检查给定范围内的子数组之和是否为完全平方数

给定一个大小为N的数组arr[]和一个数组range[] ，任务是检查子数组{range[0], .. , range[1]} 的和是否是一个完全平方数。如果总和是完全平方数，则打印总和的平方根。否则，打印-1。

例子 ：

Input: arr[] = {2, 19, 33, 48, 90, 100}, range = [1, 3] 
Output: 10 
Explanation: 
The sum of element from position 1 to position 3 is 19 + 33 + 48 = 100, which is a perfect square of 10.

Input: arr[] = {13, 15, 30, 55, 87}, range = [0, 1] 
Output: -1

朴素的方法：最简单的方法是遍历子数组并检查子数组的和是否为完全平方数。

时间复杂度： O(N)
辅助空间： O(1)

高效方法：上述方法的优化思路是使用二分查找来计算子数组和的平方根。请按照以下步骤操作：

1. 计算给定range[]中子数组的总和。
2. 现在，使用二分搜索在 0 到 sum 的范围内找到和的平方根。
3. 从开始和最后一个值中找到 mid 元素，并将mid ²的值与sum 进行比较。
4. 如果值 mid ²等于总和，找到一个完全平方，然后返回mid。
5. 如果mid ²的值大于总和，则答案只能位于 mid 元素之后范围的右侧。因此，向右重复并将搜索空间减少到0到mid – 1 。
6. 如果mid ²的值小于sum ，则将搜索空间缩小到mid + 1以求和。

下面是上述方法的实现。

C++

// C++ implementation of
// the above approach
#include 
using namespace std;
 
// Function to calculate the square
// root of the sum of a subarray in
// a given range
int checkForPerfectSquare(vector arr,
                          int i, int j)
{
    int mid, sum = 0;
 
    // Calculate the sum of array
    // elements within a given range
    for (int m = i; m <= j; m++) {
        sum += arr[m];
    }
 
    // Finding the square root
    int low = 0, high = sum / 2;
    while (low <= high) {
        mid = low + (high - low) / 2;
 
        // If a perfect square is found
        if (mid * mid == sum) {
            return mid;
        }
 
        // Reduce the search space if
        // the value is greater than sum
        else if (mid * mid > sum) {
            high = mid - 1;
        }
 
        // Reduce the search space if
        // the value if smaller than sum
        else {
            low = mid + 1;
        }
    }
    return -1;
}
 
// Driver Code
int main()
{
    // Given Array
    vector arr;
    arr = { 2, 19, 33, 48, 90, 100 };
 
    // Given range
    int L = 1, R = 3;
 
    // Function Call
    cout << checkForPerfectSquare(arr, L, R);
    return 0;
}

Java

// Java implementation of
// the above approach
import java.util.*;
class GFG{
 
// Function to calculate the square
// root of the sum of a subarray in
// a given range
static int checkForPerfectSquare(int []arr,
                                 int i, int j)
{
  int mid, sum = 0;
 
  // Calculate the sum of array
  // elements within a given range
  for (int m = i; m <= j; m++)
  {
    sum += arr[m];
  }
 
  // Finding the square root
  int low = 0, high = sum / 2;
  while (low <= high)
  {
    mid = low + (high - low) / 2;
 
    // If a perfect square
    // is found
    if (mid * mid == sum)
    {
      return mid;
    }
 
    // Reduce the search space
    // if the value is greater
    // than sum
    else if (mid * mid > sum)
    {
      high = mid - 1;
    }
 
    // Reduce the search space
    // if the value if smaller
    // than sum
    else
    {
      low = mid + 1;
    }
  }
  return -1;
}
 
// Driver Code
public static void main(String[] args)
{
  // Given Array
  int []arr = {2, 19, 33,
               48, 90, 100};
 
  // Given range
  int L = 1, R = 3;
 
  // Function Call
  System.out.print(
         checkForPerfectSquare(arr, L, R));
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python3 implementation of
# the above approach
 
# Function to calculate the square
# root of the sum of a subarray in
# a given range
def checkForPerfectSquare(arr, i, j):
 
    mid, sum = 0, 0
 
    # Calculate the sum of array
    # elements within a given range
    for m in range(i, j + 1):
        sum += arr[m]
 
    # Finding the square root
    low = 0
    high = sum // 2
     
    while (low <= high):
        mid = low + (high - low) // 2
 
        # If a perfect square is found
        if (mid * mid == sum):
            return mid
 
        # Reduce the search space if
        # the value is greater than sum
        elif (mid * mid > sum):
            high = mid - 1
 
        # Reduce the search space if
        # the value if smaller than sum
        else:
            low = mid + 1
 
    return -1
 
# Driver Code
if __name__ == '__main__':
 
    # Given Array
    arr = [ 2, 19, 33, 48, 90, 100 ]
 
    # Given range
    L = 1
    R = 3
 
    # Function call
    print(checkForPerfectSquare(arr, L, R))
 
# This code is contributed by mohit kumar 29

C#

// C# implementation of
// the above approach
using System;
class GFG{
 
// Function to calculate the square
// root of the sum of a subarray in
// a given range
static int checkForPerfectSquare(int []arr,
                                 int i, int j)
{
  int mid, sum = 0;
 
  // Calculate the sum of array
  // elements within a given range
  for (int m = i; m <= j; m++)
  {
    sum += arr[m];
  }
 
  // Finding the square root
  int low = 0, high = sum / 2;
  while (low <= high)
  {
    mid = low + (high - low) / 2;
 
    // If a perfect square
    // is found
    if (mid * mid == sum)
    {
      return mid;
    }
 
    // Reduce the search space
    // if the value is greater
    // than sum
    else if (mid * mid > sum)
    {
      high = mid - 1;
    }
 
    // Reduce the search space
    // if the value if smaller
    // than sum
    else
    {
      low = mid + 1;
    }
  }
  return -1;
}
 
// Driver Code
public static void Main(String[] args)
{
  // Given Array
  int []arr = {2, 19, 33,
               48, 90, 100};
 
  // Given range
  int L = 1, R = 3;
 
  // Function Call
  Console.Write(checkForPerfectSquare(arr,
                                      L, R));}
}
 
// This code is contributed by 29AjayKumar

Javascript

 
// Javascript implementation of
// the above approach
 
// Function to calculate the square
// root of the sum of a subarray in
// a given range
function checkForPerfectSquare(arr, i, j)
{
    let mid, sum = 0;
 
    // Calculate the sum of array
    // elements within a given range
    for(let m = i; m <= j; m++)
    {
        sum += arr[m];
    }
 
    // Finding the square root
    let low = 0, high = parseInt(sum / 2);
     
    while (low <= high)
    {
        mid = low + parseInt((high - low) / 2);
 
        // If a perfect square is found
        if (mid * mid == sum)
        {
            return mid;
        }
 
        // Reduce the search space if
        // the value is greater than sum
        else if (mid * mid > sum)
        {
            high = mid - 1;
        }
 
        // Reduce the search space if
        // the value if smaller than sum
        else
        {
            low = mid + 1;
        }
    }
    return -1;
}
 
// Driver Code
 
// Given Array
let arr = [ 2, 19, 33, 48, 90, 100 ];
 
// Given range
let L = 1, R = 3;
 
// Function Call
document.write(checkForPerfectSquare(arr, L, R));
 
// This code is contributed by rishavmahato348
 

输出：

10

时间复杂度： O(max(log(sum), N))
辅助空间： O(1)

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