检查一个数组是否可以分为相等和不相等的3个子序列

给定具有N个整数的数组arr [] 。任务是确定是否可以将数组划分为相等总和的3个子序列。如果是，则打印“是” 。否则，打印“否” 。

例子：

Input: arr[] = {1, 1, 1}
Output: Yes
Explanation:
Here array can be partition into 3 equal sum. {1}    

Input: arr[] = {40}
Output: No
Explanation:
Here array cannot be partition into 3 equal sum.

递归方法：可以使用递归来解决此问题。步骤如下：

1. 将三个变量sum1 ， sum2和sum3初始化为值0。
2. 然后，将数组arr []的每个元素添加到这3个变量中的任何一个，它们给出了所有可能的组合。
3. 如果任何子序列具有3个相等的和，则可以对数组进行分区。
4. 如果阵列可以分区，则打印“是”，否则打印“否” 。

C++

// C++ program for
// the above approach
#include 
using namespace std;
 
// Utility function to check array can
// be partition to 3 subsequences of
// equal sum or not
int checkEqualSumUtil(int arr[], int N,
                      int sm1, int sm2,
                      int sm3, int j)
{   
  // Base Case
  if (j == N)
  {
    if (sm1 == sm2 && sm2 == sm3)
      return 1;
    else
      return 0;
  }
 
  else
  {
    // When element at index
    // j is added to sm1
    int l = checkEqualSumUtil(arr, N,
                              sm1 + arr[j],
                              sm2, sm3, j + 1);
 
    // When element at index
    // j is added to sm2
    int m = checkEqualSumUtil(arr, N, sm1,
                              sm2 + arr[j],
                              sm3, j + 1);
 
    // When element at index
    // j is added to sm3
    int r = checkEqualSumUtil(arr, N, sm1, sm2,
                              sm3 + arr[j], j + 1);
 
    // Return maximum value among
    // all above 3 recursive call
    return max(max(l, m), r);
  }
}
 
// Function to check array can be
// partition to 3 subsequences of
// equal sum or not
void checkEqualSum(int arr[], int N)
{
  // Initialise 3 sums to 0
  int sum1, sum2, sum3;
 
  sum1 = sum2 = sum3 = 0;
 
  // Function Call
  if (checkEqualSumUtil(arr, N, sum1,
                        sum2, sum3, 0)== 1)
  {
    cout << "Yes";
  }
  else
  {
    cout << "No";
  }
}
 
// Driver Code
int main()
{
  // Given array arr[]
  int arr[] = {17, 34, 59, 23, 17, 67,
               57, 2, 18, 59, 1 };
 
  int N = sizeof(arr) / sizeof(arr[0]);
 
  // Function Call
  checkEqualSum(arr, N);
  return 0;
}

Java

// Java program for
// the above approach
class GFG{
 
// Utility function to check array can
// be partition to 3 subsequences of
// equal sum or not
static int checkEqualSumUtil(int arr[], int N,
                             int sm1, int sm2,
                             int sm3, int j)
{
  // Base Case
  if (j == N)
  {
    if (sm1 == sm2 && sm2 == sm3)
      return 1;
    else
      return 0;
  }
  else
  {
    // When element at index
    // j is added to sm1
    int l = checkEqualSumUtil(arr, N,
                              sm1 + arr[j],
                              sm2, sm3, j + 1);
 
    // When element at index
    // j is added to sm2
    int m = checkEqualSumUtil(arr, N, sm1,
                              sm2 + arr[j],
                              sm3, j + 1);
 
    // When element at index
    // j is added to sm3
    int r = checkEqualSumUtil(arr, N, sm1, sm2,
                              sm3 + arr[j], j + 1);
 
    // Return maximum value among
    // all above 3 recursive call
    return Math.max(Math.max(l, m), r);
  }
}
 
// Function to check array can be
// partition to 3 subsequences of
// equal sum or not
static void checkEqualSum(int arr[], int N)
{
  // Initialise 3 sums to 0
  int sum1, sum2, sum3;
 
  sum1 = sum2 = sum3 = 0;
 
  // Function Call
  if (checkEqualSumUtil(arr, N, sum1,
                        sum2, sum3, 0) == 1)
  {
    System.out.print("Yes");
  }
  else
  {
    System.out.print("No");
  }
}
 
// Driver Code
public static void main(String[] args)
{
  // Given array arr[]
  int arr[] = {17, 34, 59, 23, 17,
               67, 57, 2, 18, 59, 1};       
 
  int N = arr.length;
 
  // Function Call
  checkEqualSum(arr, N);
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 program for the above approach
 
# Utility function to check array can
# be partition to 3 subsequences of
# equal sum or not
def checkEqualSumUtil(arr, N, sm1, sm2, sm3, j):
     
    # Base case
    if j == N:
        if sm1 == sm2 and sm2 == sm3:
            return 1
        else:
            return 0
    else:
         
        # When element at index
        # j is added to sm1
        l = checkEqualSumUtil(arr, N, sm1 + arr[j],
                              sm2, sm3, j + 1)
         
        # When element at index
        # j is added to sm2
        m = checkEqualSumUtil(arr, N, sm1,
                              sm2 + arr[j],
                              sm3, j + 1)
         
        # When element at index
        # j is added to sm3
        r = checkEqualSumUtil(arr, N, sm1,
                              sm2, sm3 + arr[j],
                                     j + 1)
         
        # Return maximum value among
        # all above 3 recursive call
        return max(l, m, r)
     
# Function to check array can be
# partition to 3 subsequences of
# equal sum or not
def checkEqualSum(arr, N):
     
    # Initialise 3 sums to 0
    sum1 = sum2 = sum3 = 0
     
    # Function call
    if checkEqualSumUtil(arr, N, sum1,
                         sum2, sum3, 0) == 1:
        print("Yes")
    else:
        print("No")
     
# Driver code
     
# Given array arr[]
arr = [ 17, 34, 59, 23, 17,
        67, 57, 2, 18, 59, 1 ]
N = len(arr)
 
# Function call
checkEqualSum(arr, N)
 
# This code is contributed by Stuti Pathak

C#

// C# program for
// the above approach
using System;
class GFG{
 
// Utility function to check array can
// be partition to 3 subsequences of
// equal sum or not
static int checkEqualSumUtil(int[] arr, int N,
                             int sm1, int sm2,
                             int sm3, int j)
{
  // Base Case
  if (j == N)
  {
    if (sm1 == sm2 && sm2 == sm3)
      return 1;
    else
      return 0;
  }
  else
  {
    // When element at index
    // j is added to sm1
    int l = checkEqualSumUtil(arr, N,
                              sm1 + arr[j],
                              sm2, sm3, j + 1);
 
    // When element at index
    // j is added to sm2
    int m = checkEqualSumUtil(arr, N, sm1,
                              sm2 + arr[j],
                              sm3, j + 1);
 
    // When element at index
    // j is added to sm3
    int r = checkEqualSumUtil(arr, N,
                              sm1, sm2,
                              sm3 + arr[j],
                              j + 1);
 
    // Return maximum value among
    // all above 3 recursive call
    return Math.Max(Math.Max(l, m), r);
  }
}
 
// Function to check array can be
// partition to 3 subsequences of
// equal sum or not
static void checkEqualSum(int[] arr, int N)
{
 
  // Initialise 3 sums to 0
  int sum1, sum2, sum3;
  sum1 = sum2 = sum3 = 0;
 
  // Function Call
  if (checkEqualSumUtil(arr, N, sum1,
                        sum2, sum3, 0) == 1)
  {
    Console.Write("Yes");
  }
  else
  {
    Console.Write("No");
  }
}
 
// Driver Code
public static void Main()
{
  // Given array arr[]
  int[] arr = {17, 34, 59, 23, 17,
               67, 57, 2, 18, 59, 1};
  int N = arr.Length;
 
  // Function Call
  checkEqualSum(arr, N);
}
}
 
// This code is contributed by Chitranayal

输出：

Yes

时间复杂度： O(3 ^N )
辅助空间： O(1)

动态编程方法：此问题可以使用动态编程解决，其思想是将所有重叠的子问题值存储在映射中，并使用重叠的子结构的值来减少递归调用的次数。步骤如下：

• 令sum1 ， sum2和sum3为要划分的三个相等的总和。
• 创建具有密钥的映射dp 。
• 遍历给定数组并执行以下操作：
  • 基本情况：遍历数组时，如果我们到达数组的末尾，然后检查sum1 ， sum2和sum3的值是否相等，然后返回1，这将确保我们可以将给定的数组分解为相等的sum值的子序列。否则，返回0。
  • 递归调用：对于数组中的每个元素，将sum1 ， sum2和sum3中的每个元素一一包含，并返回这些递归调用中的最大值。

a = recursive_function(arr, N, sum1 + arr[j], sum2, sum3, j + 1) 
b = recursive_function(arr, N, sum1, sum2 + arr[j], sum3, j + 1) 
c = recursive_function(arr, N, sum1, sum2, sum3 + arr[j], j + 1) 

• 返回语句：在上面的递归调用中，三个值中的最大值将给出当前递归调用的结果。将dp表中的当前状态更新为：

string s = to_string(sum1) + ‘_’ + to_string(sum2) + to_string(j) 
return dp[s] = max(a, max(b, c) )

• 如果可以进行分区，则打印“是”，否则打印“否” 。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
map dp;
 
// Function to check array can be
// partition into sum of 3 equal
int checkEqualSumUtil(int arr[], int N,
                      int sm1,
                      int sm2,
                      int sm3, int j)
{
    string s = to_string(sm1)
               + "_" + to_string(sm2)
               + to_string(j);
 
    // Base Case
    if (j == N) {
        if (sm1 == sm2 && sm2 == sm3)
            return 1;
        else
            return 0;
    }
 
    // If value at particular index is not
    // -1 then return value at that index
    // which ensure no more further calls
    if (dp.find(s) != dp.end())
        return dp[s];
 
    else {
 
        // When element at index
        // j is added to sm1
        int l = checkEqualSumUtil(
            arr, N, sm1 + arr[j],
            sm2, sm3, j + 1);
 
        // When element at index
        // j is added to sm2
        int m = checkEqualSumUtil(
            arr, N, sm1, sm2 + arr[j],
            sm3, j + 1);
 
        // When element at index
        // j is added to sm3
        int r = checkEqualSumUtil(
            arr, N, sm1, sm2,
            sm3 + arr[j], j + 1);
 
        // Update the current state and
        // return that value
        return dp[s] = max(max(l, m), r);
    }
}
 
// Function to check array can be
// partition to 3 subsequences of
// equal sum or not
void checkEqualSum(int arr[], int N)
{
    // Initialise 3 sums to 0
    int sum1, sum2, sum3;
 
    sum1 = sum2 = sum3 = 0;
 
    // Function Call
    if (checkEqualSumUtil(
            arr, N, sum1,
            sum2, sum3, 0)
        == 1) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
}
 
// Driver Code
int main()
{
    // Given array arr[]
    int arr[]
        = { 17, 34, 59, 23, 17, 67,
            57, 2, 18, 59, 1 };
 
    int N = sizeof(arr) / sizeof(arr[0]);
 
    // Function Call
    checkEqualSum(arr, N);
 
    return 0;
}

Java

// Java program for
// the above approach
import java.util.*;
class GFG{
static HashMap dp = new HashMap();
 
// Function to check array can be
// partition into sum of 3 equal
static int checkEqualSumUtil(int arr[], int N,
                             int sm1, int sm2,
                             int sm3, int j)
{
  String s = String.valueOf(sm1) + "_" +
  String.valueOf(sm2) + String.valueOf(j);
 
  // Base Case
  if (j == N)
  {
    if (sm1 == sm2 && sm2 == sm3)
      return 1;
    else
      return 0;
  }
 
  // If value at particular index is not
  // -1 then return value at that index
  // which ensure no more further calls
  if (dp.containsKey(s))
    return dp.get(s);
 
  else
  {
    // When element at index
    // j is added to sm1
    int l = checkEqualSumUtil(arr, N, sm1 + arr[j],
                              sm2, sm3, j + 1);
 
    // When element at index
    // j is added to sm2
    int m = checkEqualSumUtil(arr, N, sm1,
                              sm2 + arr[j],
                              sm3, j + 1);
 
    // When element at index
    // j is added to sm3
    int r = checkEqualSumUtil(arr, N, sm1, sm2,
                              sm3 + arr[j], j + 1);
 
    // Update the current state and
    // return that value
    dp.put(s, Math.max(Math.max(l, m), r));
    return dp.get(s);
  }
}
 
// Function to check array can be
// partition to 3 subsequences of
// equal sum or not
static void checkEqualSum(int arr[], int N)
{
  // Initialise 3 sums to 0
  int sum1, sum2, sum3;
 
  sum1 = sum2 = sum3 = 0;
 
  // Function Call
  if (checkEqualSumUtil(arr, N, sum1,
                        sum2, sum3, 0) == 1)
  {
    System.out.print("Yes");
  }
  else
  {
    System.out.print("No");
  }
}
 
// Driver Code
public static void main(String[] args)
{
  // Given array arr[]
  int arr[] = {17, 34, 59, 23, 17,
               67, 57, 2, 18, 59, 1};
 
  int N = arr.length;
 
  // Function Call
  checkEqualSum(arr, N);
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python3 program for the above approach
dp = {}
 
# Function to check array can be
# partition into sum of 3 equal
def checkEqualSumUtil(arr, N, sm1, sm2, sm3, j):
     
    s = str(sm1) + "_" + str(sm2) + str(j)
     
    # Base Case
    if j == N:
        if sm1 == sm2 and sm2 == sm3:
            return 1
        else:
            return 0
         
    # If value at particular index is not
    # -1 then return value at that index
    # which ensure no more further calls
    if s in dp:
        return dp[s]
     
    # When element at index
    # j is added to sm1
    l = checkEqualSumUtil(arr, N, sm1 + arr[j],
                          sm2, sm3, j + 1)
     
    # When element at index
    # j is added to sm2
    m = checkEqualSumUtil(arr, N, sm1,
                          sm2 + arr[j], sm3,
                            j + 1)
     
    # When element at index
    # j is added to sm3
    r = checkEqualSumUtil(arr, N, sm1,
                          sm2, sm3 + arr[j],
                                 j + 1)
     
    # Update the current state and
    # return that value
    dp[s] = max(l, m, r)
     
    return dp[s]
 
# Function to check array can be
# partition to 3 subsequences of
# equal sum or not
def checkEqualSum(arr, N):
     
    # Initialise 3 sums to 0
    sum1 = sum2 = sum3 = 0
     
    # Function Call
    if checkEqualSumUtil(arr, N, sum1,
                         sum2, sum3, 0) == 1:
        print("Yes")
    else:
        print("No")
         
# Driver code
 
# Given array arr[]
arr = [ 17, 34, 59, 23, 17,
        67, 57, 2, 18, 59, 1 ]
N = len(arr)
 
# Function call
checkEqualSum(arr, N)
 
# This code is contributed by Stuti Pathak

C＃

// C# program for the above approach
using System;
using System.Collections.Generic;
 
class GFG{
     
static Dictionary dp = new Dictionary();
  
// Function to check array can be
// partition into sum of 3 equal
static int checkEqualSumUtil(int []arr, int N,
                             int sm1, int sm2,
                             int sm3, int j)
{
    string s = sm1.ToString() + "_" +
               sm2.ToString() + j.ToString();
     
    // Base Case
    if (j == N)
    {
        if (sm1 == sm2 && sm2 == sm3)
            return 1;
        else
            return 0;
    }
     
    // If value at particular index is not
    // -1 then return value at that index
    // which ensure no more further calls
    if (dp.ContainsKey(s))
        return dp[s];
     
    else
    {
         
        // When element at index
        // j is added to sm1
        int l = checkEqualSumUtil(arr, N, sm1 + arr[j],
                                  sm2, sm3, j + 1);
         
        // When element at index
        // j is added to sm2
        int m = checkEqualSumUtil(arr, N, sm1,
                                  sm2 + arr[j],
                                  sm3, j + 1);
         
        // When element at index
        // j is added to sm3
        int r = checkEqualSumUtil(arr, N, sm1, sm2,
                                  sm3 + arr[j], j + 1);
         
        // Update the current state and
        // return that value
        dp[s] = Math.Max(Math.Max(l, m), r);
         
        return dp[s];
    }
}
  
// Function to check array can be
// partition to 3 subsequences of
// equal sum or not
static void checkEqualSum(int []arr, int N)
{
     
    // Initialise 3 sums to 0
    int sum1, sum2, sum3;
     
    sum1 = sum2 = sum3 = 0;
     
    // Function call
    if (checkEqualSumUtil(arr, N, sum1,
                          sum2, sum3, 0) == 1)
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
  
// Driver Code
public static void Main(string[] args)
{
     
    // Given array arr[]
    int []arr = { 17, 34, 59, 23, 17,
                  67, 57, 2, 18, 59, 1 };
     
    int N = arr.Length;
     
    // Function call
    checkEqualSum(arr, N);
}
}
 
// This code is contributed by rutvik_56

输出：

Yes

时间复杂度： O(N * K ² )
辅助空间： O(N * K ² )其中K是数组的总和。

(to_string(sum1) + “_” + to_string(sum2) + “_” + to_string(sum3))

• 如果找到3个相等的子集，则value为1 ，否则value为0 。