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📜  将排序数组转换为幂序列所需的最小增量或减量

📅  最后修改于: 2021-09-07 02:31:04             🧑  作者: Mango

给定一个由N 个正整数组成的排序数组arr[] ,任务是最小化将给定数组转换为任意整数X的幂序列所需的每个数组元素的增量或减量总数。

例子:

方法:根据以下观察可以解决给定的问题:

  • 由于给定数组需要转换为任意整数X的幂序列,那么数学关系可以写成:
  • F(X)的最小值是将其转化成X的功率序列和X的最大值需要可以如下计算操作的最小数量:
  • 因此,想法是从1开始迭代X 的所有可能值,并检查以下等式是否满足:
    X^{N - 1} \le f(1) + a{[N - 1]}
    如果发现为真,则找出所有可能的值f(X) = \sum{\lvert a_i - X^i \rvert}          并返回所有获得的值中的最小值。

请按照以下步骤解决给定的问题:

  • 初始化一个变量,比如ans as (数组元素的总和– N) ,它存储使数组成为幂序列所需的最小增量或减量。
  • 迭代一个循环,从1开始,使用变量X并执行以下步骤:
    • 初始化两个变量,比如currCost0currPower1 ,它们存储表达式的总和f(X) = \sum{\lvert a_i - X^i \rvert}          和整数X的幂。
    • 迭代范围[0, N – 1]并将currCost的值更新为currCost + abs(arr[i] – currPower)并将currPower的值更新X * currPower
    • 如果表达式X^{N - 1} \le ans + a{[N - 1]}          不满意,则跳出循环。否则,将ans的值更新为anscurrCost的最小值。
  • 完成上述步骤后,打印ans的值作为所需的最小操作次数。

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 minimum number
// of increments or decrements required
// to convert array into a power sequence
int minOperations(int a[], int n)
{
    // Initialize the count to f(X) for X = 1
    int ans = accumulate(a, a + n, 0) - n;
 
    // Calculate the value of f(X)
    // X ^ (n - 1) <= f(1) + a[n - 1]
    for (int x = 1;; x++) {
 
        int curPow = 1, curCost = 0;
 
        // Calculate F(x)
        for (int i = 0; i < n; i++) {
            curCost += abs(a[i] - curPow);
            curPow *= x;
        }
 
        // Check if X ^ (n - 1) > f(1) + a[n - 1]
        if (curPow / x > ans + a[n - 1])
            break;
 
        // Update ans to store the
        // minimum of ans and F(x)
        ans = min(ans, curCost);
    }
 
    // Return the minimum number
    // of operations required
    return ans;
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 5, 7 };
    int N = sizeof(arr) / sizeof(arr[0]);
 
    cout << minOperations(arr, N);
 
    return 0;
}


Java
// Java program for the above approach
class GFG{
 
// Function to find the minimum number
// of increments or decrements required
// to convert array into a power sequence
static int minOperations(int a[], int n)
{
     
    // Initialize the count to f(X) for X = 1
    int ans = 0;
    for(int i = 0; i < n; i++)
    {
        ans += a[i];
    }
    ans -= n;
 
    // Calculate the value of f(X)
    // X ^ (n - 1) <= f(1) + a[n - 1]
    for(int x = 1;; x++)
    {
        int curPow = 1, curCost = 0;
 
        // Calculate F(x)
        for(int i = 0; i < n; i++)
        {
            curCost += Math.abs(a[i] - curPow);
            curPow *= x;
        }
 
        // Check if X ^ (n - 1) > f(1) + a[n - 1]
        if (curPow / x > ans + a[n - 1])
            break;
 
        // Update ans to store the
        // minimum of ans and F(x)
        ans = Math.min(ans, curCost);
    }
 
    // Return the minimum number
    // of operations required
    return ans;
}
 
// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 5, 7 };
    int N = arr.length;
 
    System.out.print(minOperations(arr, N));
}
}
 
// This code is contributed by avijitmondal1998


Python3
# Python3 program for the above approach
 
# Function to find the minimum number
# of increments or decrements required
# to convert array into a power sequence
def minOperations(a, n):
     
    # Initialize the count to f(X) for X = 1
    ans = 0
    for i in range(n):
        ans += a[i]
         
    ans -= n
 
    # Calculate the value of f(X)
    # X ^ (n - 1) <= f(1) + a[n - 1]
    x = 1
    while(1):
        curPow = 1
        curCost = 0
 
        # Calculate F(x)
        for i in range(n):
            curCost += abs(a[i] - curPow)
            curPow *= x
 
        # Check if X ^ (n - 1) > f(1) + a[n - 1]
        if (curPow / x > ans + a[n - 1]):
            break
 
        # Update ans to store the
        # minimum of ans and F(x)
        ans = min(ans, curCost)
        x += 1
 
    # Return the minimum number
    # of operations required
    return ans
 
# Driver Code
if __name__ == '__main__':
     
    arr =  [1, 5, 7]
    N = len(arr)
     
    print(minOperations(arr, N))
     
# This code is contributed by ipg2016107


C#
// C# program for the above approach
using System;
 
class GFG{
 
// Function to find the minimum number
// of increments or decrements required
// to convert array into a power sequence
static int minOperations(int []a, int n)
{
     
    // Initialize the count to f(X) for X = 1
    int ans = 0;
    for(int i = 0; i < n; i++)
    {
        ans += a[i];
    }
    ans -= n;
 
    // Calculate the value of f(X)
    // X ^ (n - 1) <= f(1) + a[n - 1]
    for(int x = 1;; x++)
    {
        int curPow = 1, curCost = 0;
 
        // Calculate F(x)
        for(int i = 0; i < n; i++)
        {
            curCost += Math.Abs(a[i] - curPow);
            curPow *= x;
        }
 
        // Check if X ^ (n - 1) > f(1) + a[n - 1]
        if (curPow / x > ans + a[n - 1])
            break;
 
        // Update ans to store the
        // minimum of ans and F(x)
        ans = Math.Min(ans, curCost);
    }
 
    // Return the minimum number
    // of operations required
    return ans;
}
 
// Driver Code
public static void Main()
{
    int []arr = { 1, 5, 7 };
    int N = arr.Length;
 
    Console.WriteLine(minOperations(arr, N));
}
}
 
// This code is contributed by mohit kumar 29


Javascript


输出:
4

时间复杂度: O(N*(S) (1/(N – 1)) )
辅助空间: O(1)

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