LeetCode #3326 — MEDIUM

Minimum Division Operations to Make Array Non Decreasing

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

You are given an integer array nums.

Any positive divisor of a natural number x that is strictly less than x is called a proper divisor of x. For example, 2 is a proper divisor of 4, while 6 is not a proper divisor of 6.

You are allowed to perform an operation any number of times on nums, where in each operation you select any one element from nums and divide it by its greatest proper divisor.

Return the minimum number of operations required to make the array non-decreasing.

If it is not possible to make the array non-decreasing using any number of operations, return -1.

Example 1:

Input: nums = [25,7]

Output: 1

Explanation:

Using a single operation, 25 gets divided by 5 and nums becomes [5, 7].

Example 2:

Input: nums = [7,7,6]

Output: -1

Example 3:

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

Output: 0

Constraints:

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

Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums. Any positive divisor of a natural number x that is strictly less than x is called a proper divisor of x. For example, 2 is a proper divisor of 4, while 6 is not a proper divisor of 6. You are allowed to perform an operation any number of times on nums, where in each operation you select any one element from nums and divide it by its greatest proper divisor. Return the minimum number of operations required to make the array non-decreasing. If it is not possible to make the array non-decreasing using any number of operations, return -1.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Math · Greedy

Example 1

[25,7]

Example 2

[7,7,6]

Example 3

[1,1,1,1]

Core Insight

What unlocks the optimal approach

Iterate backward from the last index.
Each number can be divided by its largest proper divisor to yield its smallest prime divisor.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #3326: Minimum Division Operations to Make Array Non Decreasing
class Solution {
    private static final int MX = (int) 1e6 + 1;
    private static final int[] LPF = new int[MX + 1];
    static {
        for (int i = 2; i <= MX; ++i) {
            for (int j = i; j <= MX; j += i) {
                if (LPF[j] == 0) {
                    LPF[j] = i;
                }
            }
        }
    }
    public int minOperations(int[] nums) {
        int ans = 0;
        for (int i = nums.length - 2; i >= 0; i--) {
            if (nums[i] > nums[i + 1]) {
                nums[i] = LPF[nums[i]];
                if (nums[i] > nums[i + 1]) {
                    return -1;
                }
                ans++;
            }
        }
        return ans;
    }
}

// Accepted solution for LeetCode #3326: Minimum Division Operations to Make Array Non Decreasing
const mx int = 1e6

var lpf = [mx + 1]int{}

func init() {
	for i := 2; i <= mx; i++ {
		if lpf[i] == 0 {
			for j := i; j <= mx; j += i {
				if lpf[j] == 0 {
					lpf[j] = i
				}
			}
		}
	}
}

func minOperations(nums []int) (ans int) {
	for i := len(nums) - 2; i >= 0; i-- {
		if nums[i] > nums[i+1] {
			nums[i] = lpf[nums[i]]
			if nums[i] > nums[i+1] {
				return -1
			}
			ans++
		}
	}
	return
}

# Accepted solution for LeetCode #3326: Minimum Division Operations to Make Array Non Decreasing
mx = 10**6 + 1
lpf = [0] * (mx + 1)
for i in range(2, mx + 1):
    if lpf[i] == 0:
        for j in range(i, mx + 1, i):
            if lpf[j] == 0:
                lpf[j] = i


class Solution:
    def minOperations(self, nums: List[int]) -> int:
        ans = 0
        for i in range(len(nums) - 2, -1, -1):
            if nums[i] > nums[i + 1]:
                nums[i] = lpf[nums[i]]
                if nums[i] > nums[i + 1]:
                    return -1
                ans += 1
        return ans

// Accepted solution for LeetCode #3326: Minimum Division Operations to Make Array Non Decreasing
// Rust example auto-generated from java reference.
// Replace the signature and local types with the exact LeetCode harness for this problem.
impl Solution {
    pub fn rust_example() {
        // Port the logic from the reference block below.
    }
}

// Reference (java):
// // Accepted solution for LeetCode #3326: Minimum Division Operations to Make Array Non Decreasing
// class Solution {
//     private static final int MX = (int) 1e6 + 1;
//     private static final int[] LPF = new int[MX + 1];
//     static {
//         for (int i = 2; i <= MX; ++i) {
//             for (int j = i; j <= MX; j += i) {
//                 if (LPF[j] == 0) {
//                     LPF[j] = i;
//                 }
//             }
//         }
//     }
//     public int minOperations(int[] nums) {
//         int ans = 0;
//         for (int i = nums.length - 2; i >= 0; i--) {
//             if (nums[i] > nums[i + 1]) {
//                 nums[i] = LPF[nums[i]];
//                 if (nums[i] > nums[i + 1]) {
//                     return -1;
//                 }
//                 ans++;
//             }
//         }
//         return ans;
//     }
// }

// Accepted solution for LeetCode #3326: Minimum Division Operations to Make Array Non Decreasing
const mx = 10 ** 6;
const lpf = Array(mx + 1).fill(0);
for (let i = 2; i <= mx; ++i) {
    for (let j = i; j <= mx; j += i) {
        if (lpf[j] === 0) {
            lpf[j] = i;
        }
    }
}

function minOperations(nums: number[]): number {
    let ans = 0;
    for (let i = nums.length - 2; ~i; --i) {
        if (nums[i] > nums[i + 1]) {
            nums[i] = lpf[nums[i]];
            if (nums[i] > nums[i + 1]) {
                return -1;
            }
            ++ans;
        }
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n log n)

Space

O(1)

Approach Breakdown

EXHAUSTIVE

O(2ⁿ) time

O(n) space

Try every possible combination of choices. With n items each having two states (include/exclude), the search space is 2ⁿ. Evaluating each combination takes O(n), giving O(n × 2ⁿ). The recursion stack or subset storage uses O(n) space.

GREEDY

O(n log n) time

O(1) space

Greedy algorithms typically sort the input (O(n log n)) then make a single pass (O(n)). The sort dominates. If the input is already sorted or the greedy choice can be computed without sorting, time drops to O(n). Proving greedy correctness (exchange argument) is harder than the implementation.

Shortcut: Sort + single pass → O(n log n). If no sort needed → O(n). The hard part is proving it works.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

Using greedy without proof

Wrong move: Locally optimal choices may fail globally.

Usually fails on: Counterexamples appear on crafted input orderings.

Fix: Verify with exchange argument or monotonic objective before committing.