将排序数组转换为幂序列所需的最小增量或减量

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。

A sequence is called a power sequence of any integer X, if and only if for every i^th element (0 ≤ i < N), arr[i] = Xⁱ, where N is length of the given array.

例子：

Input: arr[] = {1, 3, 4}
Output: 1
Explanation: Decreasing arr[1] by 1 modifies array to {1, 2, 4}, which is a power sequence of 2. Therefore, the total number of increments or decrements required is 1.

Input: arr[] = {1, 5, 7}
Output: 6
Explanation:
Operation 1: Decreasing arr[1] by 1 modifies array to {1, 4, 7}
Operation 2: Decreasing arr[1] by 1 modifies array to {1, 3, 7}
Operation 3: Increasing arr[2] by 1 modifies array to {1, 3, 8}
Operation 4: Increasing arr[2] by 1 modifies array to {1, 3, 9}, which is the power sequence of 3. Therefore, the total number of increments or decrements required is 4. 
 

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：

$f(X) = \sum{\lvert a_i - X^i \rvert}$
where, 0 <= i < N, N is the number of elements in the array.

• 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：

=> $X^{N - 1} - a[N - 1] \le f(c) \le f(1)$
=>$X^{N - 1} \le f(1) + a{[N - 1]}$

• 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currCost为0和currPower为1 ，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的值更新为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ans和currCost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

 
// JavaScript program for the above approach   
 
// Function to find the minimum number
// of increments or decrements required
// to convert array into a power sequence
    function minOperations(a , n) {
 
        // Initialize the count to f(X) for X = 1
        var ans = 0;
        for (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 (x = 1;; x++) {
            var curPow = 1, curCost = 0;
 
            // Calculate F(x)
            for (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
     
        var arr = [ 1, 5, 7 ];
        var N = arr.length;
 
        document.write(minOperations(arr, N));
 
// This code contributed by aashish1995
 

输出：

4

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

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