总和等于给定数 N 的最小立方体数

给定一个整数n ，任务是找到总和等于N的立方体的最小数量。
例子：

Input: N = 496 
Output: 3 
4³ + 6³ + 6³ = 496 
Note that 1³ + 3³ + 5³ + 7³ = 496 but it requires 4 cubes.
Input: N = 15 
Output: 8 
 

朴素的方法：编写一个递归方法，将每个小于N 的完美立方体(比如X)作为求和的一部分，然后递归求和N – X所需的立方体数量。该解决方案的时间复杂度是指数级的。
下面是上述方法的实现：

C++

// C++ implementation of the approach
#include 
using namespace std;
 
// Function to return the minimum
// number of cubes whose sum is k
int MinOfCubed(int k)
{
    // If k is less than the 2^3
    if (k < 8)
        return k;
 
    // Initialize with the maximum
    // number of cubes required
    int res = k;
    for (int i = 1; i <= k; i++) {
        if ((i * i * i) > k)
            return res;
        res = min(res, MinOfCubed(k - (i * i * i)) + 1);
    }
    return res;
}
 
// Driver code
int main()
{
    int num = 15;
    cout << MinOfCubed(num);
 
    return 0;
}

Java

// Java implementation of the approach
class GFG {
 
    // Function to return the minimum
    // number of cubes whose sum is k
    static int MinOfCubed(int k)
    {
        // If k is less than the 2^3
        if (k < 8)
            return k;
 
        // Initialize with the maximum
        // number of cubes required
        int res = k;
        for (int i = 1; i <= k; i++) {
            if ((i * i * i) > k)
                return res;
            res = Math.min(res, MinOfCubed(k - (i * i * i)) + 1);
        }
        return res;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int num = 15;
        System.out.println(MinOfCubed(num));
    }
}
 
// This code has been contributed by 29AjayKumar

Python3

# Python3 implementation of the approach
 
# Function to return the minimum
# number of cubes whose sum is k
def MinOfCubed(k):
     
    # If k is less than the 2 ^ 3
    if (k < 8):
        return k;
 
    # Initialize with the maximum
    # number of cubes required
    res = k;
    for i in range(1, k + 1):
        if ((i * i * i) > k):
            return res;
        res = min(res, MinOfCubed(k - (i * i * i)) + 1);
    return res;
 
# Driver code
num = 15;
print(MinOfCubed(num));
 
# This code contributed by PrinciRaj1992

C#

// C# implementation of the approach
using System;
 
class GFG {
 
    // Function to return the minimum
    // number of cubes whose sum is k
    static int MinOfCubed(int k)
    {
        // If k is less than the 2^3
        if (k < 8)
            return k;
 
        // Initialize with the maximum
        // number of cubes required
        int res = k;
        for (int i = 1; i <= k; i++) {
            if ((i * i * i) > k)
                return res;
            res = Math.Min(res, MinOfCubed(k - (i * i * i)) + 1);
        }
        return res;
    }
 
    // Driver code
    static public void Main()
    {
        int num = 15;
        Console.WriteLine(MinOfCubed(num));
    }
}
 
// This code has been contributed by ajit.

PHP

 $k)
            return $res;
        $res = min($res, MinOfCubed($k - ($i *$i * $i)) + 1);
    }
    return $res;
}
 
    // Driver code
    $num = 15;
    echo MinOfCubed($num);
     
    // This code is contributed by Ryuga
 
?>

Javascript

    // Javascript implementation of the approach
     
    // Function to return the minimum
    // number of cubes whose sum is k
    function MinOfCubed(k)
    {
        // If k is less than the 2^3
        if (k < 8)
            return k;
   
        // Initialize with the maximum
        // number of cubes required
        let res = k;
        for (let i = 1; i <= k; i++) {
            if ((i * i * i) > k)
                return res;
            res = Math.min(res, MinOfCubed(k - (i * i * i)) + 1);
        }
        return res;
    }
     
    let num = 15;
      document.write(MinOfCubed(num));
 

输出：

8

有效的方法：如果我们为上述解决方案绘制完整的递归树，我们可以看到许多子问题一次又一次地解决，因此我们可以看到这个问题具有重叠子问题的性质。这导致我们使用动态规划范式解决问题。
下面是上述方法的实现：

C++

// C++ implementation of the approach
#include 
using namespace std;
 
// Function to return the minimum
// number of cubes whose sum is k
int MinOfCubedDP(int k)
{
    int *DP = new int[k + 1], j = 1, t = 1;
    DP[0] = 0;
    for (int i = 1; i <= k; i++) {
        DP[i] = INT_MAX;
 
        // While current perfect cube
        // is less than current element
        while (j <= i) {
 
            // If i is a perfect cube
            if (j == i)
                DP[i] = 1;
 
            // i = (i - 1) + 1^3
            else if (DP[i] > DP[i - j])
                DP[i] = DP[i - j] + 1;
 
            // Next perfect cube
            t++;
            j = t * t * t;
        }
 
        // Re-initialization for next element
        t = j = 1;
    }
    return DP[k];
}
 
// Driver code
int main()
{
    int num = 15;
    cout << MinOfCubedDP(num);
 
    return 0;
}

Java

// Java implementation of the approach
class GFG {
 
    // Function to return the minimum
    // number of cubes whose sum is k
    static int MinOfCubedDP(int k)
    {
        int[] DP = new int[k + 1];
        int j = 1, t = 1;
        DP[0] = 0;
        for (int i = 1; i <= k; i++) {
            DP[i] = Integer.MAX_VALUE;
 
            // While current perfect cube
            // is less than current element
            while (j <= i) {
 
                // If i is a perfect cube
                if (j == i)
                    DP[i] = 1;
 
                // i = (i - 1) + 1^3
                else if (DP[i] > DP[i - j])
                    DP[i] = DP[i - j] + 1;
 
                // Next perfect cube
                t++;
                j = t * t * t;
            }
 
            // Re-initialization for next element
            t = j = 1;
        }
        return DP[k];
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int num = 15;
        System.out.println(MinOfCubedDP(num));
    }
}
 
/* This code contributed by PrinciRaj1992 */

蟒蛇3

# Python implementation of the approach
import sys
 
# Function to return the minimum
# number of cubes whose sum is k
def MinOfCubedDP(k):
    DP = [0] * (k + 1);
    j = 1;
    t = 1;
    DP[0] = 0;
    for i in range(1, k + 1):
        DP[i] = sys.maxsize;
 
        # While current perfect cube
        # is less than current element
        while (j <= i):
 
            # If i is a perfect cube
            if (j == i):
                DP[i] = 1;
 
            # i = (i - 1) + 1^3
            elif (DP[i] > DP[i - j]):
                DP[i] = DP[i - j] + 1;
 
            # Next perfect cube
            t += 1;
            j = t * t * t;
 
        # Re-initialization for next element
        t = j = 1;
    return DP[k];
 
 
# Driver code
num = 15;
print(MinOfCubedDP(num));
 
# This code contributed by Rajput-Ji

C＃

// C# implementation of the approach
using System;
 
class GFG {
 
    // Function to return the minimum
    // number of cubes whose sum is k
    static int MinOfCubedDP(int k)
    {
        int[] DP = new int[k + 1];
        int j = 1, t = 1;
        DP[0] = 0;
        for (int i = 1; i <= k; i++) {
            DP[i] = int.MaxValue;
 
            // While current perfect cube
            // is less than current element
            while (j <= i) {
 
                // If i is a perfect cube
                if (j == i)
                    DP[i] = 1;
 
                // i = (i - 1) + 1^3
                else if (DP[i] > DP[i - j])
                    DP[i] = DP[i - j] + 1;
 
                // Next perfect cube
                t++;
                j = t * t * t;
            }
 
            // Re-initialization for next element
            t = j = 1;
        }
        return DP[k];
    }
 
    // Driver code
    public static void Main()
    {
        int num = 15;
        Console.WriteLine(MinOfCubedDP(num));
    }
}
 
/* This code contributed by Code_Mech */

PHP

 $DP[$i - $j])
                $DP[$i] = $DP[$i - $j] + 1;
 
            // Next perfect cube
            $t++;
            $j = $t * $t * $t;
        }
 
        // Re-initialization for next element
        $t = $j = 1;
    }
    return $DP[$k];
}
 
// Driver code
$num = 15;
echo(MinOfCubedDP($num));
 
// This code contributed by Code_Mech
?>

Javascript

 
    // Javascript implementation of the approach
     
    // Function to return the minimum
    // number of cubes whose sum is k
    function MinOfCubedDP(k)
    {
        let DP = new Array(k + 1);
        DP.fill(0);
        let j = 1, t = 1;
        DP[0] = 0;
        for (let i = 1; i <= k; i++) {
            DP[i] = Number.MAX_VALUE;
  
            // While current perfect cube
            // is less than current element
            while (j <= i) {
  
                // If i is a perfect cube
                if (j == i)
                    DP[i] = 1;
  
                // i = (i - 1) + 1^3
                else if (DP[i] > DP[i - j])
                    DP[i] = DP[i - j] + 1;
  
                // Next perfect cube
                t++;
                j = t * t * t;
            }
  
            // Re-initialization for next element
            t = j = 1;
        }
        return DP[k];
    }
     
    let num = 15;
    document.write(MinOfCubedDP(num));
 

输出：

8

乍一看，该算法似乎在多项式时间内工作，因为我们有两个嵌套循环，外循环使用 O(n)，内循环使用 O(n^(1/3))。所以整个算法需要O(n*n^(1/3))。但是作为输入长度的函数的复杂度是多少?要表示一个大小为 n 的数字，我们需要 m=log(n) –(以 2 为底的对数)位。在这种情况下，n=2^m。如果我们在公式 O(n*n^(1/3)) 中将 2^m 设置为 n，我们看到时间复杂度仍然是指数级的。该算法称为伪多项式。

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