总和可被 M 整除的子集数 | 2套

给定一个由N 个整数组成的数组arr[]和一个整数M ，任务是找到非空子序列的数量，使得子序列的总和可以被M整除。

例子：

Input: arr[] = {1, 2, 3}, M = 1 
Output: 7 
Number of non-empty subsets of this array are 7. 
Since m = 1, m will divide all the possible subsets.
Input: arr[] = {1, 2, 3}, M = 2 
Output: 3  

方法：本文讨论了一种基于动态规划的方法，其时间复杂度为O(N * SUM) ，其中N是数组的长度， SUM是所有数组元素的总和。
在本文中，将讨论对先前方法的改进。代替 sum 作为 DP 的状态之一，(sum % m) 将用作 DP 的状态之一，因为它足够了。因此，时间复杂度归结为 O(m * N)。
递归关系：

dp[i][curr] = dp[i + 1][(curr + arr[i]) % m] + dp[i + 1][curr]  

现在让我们定义状态， dp[i][curr]仅表示子数组arr[i…N-1]的子集数，使得(其元素的总和 + curr) % m = 0 。

下面是上述方法的实现：

C++

// C++ implementation of the approach
#include 
using namespace std;
#define maxN 20
#define maxM 10
 
// To store the states of DP
int dp[maxN][maxM];
bool v[maxN][maxM];
 
// Function to find the required count
int findCnt(int* arr, int i, int curr, int n, int m)
{
    // Base case
    if (i == n) {
        if (curr == 0)
            return 1;
        else
            return 0;
    }
 
    // If the state has been solved before
    // return the value of the state
    if (v[i][curr])
        return dp[i][curr];
 
    // Setting the state as solved
    v[i][curr] = 1;
 
    // Recurrence relation
    return dp[i][curr] = findCnt(arr, i + 1,
                                 curr, n, m)
                         + findCnt(arr, i + 1,
                                   (curr + arr[i]) % m,
                                   n, m);
}
 
// Driver code
int main()
{
    int arr[] = { 3, 3, 3, 3 };
    int n = sizeof(arr) / sizeof(int);
    int m = 6;
 
    cout << findCnt(arr, 0, 0, n, m) - 1;
 
    return 0;
}

Java

// Java implementation of the approach
class GFG
{
static int maxN = 20;
static int maxM = 10;
 
// To store the states of DP
static int [][]dp = new int[maxN][maxM];
static boolean [][]v = new boolean[maxN][maxM];
 
// Function to find the required count
static int findCnt(int[] arr, int i,
                   int curr, int n, int m)
{
    // Base case
    if (i == n)
    {
        if (curr == 0)
            return 1;
        else
            return 0;
    }
 
    // If the state has been solved before
    // return the value of the state
    if (v[i][curr])
        return dp[i][curr];
 
    // Setting the state as solved
    v[i][curr] = true;
 
    // Recurrence relation
    return dp[i][curr] = findCnt(arr, i + 1,
                                 curr, n, m) +
                         findCnt(arr, i + 1,
                                (curr + arr[i]) % m,
                                 n, m);
}
 
// Driver code
public static void main(String[] args)
{
    int arr[] = { 3, 3, 3, 3 };
    int n = arr.length;
    int m = 6;
 
    System.out.println(findCnt(arr, 0, 0, n, m) - 1);
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python3 implementation of the approach
maxN = 20
maxM = 10
 
# To store the states of DP
dp = [[0 for i in range(maxN)]
         for i in range(maxM)]
v = [[0 for i in range(maxN)]
        for i in range(maxM)]
 
# Function to find the required count
def findCnt(arr, i, curr, n, m):
     
    # Base case
    if (i == n):
        if (curr == 0):
            return 1
        else:
            return 0
 
    # If the state has been solved before
    # return the value of the state
    if (v[i][curr]):
        return dp[i][curr]
 
    # Setting the state as solved
    v[i][curr] = 1
 
    # Recurrence relation
    dp[i][curr] = findCnt(arr, i + 1,
                          curr, n, m) + \
                  findCnt(arr, i + 1,
                         (curr + arr[i]) % m, n, m)
    return dp[i][curr]
 
# Driver code
arr = [3, 3, 3, 3]
n = len(arr)
m = 6
 
print(findCnt(arr, 0, 0, n, m) - 1)
 
# This code is contributed by Mohit Kumar

C#

// C# implementation of the approach
using System;
 
class GFG
{
    static int maxN = 20;
    static int maxM = 10;
     
    // To store the states of DP
    static int [,]dp = new int[maxN, maxM];
    static bool [,]v = new bool[maxN, maxM];
     
    // Function to find the required count
    static int findCnt(int[] arr, int i,
                       int curr, int n, int m)
    {
        // Base case
        if (i == n)
        {
            if (curr == 0)
                return 1;
            else
                return 0;
        }
     
        // If the state has been solved before
        // return the value of the state
        if (v[i, curr])
            return dp[i, curr];
     
        // Setting the state as solved
        v[i, curr] = true;
     
        // Recurrence relation
        return dp[i, curr] = findCnt(arr, i + 1,
                                     curr, n, m) +
                             findCnt(arr, i + 1,
                                    (curr + arr[i]) % m, n, m);
    }
     
    // Driver code
    public static void Main()
    {
        int []arr = { 3, 3, 3, 3 };
        int n = arr.Length;
        int m = 6;
     
        Console.WriteLine(findCnt(arr, 0, 0, n, m) - 1);
    }
}
 
// This code is contributed by kanugargng

Javascript

 
// Javascript implementation of the approach
var maxN = 20
var maxM = 10
 
// To store the states of DP
var dp = Array.from(Array(maxN), () => Array(maxM));
var v = Array.from(Array(maxN), () => Array(maxM));
 
// Function to find the required count
function findCnt(arr, i, curr, n, m)
{
    // Base case
    if (i == n) {
        if (curr == 0)
            return 1;
        else
            return 0;
    }
 
    // If the state has been solved before
    // return the value of the state
    if (v[i][curr])
        return dp[i][curr];
 
    // Setting the state as solved
    v[i][curr] = 1;
 
    // Recurrence relation
    dp[i][curr] = findCnt(arr, i + 1,
                                 curr, n, m)
                         + findCnt(arr, i + 1,
                                   (curr + arr[i]) % m,
                                   n, m);
 
    return dp[i][curr];
}
 
// Driver code
var arr = [3, 3, 3, 3];
var n = arr.length;
var m = 6;
document.write( findCnt(arr, 0, 0, n, m) - 1);
 

输出：

7

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