] += f[i - 1][j - k * v];
        }
        System.out.println(f[n][m]);
    }
}

java 二维优化

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;

public class Main {
    static int N = 3010, n, m;
    static long[][] f = new long[20][N];
    static BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

    public static void main(String[] args) throws IOException {
        String[] s1 = br.readLine().split(" ");
        n = Integer.parseInt(s1[0]);
        m = Integer.parseInt(s1[1]);
        f[0][0] = 1;
        for (int i = 1; i <= n; i++) {
            int v = Integer.parseInt(br.readLine());
            for (int j = 0; j <= m; j++) {
                f[i][j] = f[i-1][j];
                if(j >= v)
                    f[i][j] += f[i][j - v];
            }
        }
        System.out.println(f[n][m]);
    }
}

java 一维优化

能减少如此多的条件，关键就是在于
f[i][j] = f[i-1][j] + f[i][j - v]
我们for循环可以采取直接选择v开始，就是因为计算f[j] += f[j - v]此时调用的f[j]其实就是上一个i-1状态下的f[i-1][j]因此省却了先赋值上一层，然后再加错位项的情况

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;

public class Main {
    static int N = 3010, n, m;
    static long []f = new long[N];
    static BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

    public static void main(String[] args) throws IOException {
        String [] str = br.readLine().split(" ");
        n = Integer.parseInt(str[0]);
        m = Integer.parseInt(str[1]);
        f[0] = 1;
        for (int i = 1; i <= n; i++) {
            int v = Integer.parseInt(br.readLine());
            for (int j = v; j <= m; j++) {
                f[j] += f[j - v];
            }
        }
        System.out.println(f[m]);
    }
}