总和小于相应索引总和的最大子集

给定一个数组arr[] ，任务是找到元素总和小于或等于其索引总和的最大子集的长度(基于 1 的索引)。

例子：

Input: arr[] = {1, 7, 3, 5, 9, 6, 6} 
Output: 5 
Explanation: 
Largest Subset is {1, 3, 5, 6, 6} 
Sum of indexes = 1 + 3 + 4 + 6 + 7 = 21 
Sum of elements = 1 + 3 + 5 + 6 + 6 = 21

Input: arr[] = {4, 1, 6, 7, 8, 2} 
Output: 3

天真的方法：
解决问题的最简单方法是生成所有可能的子集并计算元素总和小于或等于其各自索引之和的子集的长度。

时间复杂度： O(N*2 ^N )
辅助空间： O(N)

有效的方法：
请按照以下步骤解决问题：

• 迭代所有索引，只考虑那些值大于或等于存储在其中的各个值的值的索引。
• 不断更新上一步得到的差值的总和。
• 对于剩下的元素，用它们各自的索引存储它们的差异。对差异进行排序。
• 将元素一一包含在子集中，并从总和中减去差值。继续包含，直到遇到与其索引的差值超过剩余总和的元素或已包含所有数组元素。

下面是上述方法的实现：

C++

// C++ Program to implement
// the above approach
 
#include 
using namespace std;
 
// Function to find the length
// of the longest subset
int findSubset(int* a, int n)
{
    // Stores the sum of differences
    // between elements and
    // their respective index
    int sum = 0;
 
    // Stores the size of
    // the subset
    int cnt = 0;
 
    vector v;
 
    // Iterate over the array
    for (int i = 1; i <= n; i++) {
 
        // If an element which is
        // smaller than or equal
        // to its index is encountered
        if (a[i - 1] - i <= 0) {
 
            // Increase count and sum
            sum += a[i - 1] - i;
            cnt += 1;
        }
 
        // Store the difference with
        // index of the remaining
        // elements
        else {
            v.push_back(a[i - 1] - i);
        }
    }
 
    // Sort the differences
    // in increasing order
    sort(v.begin(), v.end());
 
    int ptr = 0;
 
    // Include the differences while
    // sum remains positive or
    while (ptr < v.size()
           && sum + v[ptr] <= 0) {
        cnt += 1;
        ptr += 1;
        sum += v[ptr];
    }
 
    // Return the size
    return cnt;
}
 
// Driver Code
int main()
{
    int arr[] = { 4, 1, 6, 7,
                  8, 2 };
 
    int n = sizeof(arr)
            / sizeof(arr[0]);
 
    // Function Calling
    cout << findSubset(arr, n)
         << endl;
}

Java

// Java program to implement
// the above approach
import java.util.*;
class GFG{
     
// Function to find the length
// of the longest subset
public static int findSubset(int[] a, int n)
{
     
    // Stores the sum of differences
    // between elements and
    // their respective index
    int sum = 0;
 
    // Stores the size of
    // the subset
    int cnt = 0;
 
    Vector v = new Vector<>();
 
    // Iterate over the array
    for(int i = 1; i <= n; i++)
    {
         
        // If an element which is
        // smaller than or equal
        // to its index is encountered
        if (a[i - 1] - i <= 0)
        {
             
            // Increase count and sum
            sum += a[i - 1] - i;
            cnt += 1;
        }
 
        // Store the difference with
        // index of the remaining
        // elements
        else
        {
            v.add(a[i - 1] - i);
        }
    }
 
    // Sort the differences
    // in increasing order
    Collections.sort(v);
 
    int ptr = 0;
 
    // Include the differences while
    // sum remains positive or
    while (ptr < v.size() &&
           sum + v.get(ptr) <= 0)
    {
        cnt += 1;
        ptr += 1;
        sum += v.get(ptr);
    }
 
    // Return the size
    return cnt;
}
 
// Driver code
public static void main(String[] args)
{
    int arr[] = { 4, 1, 6, 7, 8, 2 };
    int n = arr.length;
 
    // Function Calling
    System.out.println(findSubset(arr, n));
}
}
 
// This code is contributed by divyeshrabadiya07

Python3

# Python3 program to implement 
# the above approach
 
# Function to find the length 
# of the longest subset
def findSubset(a, n):
     
    # Stores the sum of differences 
    # between elements and 
    # their respective index 
    sum = 0
   
    # Stores the size of 
    # the subset 
    cnt = 0
   
    v = [] 
   
    # Iterate over the array 
    for i in range(1, n + 1):
         
        # If an element which is 
        # smaller than or equal 
        # to its index is encountered 
        if (a[i - 1] - i <= 0):
             
            # Increase count and sum 
            sum += a[i - 1] - i 
            cnt += 1
     
        # Store the difference with 
        # index of the remaining 
        # elements 
        else:
            v.append(a[i - 1] - i)
             
    # Sort the differences 
    # in increasing order 
    v.sort()
   
    ptr = 0
   
    # Include the differences while 
    # sum remains positive or 
    while (ptr < len(v) and
           sum + v[ptr] <= 0):
        cnt += 1
        ptr += 1 
        sum += v[ptr] 
     
    # Return the size 
    return cnt
 
# Driver code
if __name__=="__main__":
     
    arr = [ 4, 1, 6, 7, 8, 2 ] 
    n = len(arr)
   
    # Function calling 
    print(findSubset(arr, n))
 
# This code is contributed by rutvik_56

C#

// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
 
class GFG{
     
// Function to find the length
// of the longest subset
public static int findSubset(int[] a, int n)
{
     
    // Stores the sum of differences
    // between elements and
    // their respective index
    int sum = 0;
 
    // Stores the size of
    // the subset
    int cnt = 0;
 
    List v = new List();
 
    // Iterate over the array
    for(int i = 1; i <= n; i++)
    {
         
        // If an element which is
        // smaller than or equal
        // to its index is encountered
        if (a[i - 1] - i <= 0)
        {
             
            // Increase count and sum
            sum += a[i - 1] - i;
            cnt += 1;
        }
 
        // Store the difference with
        // index of the remaining
        // elements
        else
        {
            v.Add(a[i - 1] - i);
        }
    }
 
    // Sort the differences
    // in increasing order
    v.Sort();
 
    int ptr = 0;
 
    // Include the differences while
    // sum remains positive or
    while (ptr < v.Count &&
           sum + v[ptr] <= 0)
    {
        cnt += 1;
        ptr += 1;
        sum += v[ptr];
    }
 
    // Return the size
    return cnt;
}
 
// Driver code
public static void Main(String[] args)
{
    int []arr = { 4, 1, 6, 7, 8, 2 };
    int n = arr.Length;
 
    // Function calling
    Console.WriteLine(findSubset(arr, n));
}
}
 
// This code is contributed by amal kumar choubey

Javascript

 
// Javascript program for the above approach
 
// Function to find the length
// of the longest subset
function findSubset(a, n)
{
      
    // Stores the sum of differences
    // between elements and
    // their respective index
    let sum = 0;
  
    // Stores the size of
    // the subset
    let cnt = 0;
  
    let v = [];
  
    // Iterate over the array
    for(let i = 1; i <= n; i++)
    {
          
        // If an element which is
        // smaller than or equal
        // to its index is encountered
        if (a[i - 1] - i <= 0)
        {
              
            // Increase count and sum
            sum += a[i - 1] - i;
            cnt += 1;
        }
  
        // Store the difference with
        // index of the remaining
        // elements
        else
        {
            v.push(a[i - 1] - i);
        }
    }
  
    // Sort the differences
    // in increasing order
    v.sort();
  
    let ptr = 0;
  
    // Include the differences while
    // sum remains positive or
    while (ptr < v.length &&
           sum + v[ptr] <= 0)
    {
        cnt += 1;
        ptr += 1;
        sum += v[ptr];
    }
  
    // Return the size
    return cnt;
}
 
// Driver Code
 
    let arr = [ 4, 1, 6, 7, 8, 2 ];
    let n = arr.length;
  
    // Function Calling
    document.write(findSubset(arr, n));
 

输出：

3

时间复杂度： O(N)
空间复杂度： O(N)

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