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#3139Hard~50 min

Minimum Cost to Equalize Array

Time O(n) · Space O(1) · Official statement on LeetCode

constructive algorithmsconstructive algorithms
cpppython

Solutions

// Time:  O(n)
// Space: O(1)

// constructive algorithms, math
class Solution {
public:
    int minCostToEqualizeArray(vector<int>& nums, int cost1, int cost2) {
        static const int MOD = 1e9 + 7;

        const int n = size(nums);
        const int64_t mx = ranges::max(nums);
        int64_t total = mx * n - accumulate(cbegin(nums), cend(nums), 0ll);
        // fill until mx with only cost1 operations
        if (n <= 2 || 2 * cost1 <= cost2) {
            return total * cost1 % MOD;
        }

        int64_t result = numeric_limits<int64_t>::max();
        // fill until mx with more cost2 operations and fewer cost1 operations
        const int64_t mn = ranges::min(nums);
        int64_t cnt1 = max((mx - mn) - (total - (mx - mn)), static_cast<int64_t>(0));
        int64_t cnt2 = total - cnt1;
        result = min(result, (cnt1 + cnt2 % 2) * cost1 + cnt2 / 2 * cost2);

        // fill until mx+x with most cost2 operations and fewest cost1 operations,
        // where x is the max of x s.t. cnt1+x >= (n-1)*x => cnt1 >= (n-2)*x
        const int64_t x = cnt1 / (n - 2);
        cnt1 %= n - 2;
        total += n * x;
        cnt2 = total - cnt1;
        result = min(result, (cnt1 + cnt2 % 2) * cost1 + (cnt2 / 2) * cost2);

        // fill until mx+x+1 or mx+x+2 with nearly all cost2 operations and at most one cost1 operation
        for (int _ = 0; _ < 2; ++_) {  // increase twice is for odd n
            total += n;
            result = min(result, total % 2 * cost1 + total / 2 * cost2);
        }
        return result % MOD;
    }
};

Beginner Explanation

What is Minimum Cost to Equalize Array?

Minimum Cost to Equalize Array (LeetCode #3139) is a Hard problem that primarily trains constructive algorithms.

How to think about it

  1. Restate the goal in your own words before coding.
  2. Work a tiny example by hand so the invariant becomes obvious.
  3. Identify the pattern — this problem aligns with constructive algorithms.
  4. Only then translate the idea into code.

Why this problem matters

Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: Constructive Algorithms, Math.

AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.

Interview Walkthrough

Interview approach for Minimum Cost to Equalize Array

Opening (30–60 seconds)

  • Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
  • State a brute force so the interviewer knows you can solve it naively.
  • Propose the optimal direction tied to constructive algorithms.

Core solution narrative

  1. Define the state you track (pointers, DP cell, set membership, stack top, etc.).
  2. Explain the transition when you process the next element.
  3. Call out time (O(n)) and space (O(1)) before coding.
  4. Code cleanly; narrate variable names.

What interviewers listen for

  • Correctness on edge cases
  • Complexity honesty
  • Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)

Follow-up questions they may ask

  • Can you solve it with less memory?
  • What if the input stream is infinite / doesn't fit in RAM?
  • How would tests look for adversarial inputs?

Optimized Approach

Optimized solution notes

The reference solutions on AlgoForge target O(n) time and O(1) space.

Pattern focus: constructive algorithms

Use the pattern as a checklist:

  • constructive algorithms — confirm the invariant holds after each step

Start from the primary solution, then rewrite from memory to lock it in.

Implementation tips

  • Prefer readable names over micro-optimizations in interviews.
  • Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
  • After AC-level logic, re-scan for off-by-one and null checks.

Complexity Analysis

Complexity

Measure Bound
Time O(n)
Space O(1)

How to justify this in an interview

  • Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
  • Space: include hash maps, recursion stack, and output allocation when the problem asks for it.

If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.

Common Mistakes

Common mistakes on Minimum Cost to Equalize Array

  1. Skipping edge cases — empty collections, single-element inputs, max constraints.
  2. Wrong invariant for constructive algorithms — updating state too early or too late.
  3. Mutating input unexpectedly when the problem forbids it.
  4. Off-by-one in windows, ranges, or binary search bounds.
  5. Ignoring overflow / precision for integer arithmetic problems.
  6. Overengineering — jumping to an advanced structure when a simpler approach works.

Alternative Approaches

AI expand later

Alternatives

Placeholder for multi-approach comparison. Future AI content generation can expand:

  • Brute force baseline
  • Optimal constructive algorithms solution
  • Space-optimized rewrite

Prompt slot: expand alternatives for minimum-cost-to-equalize-array.

Edge Cases

Edge cases checklist

  • Minimum input size
  • Maximum input size / time limits
  • Duplicates and already-sorted input
  • Negative numbers / zeros (if applicable)
  • Disconnected structures (graphs/trees)
  • Single path vs branching recursion depth

Pattern Recognition

Spotting this pattern

Signal phrases that point to constructive algorithms:

  • Sorted input or ability to sort without changing the answer class
  • Need for contiguous subarray / substring → consider sliding window
  • Need for O(1) membership → hash set/map
  • Optimal substructure + overlapping subproblems → DP
  • Connectivity / components → graph DFS/BFS or Union-Find

Primary topics: constructive algorithms.

Follow-up Interview Questions

Follow-ups

  1. How does the solution change if the input is a stream?
  2. Can you solve it in-place?
  3. What if duplicates must be handled differently?
  4. How would you parallelize the approach?
  5. Design tests that would break a buggy implementation.

Practice Recommendations

What to practice next

  1. Re-solve Minimum Cost to Equalize Array in a second language (cpp, python).
  2. Drill 3–5 more problems tagged constructive algorithms.
  3. Teach the solution out loud in under 5 minutes.
  4. Add this problem to your revision calendar in 3 days and 14 days.

Visualization

Conceptual diagram for Minimum Cost to Equalize Array: show input structure (constructive algorithms), highlight the moving parts of the constructive algorithms approach, and annotate each step with the maintained invariant and complexity.

Study checklist

  • Read the official problem statement on LeetCode
  • Solve on paper / whiteboard first
  • Implement the constructive algorithms approach
  • Verify edge cases from the checklist
  • State time and space complexity aloud
  • Compare with the AlgoForge reference solution
  • Schedule a revision session

Revision notes

Minimum Cost to Equalize Array (#3139) — Hard. Pattern: constructive algorithms. Complexity: O(n) time / O(1) space. Re-derive the invariant before coding.

FAQs

What is the time complexity of Minimum Cost to Equalize Array?+

The reference solutions aim for O(n) time and O(1) space. Always re-derive complexity from the code you write in the interview.

What pattern does Minimum Cost to Equalize Array use?+

It primarily maps to constructive algorithms, within the broader topic of constructive algorithms.

Is Minimum Cost to Equalize Array good for interviews?+

Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.

Where can I read the official statement?+

Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/minimum-cost-to-equalize-array/