给定一个容量为C的背包和两个数组w[]和val[]代表N 个不同物品的重量和值，任务是找到可以放入背包的最大值。物品不能被破坏，重量为X的物品需要背包的X容量。
例子：
Input: w[] = {3, 4, 5}, val[] = {30, 50, 60}, C = 8
Output: 90
We take objects ‘1’ and ‘3’.
The total value we get is (30 + 60) = 90.
Total weight is 8, thus it fits in the given capacity
Input: w[] = {10000}, val[] = {10}, C = 100000
Output: 10
方法：传统著名的0-1背包问题可以在O(N*C)时间内解决，但如果背包的容量很大，则无法制作2D N*C阵列。幸运的是，它可以通过重新定义 dp 的状态来解决。
我们先来看看DP的状态。
dp[V][i]表示获得至少为V的值所需的子数组 arr[i…N-1]的最小权重子集。递推关系为：
dp[V][i] = min(dp[V][i+1], w[i] + dp[V – val[i]][i + 1])
因此，对于从0到V可能的最大值的每个V ，尝试找出给定的V 是否可以用给定的数组表示。可以表示的最大这样的V成为所需的答案。
下面是上述方法的实现：
C++
// C++ implementation of the approach
#include
using namespace std;
#define V_SUM_MAX 1000
#define N_MAX 100
#define W_MAX 10000000
// To store the states of DP
int dp[V_SUM_MAX + 1][N_MAX];
bool v[V_SUM_MAX + 1][N_MAX];
// Function to solve the recurrence relation
int solveDp(int r, int i, int* w, int* val, int n)
{
// Base cases
if (r <= 0)
return 0;
if (i == n)
return W_MAX;
if (v[r][i])
return dp[r][i];
// Marking state as solved
v[r][i] = 1;
// Recurrence relation
dp[r][i]
= min(solveDp(r, i + 1, w, val, n),
w[i] + solveDp(r - val[i],
i + 1, w, val, n));
return dp[r][i];
}
// Function to return the maximum weight
int maxWeight(int* w, int* val, int n, int c)
{
// Iterating through all possible values
// to find the the largest value that can
// be represented by the given weights
for (int i = V_SUM_MAX; i >= 0; i--) {
if (solveDp(i, 0, w, val, n) <= c) {
return i;
}
}
return 0;
}
// Driver code
int main()
{
int w[] = { 3, 4, 5 };
int val[] = { 30, 50, 60 };
int n = sizeof(w) / sizeof(int);
int C = 8;
cout << maxWeight(w, val, n, C);
return 0;
} Java
// Java implementation of the approach
class GFG
{
static final int V_SUM_MAX = 1000;
static final int N_MAX = 100;
static final int W_MAX = 10000000;
// To store the states of DP
static int dp[][] = new int[V_SUM_MAX + 1][N_MAX];
static boolean v[][] = new boolean [V_SUM_MAX + 1][N_MAX];
// Function to solve the recurrence relation
static int solveDp(int r, int i, int w[],
int val[], int n)
{
// Base cases
if (r <= 0)
return 0;
if (i == n)
return W_MAX;
if (v[r][i])
return dp[r][i];
// Marking state as solved
v[r][i] = true;
// Recurrence relation
dp[r][i] = Math.min(solveDp(r, i + 1, w, val, n),
w[i] + solveDp(r - val[i],
i + 1, w, val, n));
return dp[r][i];
}
// Function to return the maximum weight
static int maxWeight(int w[], int val[],
int n, int c)
{
// Iterating through all possible values
// to find the the largest value that can
// be represented by the given weights
for (int i = V_SUM_MAX; i >= 0; i--)
{
if (solveDp(i, 0, w, val, n) <= c)
{
return i;
}
}
return 0;
}
// Driver code
public static void main (String[] args)
{
int w[] = { 3, 4, 5 };
int val[] = { 30, 50, 60 };
int n = w.length;
int C = 8;
System.out.println(maxWeight(w, val, n, C));
}
}
// This code is contributed by AnkitRai01Python3
# Python3 implementation of the approach
V_SUM_MAX = 1000
N_MAX = 100
W_MAX = 10000000
# To store the states of DP
dp = [[ 0 for i in range(N_MAX)]
for i in range(V_SUM_MAX + 1)]
v = [[ 0 for i in range(N_MAX)]
for i in range(V_SUM_MAX + 1)]
# Function to solve the recurrence relation
def solveDp(r, i, w, val, n):
# Base cases
if (r <= 0):
return 0
if (i == n):
return W_MAX
if (v[r][i]):
return dp[r][i]
# Marking state as solved
v[r][i] = 1
# Recurrence relation
dp[r][i] = min(solveDp(r, i + 1, w, val, n),
w[i] + solveDp(r - val[i], i + 1,
w, val, n))
return dp[r][i]
# Function to return the maximum weight
def maxWeight( w, val, n, c):
# Iterating through all possible values
# to find the the largest value that can
# be represented by the given weights
for i in range(V_SUM_MAX, -1, -1):
if (solveDp(i, 0, w, val, n) <= c):
return i
return 0
# Driver code
if __name__ == '__main__':
w = [3, 4, 5]
val = [30, 50, 60]
n = len(w)
C = 8
print(maxWeight(w, val, n, C))
# This code is contributed by Mohit KumarC#
// C# implementation of the approach
using System;
class GFG
{
static readonly int V_SUM_MAX = 1000;
static readonly int N_MAX = 100;
static readonly int W_MAX = 10000000;
// To store the states of DP
static int [,]dp = new int[V_SUM_MAX + 1, N_MAX];
static bool [,]v = new bool [V_SUM_MAX + 1, N_MAX];
// Function to solve the recurrence relation
static int solveDp(int r, int i, int []w,
int []val, int n)
{
// Base cases
if (r <= 0)
return 0;
if (i == n)
return W_MAX;
if (v[r, i])
return dp[r, i];
// Marking state as solved
v[r, i] = true;
// Recurrence relation
dp[r, i] = Math.Min(solveDp(r, i + 1, w, val, n),
w[i] + solveDp(r - val[i],
i + 1, w, val, n));
return dp[r, i];
}
// Function to return the maximum weight
static int maxWeight(int []w, int []val,
int n, int c)
{
// Iterating through all possible values
// to find the the largest value that can
// be represented by the given weights
for (int i = V_SUM_MAX; i >= 0; i--)
{
if (solveDp(i, 0, w, val, n) <= c)
{
return i;
}
}
return 0;
}
// Driver code
public static void Main(String[] args)
{
int []w = { 3, 4, 5 };
int []val = { 30, 50, 60 };
int n = w.Length;
int C = 8;
Console.WriteLine(maxWeight(w, val, n, C));
}
}
// This code is contributed by 29AjayKumarJavascript
输出：
90时间复杂度： O(V_sum * N) 其中 V_sum 是数组 val[] 中所有值的总和。
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