Minimum Cost to Make Array Equalindromic

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ArraysGreedy Algorithms

You are given a 0-indexed integer array nums having length n.

You are allowed to perform a special move any number of times (including zero) on nums. In one special move you perform the following steps in order:

Choose an index i in the range [0, n - 1], and a positive integer x.
Add |nums[i] - x| to the total cost.
Change the value of nums[i] to x.

A palindromic number is a positive integer that remains the same when its digits are reversed. For example, 121, 2552 and 65756 are palindromic numbers whereas 24, 46, 235 are not palindromic numbers.

An array is considered equalindromic if all the elements in the array are equal to an integer y, where y is a palindromic number less than 10⁹.

Return an integer denoting the minimum possible total cost to make nums equalindromic by performing any number of special moves.

Example 1:

Input: nums = [1,2,3,4,5]
Output: 6
Explanation: We can make the array equalindromic by changing all elements to 3 which is a palindromic number. The cost of changing the array to [3,3,3,3,3] using 4 special moves is given by |1 - 3| + |2 - 3| + |4 - 3| + |5 - 3| = 6.
It can be shown that changing all elements to any palindromic number other than 3 cannot be achieved at a lower cost.

Example 2:

Input: nums = [10,12,13,14,15]
Output: 11
Explanation: We can make the array equalindromic by changing all elements to 11 which is a palindromic number. The cost of changing the array to [11,11,11,11,11] using 5 special moves is given by |10 - 11| + |12 - 11| + |13 - 11| + |14 - 11| + |15 - 11| = 11.
It can be shown that changing all elements to any palindromic number other than 11 cannot be achieved at a lower cost.

Example 3:

Input: nums = [22,33,22,33,22]
Output: 22
Explanation: We can make the array equalindromic by changing all elements to 22 which is a palindromic number. The cost of changing the array to [22,22,22,22,22] using 2 special moves is given by |33 - 22| + |33 - 22| = 22.
It can be shown that changing all elements to any palindromic number other than 22 cannot be achieved at a lower cost.

Constraints:

1 <= n <= 10⁵
1 <= nums[i] <= 10⁹

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the size of the input array and the range of values within it?
Can the array contain negative numbers, zeros, or floating-point numbers?
What should I return if the array is already equalindromic?
By 'equalindromic', do you mean each element must equal the element in the mirrored position across the center of the array, or that each element must be the same value after some operations?
Is there a cost associated with increasing a number different from decreasing it and if so, are those costs specified?

Brute Force Solution

Approach

The brute force approach to making an array equalindromic means we're going to try every possible change to the numbers. We will check after each possible change if the array is now equalindromic and calculate the cost. Then, we just pick the lowest cost from all the options we tried.

Here's how the algorithm would work step-by-step:

Consider making no changes to the numbers in the array.
Check if the array is already equalindromic, and if it is, the cost is zero.
Now, imagine increasing or decreasing each number in the array by 1.
For each of these changes, check if the array is equalindromic, and calculate the cost of that change.
Next, try increasing or decreasing each number in the array by 2. Keep checking and calculating costs.
Continue this process of increasing or decreasing the numbers, either one by one or multiple at a time. You're changing the array in every way you can think of.
Every time you make a change, calculate the cost, and see if it's the lowest cost you've found so far.
After trying a very large range of possible changes to the numbers, select the change that resulted in an equalindromic array at the very lowest cost.

Code Implementation

def min_cost_equalindromic_brute_force(numbers):
    minimum_cost = float('inf')

    # Iterate through a range of possible changes to the numbers
    for i in range(-10, 11): 
        modified_numbers = [number + i for number in numbers]

        # Check if the modified array is equalindromic
        if is_equalindromic(modified_numbers):
            cost = calculate_cost(numbers, modified_numbers)
            
            # Update the minimum cost if the current cost is lower
            minimum_cost = min(minimum_cost, cost)

    return minimum_cost

def is_equalindromic(numbers):
    stringified_numbers = [str(number) for number in numbers]
    concatenated_string = ''.join(stringified_numbers)

    # Check if the concatenated string is a palindrome
    return concatenated_string == concatenated_string[::-1]

def calculate_cost(original_numbers, modified_numbers):
    cost = 0
    for i in range(len(original_numbers)):
        cost += abs(original_numbers[i] - modified_numbers[i])
    return cost

Big(O) Analysis

Time Complexity

O(∞) – The described brute force approach explores all possible changes to the numbers in the array to find an equalindromic configuration. It considers increasing or decreasing each number by increasing amounts until it finds a solution, or until some arbitrary limit is reached. Since the space of possible changes is theoretically infinite (you could keep increasing/decreasing numbers indefinitely), and there's no guarantee of finding a solution or a limit to the search, the time complexity is effectively unbounded, represented as O(∞). This approach has the potential to run indefinitely if a valid solution is not encountered or the limits of the search are not properly defined.

Space Complexity

O(1) – The brute force approach described does not use any significant auxiliary space. It only involves modifying the input array in place and keeping track of the minimum cost found so far. Therefore, the space required remains constant, irrespective of the size of the input array N.

Optimal Solution

Approach

The core idea is to find the best value to make all numbers in the array equal to, since that's what makes an array an 'equalindromic' array. We can use the median as a strategic point to minimize the total cost, since it balances the increases and decreases needed.

Here's how the algorithm would work step-by-step:

First, sort the numbers in the array from smallest to largest.
Next, find the middle number (or the average of the two middle numbers if there's an even number of elements). This is the median.
For each number in the original array, calculate the difference between it and the median. This difference represents the cost of changing that number to the median.
Add up all of these differences. This sum is the minimum cost to make all the numbers in the array equal to the median, and hence the array 'equalindromic'.

Code Implementation

def min_cost_equalindromic(numbers):
    sorted_numbers = sorted(numbers)

    array_length = len(sorted_numbers)

    # Calculating the median is crucial for minimizing total cost.
    if array_length % 2 == 0:
        median = (sorted_numbers[array_length // 2 - 1] + sorted_numbers[array_length // 2]) / 2
    else:
        median = sorted_numbers[array_length // 2]

    total_cost = 0

    # Summing absolute differences from the median yields the minimum cost
    for number in numbers:
        cost = abs(number - median)
        total_cost += cost

    return total_cost

Big(O) Analysis

Time Complexity

O(n log n) – The algorithm begins by sorting the input array of size n. Sorting algorithms like merge sort or quicksort typically have a time complexity of O(n log n). After sorting, finding the median takes constant time, O(1). Finally, iterating through the array to calculate the cost involves visiting each of the n elements once, resulting in O(n) time complexity. Since O(n log n) dominates O(n) and O(1), the overall time complexity is O(n log n).

Space Complexity

O(1) – The algorithm sorts the input array in place, implying no auxiliary space for sorting. It calculates the median, which may involve storing a few constant number of variables, independent of the input size N. Furthermore, the cost calculation iterates through the array but only stores the cumulative sum in a single variable. Therefore, the auxiliary space used is constant, resulting in O(1) space complexity.

Edge Cases

Case	How to Handle
Empty or null input array	Return 0 since no modifications are needed for an empty array.
Array with only one element	Return 0 as a single-element array is already equalindromic.
Array with two elements that are already equal	Return 0 as no changes are needed.
Array with two elements and the optimal equalindromic value is very large	Ensure that the chosen median can handle large numerical values to avoid overflow during cost calculation.
Array with negative numbers	Handle negative numbers correctly when calculating the cost to transform elements to the target value, potentially requiring absolute value.
Array with very large numbers that can cause integer overflow	Use a data type that can accommodate large numbers (e.g., long long in C++, long in Java, or equivalent in other languages) to prevent overflow during cost calculation.
Array with many duplicate values and only a few unique values	The median calculation will still work correctly as the algorithm focuses on the sorted array.
Maximum-sized input array that could lead to memory exhaustion during sorting	Consider using an in-place sorting algorithm or a more memory-efficient approach if memory becomes a constraint.

Empty or null input array

How to Handle:

Return 0 since no modifications are needed for an empty array.

Array with only one element

How to Handle:

Return 0 as a single-element array is already equalindromic.

Array with two elements that are already equal

How to Handle:

Return 0 as no changes are needed.

Array with two elements and the optimal equalindromic value is very large

How to Handle:

Ensure that the chosen median can handle large numerical values to avoid overflow during cost calculation.

Array with negative numbers

How to Handle:

Handle negative numbers correctly when calculating the cost to transform elements to the target value, potentially requiring absolute value.

Array with very large numbers that can cause integer overflow

How to Handle:

Use a data type that can accommodate large numbers (e.g., long long in C++, long in Java, or equivalent in other languages) to prevent overflow during cost calculation.

Array with many duplicate values and only a few unique values

How to Handle:

The median calculation will still work correctly as the algorithm focuses on the sorted array.

Maximum-sized input array that could lead to memory exhaustion during sorting

How to Handle:

Consider using an in-place sorting algorithm or a more memory-efficient approach if memory becomes a constraint.