#3010

Easy

LeetCode Problem Workspace

Divide an Array Into Subarrays With Minimum Cost I

Divide an array into three contiguous subarrays with minimum cost by leveraging array and sorting principles.

Problem Statement

You are given an integer array nums of length n. You need to divide this array into exactly three disjoint contiguous subarrays, minimizing the total cost of the division. The cost of a subarray is defined as the value of its first element.

Return the minimum possible cost of dividing nums into three subarrays. The array can be split at different points, but you are constrained to use exactly three subarrays. The solution requires careful attention to sorting and array manipulation techniques.

Examples

Example 1

Input: nums = [1,2,3,12]

Output: 6

The best possible way to form 3 subarrays is: [1], [2], and [3,12] at a total cost of 1 + 2 + 3 = 6. The other possible ways to form 3 subarrays are:

[1], [2,3], and [12] at a total cost of 1 + 2 + 12 = 15.
[1,2], [3], and [12] at a total cost of 1 + 3 + 12 = 16.

Example 2

Input: nums = [5,4,3]

Output: 12

The best possible way to form 3 subarrays is: [5], [4], and [3] at a total cost of 5 + 4 + 3 = 12. It can be shown that 12 is the minimum cost achievable.

Example 3

Input: nums = [10,3,1,1]

Output: 12

The best possible way to form 3 subarrays is: [10,3], [1], and [1] at a total cost of 10 + 1 + 1 = 12. It can be shown that 12 is the minimum cost achievable.

Constraints

3 <= n <= 50
1 <= nums[i] <= 50

Solution Approach

Sorting the Array

First, sort the array in ascending order to minimize the cost when selecting the first element of each subarray. Sorting is essential for ensuring the lowest possible cost when splitting the array.

Enumeration of Possible Splits

Next, enumerate all possible ways to split the sorted array into three subarrays. For each split, calculate the total cost by summing the first element of each subarray.

Choosing the Optimal Split

Finally, choose the split that results in the lowest total cost. The optimal split minimizes the sum of the first elements of the subarrays, providing the required minimum cost.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this problem depends on the sorting step, which is O(n log n). After sorting, enumerating the possible splits is linear in terms of the array length, so the total complexity is dominated by the sorting step. The space complexity is O(n) due to the storage needed for the array and any intermediate results.

What Interviewers Usually Probe

The candidate demonstrates an understanding of sorting-based optimizations for minimizing costs.
The candidate shows proficiency in breaking down the problem into smaller, manageable subproblems.
The candidate efficiently uses enumeration to handle different possible splits of the array.

Common Pitfalls or Variants

Common pitfalls

Forgetting to sort the array before attempting to split it into subarrays, which leads to suboptimal cost results.
Not considering all possible splits of the array, leading to missed opportunities for minimizing the total cost.
Overcomplicating the problem by attempting to use advanced data structures instead of sorting and simple array manipulations.

Follow-up variants

Change the number of subarrays to be split into, e.g., 4 or 5 subarrays, adjusting the approach accordingly.
Consider cases where the cost is defined by a different metric, such as the sum of the elements in the subarray instead of the first element.
Limit the possible values of elements in the array to a specific range to explore how the problem scales with constraints.

FAQ

What is the pattern for solving the problem "Divide an Array Into Subarrays With Minimum Cost I"?

The key pattern is sorting the array first and then enumerating all possible splits of the array into three subarrays, choosing the split that minimizes the total cost based on the first element of each subarray.

How do I handle multiple possible splits in the array?

You need to evaluate each possible split by calculating the total cost, then choose the split that minimizes the sum of the first elements of the subarrays.

What is the time complexity of this problem?

The time complexity is O(n log n) due to the sorting step, followed by a linear pass to evaluate all possible splits.

Can this problem be solved with a greedy approach?

A greedy approach can be used to some extent, but the optimal solution requires considering all possible splits, not just local decisions, making sorting a crucial step.

What if the array is already sorted?

If the array is already sorted, the problem becomes simpler as the best splits are likely to be closer to each other, but you still need to evaluate possible divisions.

terminal

Solution

Solution 1: Traverse to Find the Smallest and Second Smallest Values

We set the first element of the array $nums$ as $a$, the smallest element among the remaining elements as $b$, and the second smallest element as $c$. The answer is $a+b+c$.

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class Solution:
    def minimumCost(self, nums: List[int]) -> int:
        a, b, c = nums[0], inf, inf
        for x in nums[1:]:
            if x < b:
                c, b = b, x
            elif x < c:
                c = x
        return a + b + c

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