Minimum Cost to Equalize Array

Hard

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ArraysGreedy Algorithms

You are given an integer array nums and two integers cost1 and cost2. You are allowed to perform either of the following operations any number of times:

Choose an index i from nums and increase nums[i] by 1 for a cost of cost1.
Choose two different indices i, j, from nums and increase nums[i] and nums[j] by 1 for a cost of cost2.

Return the minimum cost required to make all elements in the array equal.

Since the answer may be very large, return it modulo 10⁹ + 7.

Example 1:

Input: nums = [4,1], cost1 = 5, cost2 = 2

Output: 15

Explanation:

The following operations can be performed to make the values equal:

Increase nums[1] by 1 for a cost of 5. nums becomes [4,2].
Increase nums[1] by 1 for a cost of 5. nums becomes [4,3].
Increase nums[1] by 1 for a cost of 5. nums becomes [4,4].

The total cost is 15.

Example 2:

Input: nums = [2,3,3,3,5], cost1 = 2, cost2 = 1

Output: 6

Explanation:

The following operations can be performed to make the values equal:

Increase nums[0] and nums[1] by 1 for a cost of 1. nums becomes [3,4,3,3,5].
Increase nums[0] and nums[2] by 1 for a cost of 1. nums becomes [4,4,4,3,5].
Increase nums[0] and nums[3] by 1 for a cost of 1. nums becomes [5,4,4,4,5].
Increase nums[1] and nums[2] by 1 for a cost of 1. nums becomes [5,5,5,4,5].
Increase nums[3] by 1 for a cost of 2. nums becomes [5,5,5,5,5].

The total cost is 6.

Example 3:

Input: nums = [3,5,3], cost1 = 1, cost2 = 3

Output: 4

Explanation:

The following operations can be performed to make the values equal:

Increase nums[0] by 1 for a cost of 1. nums becomes [4,5,3].
Increase nums[0] by 1 for a cost of 1. nums becomes [5,5,3].
Increase nums[2] by 1 for a cost of 1. nums becomes [5,5,4].
Increase nums[2] by 1 for a cost of 1. nums becomes [5,5,5].

The total cost is 4.

Constraints:

1 <= nums.length <= 10⁵
1 <= nums[i] <= 10⁶
1 <= cost1 <= 10⁶
1 <= cost2 <= 10⁶

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 is the range of values that can be present in the input array? Can I expect negative numbers, zeros, or floating point numbers?
What is the expected size of the array? Should I be concerned about integer overflow when calculating costs?
Are there any restrictions on the operations I can perform to equalize the array, or is the cost function the only constraint?
If it's impossible to equalize the array using the allowed operations, what should the function return?
Can you provide more details about the cost function? Is it linear, quadratic, or some other function of the difference between elements?

Brute Force Solution

Approach

The brute force approach to minimizing the cost to equalize an array involves trying every single possible target value. We'll calculate the cost associated with making every element in the array equal to that target, then pick the target that results in the lowest cost. It's like trying out every possible shade of paint until you find the one that covers the wall for the least amount of money.

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

First, look at all the numbers you have in your collection.
Consider the smallest number as the potential target value that you want to make all other numbers equal to.
Calculate the cost to turn every other number into that target number by finding the difference between each other number and the target number, and then adding up all the differences.
Now, consider the next smallest number as the target value.
Calculate the cost again, like before, to make all other numbers equal to this new target.
Repeat this process for *every* single number in the collection. Each time you're figuring out what it would cost to make all the numbers the same as that specific number.
Once you've done this for every number in the collection, compare all the costs you've calculated.
Choose the target number (and its corresponding cost) that results in the absolute lowest cost.

Code Implementation

def min_cost_to_equalize_array_brute_force(numbers):
    unique_numbers = sorted(list(set(numbers)))
    minimum_cost = float('inf')

    # Iterate through each unique number to use as a target
    for target_value in unique_numbers:

        current_cost = 0
        # Calculate cost to convert all array elements to the target
        for element in numbers:
            current_cost += abs(element - target_value)

        # Update minimum cost if current cost is lower
        if current_cost < minimum_cost:
            minimum_cost = current_cost

    return minimum_cost

Big(O) Analysis

Time Complexity

O(n²) – The provided brute force solution iterates through each of the n elements in the input array to consider it as a potential target value. For each target value, the algorithm then iterates through the entire array again (n elements) to calculate the cost of converting all other elements to that target. This nested iteration results in approximately n * n operations. Therefore, the time complexity is O(n²).

Space Complexity

O(1) – The algorithm iterates through the input array, calculating the cost for each number as a target. It maintains a single variable to store the minimum cost found so far. No additional data structures that scale with the input size N are created; temporary calculations are done using a fixed number of variables. Thus, the auxiliary space used is constant, independent of the input size N.

Optimal Solution

Approach

To find the minimum cost to equalize an array, we need to find a target value that minimizes the total cost. Instead of trying every number, we can find this value by focusing on the middle value(s) of the sorted array because cost functions usually form a U-shape when plotted.

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

First, sort the given array so that we can easily find the middle element(s).
If the array has an odd number of elements, the median (middle element) is our candidate target value. If it has an even number of elements, we have two median candidates; the two middle values.
Calculate the total cost for each candidate target value. The cost for each element is simply the absolute difference between it and the target.
If there are two target candidates, compare their respective total costs. The target value with the smaller cost is the one that minimizes the equalization cost.
The minimum total cost we calculated is the answer.

Code Implementation

def minimum_cost_to_equalize_array(input_array):
    input_array.sort()
    array_length = len(input_array)

    # Determine if array length is even or odd to find median(s).
    if array_length % 2 == 0:
        median_index_right = array_length // 2
        median_index_left = median_index_right - 1
        median_candidates = [input_array[median_index_left], input_array[median_index_right]]
    else:
        median_index = array_length // 2
        median_candidates = [input_array[median_index]]

    minimum_cost = float('inf')

    # Calculate cost for each candidate and pick minimum
    for target_value in median_candidates:
        current_cost = 0
        for element in input_array:
            current_cost += abs(element - target_value)

        if current_cost < minimum_cost:
            minimum_cost = current_cost

    return minimum_cost

Big(O) Analysis

Time Complexity

O(n log n) – The dominant operation in this algorithm is the sorting of the input array, which has a time complexity of O(n log n), where n is the number of elements in the array. Calculating the cost for each candidate involves iterating through the array once, taking O(n) time. Since we have at most two candidates, this cost calculation is performed a maximum of twice. Therefore, the overall time complexity is dominated by the sorting step, resulting in O(n log n).

Space Complexity

O(N) – The dominant space complexity comes from sorting the input array. Depending on the sorting algorithm used, such as merge sort, it may require auxiliary space proportional to the input size N to store temporary arrays during the sorting process. Calculating the cost for each candidate target value uses a constant amount of extra space for variables like total_cost and the target value itself. Therefore, the auxiliary space is primarily determined by the sorting algorithm, resulting in O(N) space complexity.

Edge Cases

Case	How to Handle
Null or empty input array	Return 0 immediately as no operations are needed to equalize an empty array.
Array with one element	Return 0 since an array with a single element is already equalized.
Array with all elements already equal	Return 0 as no cost is required to equalize an already equal array.
Large input array exceeding memory constraints	Consider an iterative approach with constant space complexity to avoid memory exhaustion.
Array with very large positive or negative numbers leading to integer overflow	Use a data type that can accommodate a larger range, such as long, or perform calculations using modular arithmetic where appropriate.
Input array contains only negative numbers	The solution should work correctly with negative numbers, the core logic of finding minimum cost should apply regardless of sign.
Significant differences in magnitude between elements (e.g., 1 and 1000000)	The chosen algorithm should handle large cost variations without leading to inefficient calculations, possibly requiring dynamic programming or careful sorting.
The cost function can yield negative values	The problem must define the nature of the cost function and its outputs; if it's theoretically possible for it to return negative costs, ensure that any cost minimization algorithm accounts for this possibility and finds a true minimum (e.g., potentially through Dijkstra's).

Null or empty input array

How to Handle:

Return 0 immediately as no operations are needed to equalize an empty array.

Array with one element

How to Handle:

Return 0 since an array with a single element is already equalized.

Array with all elements already equal

How to Handle:

Return 0 as no cost is required to equalize an already equal array.

Large input array exceeding memory constraints

How to Handle:

Consider an iterative approach with constant space complexity to avoid memory exhaustion.

Array with very large positive or negative numbers leading to integer overflow

How to Handle:

Use a data type that can accommodate a larger range, such as long, or perform calculations using modular arithmetic where appropriate.

Input array contains only negative numbers

How to Handle:

The solution should work correctly with negative numbers, the core logic of finding minimum cost should apply regardless of sign.

Significant differences in magnitude between elements (e.g., 1 and 1000000)

How to Handle:

The chosen algorithm should handle large cost variations without leading to inefficient calculations, possibly requiring dynamic programming or careful sorting.

The cost function can yield negative values

How to Handle:

The problem must define the nature of the cost function and its outputs; if it's theoretically possible for it to return negative costs, ensure that any cost minimization algorithm accounts for this possibility and finds a *true* minimum (e.g., potentially through Dijkstra's).