最大化 K 的值，使得在 [1, K] 范围内的每个整数 i 都存在一个总和为 i 的子序列

给定一个由N个整数组成的数组arr[] ，任务是找到K的最大值，使得对于[1, K]范围内的每个整数i ，都存在一个总和为i的子序列。

例子：

Input: arr[] = {1, 2, 1, 3}
Output: 7
Explanation:
Below are the possible values of the sum for all the subsequences that can be formed:

1. Subsequence {1}, the sum of elements is 1.
2. Subsequence {2}, the sum of elements is  2.
3. Subsequence {3}, the sum of elements is 3.
4. Subsequence {1, 3}, the sum of elements is 4.
5. Subsequence {1, 1, 3}, the sum of elements is 5.
6. Subsequence {1, 2, 3}, the sum of elements is 6.
7. Subsequence {1, 1, 2, 3}, the sum of elements is 7.

Hence, the maximum value of K is 7. Hence print 7.

Input: arr[] = {2, 5, 2, 3}
Output: 0

方法：按照以下步骤解决给定问题：

• 对给定的数组arr[]进行排序。
• 初始化一个变量，比如next为0以存储K的结果最大值。
• 在[0, N – 1]范围内遍历数组arr[ ] 并执行以下步骤：
  • 如果arr[i]的值小于或等于(next + 1) ，则可以使用数组元素生成子序列{0, 1, 2, ..., next + arr[i]} 。因此，将值arr[i]添加到变量next 。
  • 否则，跳出循环，因为总和(next + 1)不能由任何子序列生成 数组， arr[i]后面的所有值都大于该值。
• 完成上述步骤后，将next的值打印为K的结果最大值。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
using namespace std;
 
// Function to find the maximum value
// of K such that for all value i over
// the range [1, K] there exists a
// subsequence whose value is i
void maximumK(vector& arr)
{
    // Sort the given array
    sort(arr.begin(), arr.end());
 
    int N = arr.size();
 
    // Stores the maximum value up to
    // which subsequences can be
    // generated by the array
    int next = 0;
 
    // Traverse the given array
    for (int i = 0; i < N; i++) {
 
        // Check if the current element
        // is greater than next + 1
        if (arr[i] > next + 1)
            break;
 
        // Add the contribution from
        // the current element and
        // update the next value
        next += arr[i];
    }
 
    // Print the answer
    cout << next << endl;
}
 
// Driver Code
int main()
{
    vector arr = { 1, 2, 1, 3 };
    maximumK(arr);
 
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.Arrays;
 
class GFG
 
{
    // Function to find the maximum value
    // of K such that for all value i over
    // the range [1, K] there exists a
    // subsequence whose value is i
    static void maximumK(int[] arr)
    {
       
        // Sort the given array
        Arrays.sort(arr);
 
        int N = arr.length;
 
        // Stores the maximum value up to
        // which subsequences can be
        // generated by the array
        int next = 0;
 
        // Traverse the given array
        for (int i = 0; i < N; i++) {
 
            // Check if the current element
            // is greater than next + 1
            if (arr[i] > next + 1)
                break;
 
            // Add the contribution from
            // the current element and
            // update the next value
            next += arr[i];
        }
 
        // Print the answer
        System.out.println(next);
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int[] arr = { 1, 2, 1, 3 };
        maximumK(arr);
    }
}
 
// This code is contributed by Potta Lokesh

Python3

# Python program for the above approach
 
# Function to find the maximum value
# of K such that for all value i over
# the range [1, K] there exists a
# subsequence whose value is i
def maximumK(arr):
    # Sort the given array
    arr.sort();
 
    N = len(arr);
 
    # Stores the maximum value up to
    # which subsequences can be
    # generated by the array
    next = 0;
 
    # Traverse the given array
    for i in range(N):
 
        # Check if the current element
        # is greater than next + 1
        if (arr[i] > next + 1):
            break;
 
        # Add the contribution from
        # the current element and
        # update the next value
        next += arr[i];
 
    # Print the answer
    print(next);
 
# Driver Code
 
arr = [1, 2, 1, 3];
maximumK(arr);
 
# This code is contributed by gfgking

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
 
class GFG{
 
// Function to find the maximum value
// of K such that for all value i over
// the range [1, K] there exists a
// subsequence whose value is i
static void maximumK(List arr)
{
    // Sort the given array
    arr.Sort();
    int N = arr.Count;
 
    // Stores the maximum value up to
    // which subsequences can be
    // generated by the array
    int next = 0;
 
    // Traverse the given array
    for (int i = 0; i < N; i++) {
 
        // Check if the current element
        // is greater than next + 1
        if (arr[i] > next + 1)
            break;
 
        // Add the contribution from
        // the current element and
        // update the next value
        next += arr[i];
    }
 
    // Print the answer
    Console.Write(next);
}
 
// Driver Code
public static void Main()
{
    List arr = new List(){ 1, 2, 1, 3 };
    maximumK(arr);
}
}
 
// This code is contributed by ipg2016107.

Javascript

 
// Javascript program for the above approach
 
// Function to find the maximum value
// of K such that for all value i over
// the range [1, K] there exists a
// subsequence whose value is i
function maximumK(arr) {
    // Sort the given array
    arr.sort((a, b) => a - b);
 
    let N = arr.length;
 
    // Stores the maximum value up to
    // which subsequences can be
    // generated by the array
    let next = 0;
 
    // Traverse the given array
    for (let i = 0; i < N; i++) {
 
        // Check if the current element
        // is greater than next + 1
        if (arr[i] > next + 1)
            break;
 
        // Add the contribution from
        // the current element and
        // update the next value
        next += arr[i];
    }
 
    // Print the answer
    document.write(next);
}
 
// Driver Code
 
let arr = [1, 2, 1, 3];
maximumK(arr);
 
// This code is contributed by _saurabh_jaiswal.

输出：

7

时间复杂度： O(N* log N)
辅助空间： O(1)