LeetCode #3741 — MEDIUM

Minimum Distance Between Three Equal Elements II

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

The Problem

Problem Statement

You are given an integer array nums.

A tuple (i, j, k) of 3 distinct indices is good if nums[i] == nums[j] == nums[k].

The distance of a good tuple is abs(i - j) + abs(j - k) + abs(k - i), where abs(x) denotes the absolute value of x.

Return an integer denoting the minimum possible distance of a good tuple. If no good tuples exist, return -1.

Example 1:

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

Output: 6

Explanation:

The minimum distance is achieved by the good tuple (0, 2, 3).

(0, 2, 3) is a good tuple because nums[0] == nums[2] == nums[3] == 1. Its distance is abs(0 - 2) + abs(2 - 3) + abs(3 - 0) = 2 + 1 + 3 = 6.

Example 2:

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

Output: 8

Explanation:

The minimum distance is achieved by the good tuple (2, 4, 6).

(2, 4, 6) is a good tuple because nums[2] == nums[4] == nums[6] == 2. Its distance is abs(2 - 4) + abs(4 - 6) + abs(6 - 2) = 2 + 2 + 4 = 8.

Example 3:

Input: nums = [1]

Output: -1

Explanation:

There are no good tuples. Therefore, the answer is -1.

Constraints:

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

Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums. A tuple (i, j, k) of 3 distinct indices is good if nums[i] == nums[j] == nums[k]. The distance of a good tuple is abs(i - j) + abs(j - k) + abs(k - i), where abs(x) denotes the absolute value of x. Return an integer denoting the minimum possible distance of a good tuple. If no good tuples exist, return -1.

Baseline thinking

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

Pattern signal: Array · Hash Map

Example 1

[1,2,1,1,3]

Example 2

[1,1,2,3,2,1,2]

Example 3

[1]

Step 02

Core Insight

What unlocks the optimal approach

The distance formula <code>abs(i - j) + abs(j - k) + abs(k - i)</code> simplifies to <code>2 * (max(i, j, k) - min(i, j, k))</code>.
Group the indices for each unique number. For a number to form a good tuple, it must appear at least 3 times.
For each number that appears at least 3 times, we want to find three of its indices <code>p < q < r</code> that minimize <code>r - p</code>. This is achieved by considering every three consecutive indices in the sorted list of indices.

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 #3741: Minimum Distance Between Three Equal Elements II
class Solution {
    public int minimumDistance(int[] nums) {
        int n = nums.length;
        Map<Integer, List<Integer>> g = new HashMap<>();
        for (int i = 0; i < n; ++i) {
            g.computeIfAbsent(nums[i], k -> new ArrayList<>()).add(i);
        }
        final int inf = 1 << 30;
        int ans = inf;
        for (var ls : g.values()) {
            int m = ls.size();
            for (int h = 0; h < m - 2; ++h) {
                int i = ls.get(h);
                int k = ls.get(h + 2);
                int t = (k - i) * 2;
                ans = Math.min(ans, t);
            }
        }
        return ans == inf ? -1 : ans;
    }
}

// Accepted solution for LeetCode #3741: Minimum Distance Between Three Equal Elements II
func minimumDistance(nums []int) int {
	g := make(map[int][]int)
	for i, x := range nums {
		g[x] = append(g[x], i)
	}

	inf := 1 << 30
	ans := inf

	for _, ls := range g {
		m := len(ls)
		for h := 0; h < m-2; h++ {
			i := ls[h]
			k := ls[h+2]
			t := (k - i) * 2
			ans = min(ans, t)
		}
	}

	if ans == inf {
		return -1
	}
	return ans
}

# Accepted solution for LeetCode #3741: Minimum Distance Between Three Equal Elements II
class Solution:
    def minimumDistance(self, nums: List[int]) -> int:
        g = defaultdict(list)
        for i, x in enumerate(nums):
            g[x].append(i)
        ans = inf
        for ls in g.values():
            for h in range(len(ls) - 2):
                i, k = ls[h], ls[h + 2]
                ans = min(ans, (k - i) * 2)
        return -1 if ans == inf else ans

// Accepted solution for LeetCode #3741: Minimum Distance Between Three Equal Elements II
// 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 #3741: Minimum Distance Between Three Equal Elements II
// class Solution {
//     public int minimumDistance(int[] nums) {
//         int n = nums.length;
//         Map<Integer, List<Integer>> g = new HashMap<>();
//         for (int i = 0; i < n; ++i) {
//             g.computeIfAbsent(nums[i], k -> new ArrayList<>()).add(i);
//         }
//         final int inf = 1 << 30;
//         int ans = inf;
//         for (var ls : g.values()) {
//             int m = ls.size();
//             for (int h = 0; h < m - 2; ++h) {
//                 int i = ls.get(h);
//                 int k = ls.get(h + 2);
//                 int t = (k - i) * 2;
//                 ans = Math.min(ans, t);
//             }
//         }
//         return ans == inf ? -1 : ans;
//     }
// }

// Accepted solution for LeetCode #3741: Minimum Distance Between Three Equal Elements II
function minimumDistance(nums: number[]): number {
    const n = nums.length;
    const g = new Map<number, number[]>();

    for (let i = 0; i < n; i++) {
        if (!g.has(nums[i])) {
            g.set(nums[i], []);
        }
        g.get(nums[i])!.push(i);
    }

    const inf = 1 << 30;
    let ans = inf;

    for (const ls of g.values()) {
        const m = ls.length;
        for (let h = 0; h < m - 2; h++) {
            const i = ls[h];
            const k = ls[h + 2];
            const t = (k - i) * 2;
            ans = Math.min(ans, t);
        }
    }

    return ans === inf ? -1 : 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)

Space

O(n)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

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.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.