LeetCode #3244 — HARD

Shortest Distance After Road Addition Queries II

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

You are given an integer n and a 2D integer array queries.

There are n cities numbered from 0 to n - 1. Initially, there is a unidirectional road from city i to city i + 1 for all 0 <= i < n - 1.

queries[i] = [u_i, v_i] represents the addition of a new unidirectional road from city u_i to city v_i. After each query, you need to find the length of the shortest path from city 0 to city n - 1.

There are no two queries such that queries[i][0] < queries[j][0] < queries[i][1] < queries[j][1].

Return an array answer where for each i in the range [0, queries.length - 1], answer[i] is the length of the shortest path from city 0 to city n - 1 after processing the first i + 1 queries.

Example 1:

Input: n = 5, queries = [[2,4],[0,2],[0,4]]

Output: [3,2,1]

Explanation:

After the addition of the road from 2 to 4, the length of the shortest path from 0 to 4 is 3.

After the addition of the road from 0 to 2, the length of the shortest path from 0 to 4 is 2.

After the addition of the road from 0 to 4, the length of the shortest path from 0 to 4 is 1.

Example 2:

Input: n = 4, queries = [[0,3],[0,2]]

Output: [1,1]

Explanation:

After the addition of the road from 0 to 3, the length of the shortest path from 0 to 3 is 1.

After the addition of the road from 0 to 2, the length of the shortest path remains 1.

Constraints:

3 <= n <= 10⁵
1 <= queries.length <= 10⁵
queries[i].length == 2
0 <= queries[i][0] < queries[i][1] < n
1 < queries[i][1] - queries[i][0]
There are no repeated roads among the queries.
There are no two queries such that i != j and queries[i][0] < queries[j][0] < queries[i][1] < queries[j][1].

Step 01

Brute Force Baseline

Problem summary: You are given an integer n and a 2D integer array queries. There are n cities numbered from 0 to n - 1. Initially, there is a unidirectional road from city i to city i + 1 for all 0 <= i < n - 1. queries[i] = [ui, vi] represents the addition of a new unidirectional road from city ui to city vi. After each query, you need to find the length of the shortest path from city 0 to city n - 1. There are no two queries such that queries[i][0] < queries[j][0] < queries[i][1] < queries[j][1]. Return an array answer where for each i in the range [0, queries.length - 1], answer[i] is the length of the shortest path from city 0 to city n - 1 after processing the first i + 1 queries.

Baseline thinking

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

Pattern signal: Array · Greedy · Segment Tree

Example 1

5
[[2,4],[0,2],[0,4]]

Example 2

4
[[0,3],[0,2]]

Step 02

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

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

Largest constraint values

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 #3244: Shortest Distance After Road Addition Queries II
class Solution {
    public int[] shortestDistanceAfterQueries(int n, int[][] queries) {
        int[] nxt = new int[n - 1];
        for (int i = 1; i < n; ++i) {
            nxt[i - 1] = i;
        }
        int m = queries.length;
        int cnt = n - 1;
        int[] ans = new int[m];
        for (int i = 0; i < m; ++i) {
            int u = queries[i][0], v = queries[i][1];
            if (nxt[u] > 0 && nxt[u] < v) {
                int j = nxt[u];
                while (j < v) {
                    --cnt;
                    int t = nxt[j];
                    nxt[j] = 0;
                    j = t;
                }
                nxt[u] = v;
            }
            ans[i] = cnt;
        }
        return ans;
    }
}

// Accepted solution for LeetCode #3244: Shortest Distance After Road Addition Queries II
func shortestDistanceAfterQueries(n int, queries [][]int) (ans []int) {
	nxt := make([]int, n-1)
	for i := range nxt {
		nxt[i] = i + 1
	}
	cnt := n - 1
	for _, q := range queries {
		u, v := q[0], q[1]
		if nxt[u] > 0 && nxt[u] < v {
			i := nxt[u]
			for i < v {
				cnt--
				nxt[i], i = 0, nxt[i]
			}
			nxt[u] = v
		}
		ans = append(ans, cnt)
	}
	return
}

# Accepted solution for LeetCode #3244: Shortest Distance After Road Addition Queries II
class Solution:
    def shortestDistanceAfterQueries(
        self, n: int, queries: List[List[int]]
    ) -> List[int]:
        nxt = list(range(1, n))
        ans = []
        cnt = n - 1
        for u, v in queries:
            if 0 < nxt[u] < v:
                i = nxt[u]
                while i < v:
                    cnt -= 1
                    nxt[i], i = 0, nxt[i]
                nxt[u] = v
            ans.append(cnt)
        return ans

// Accepted solution for LeetCode #3244: Shortest Distance After Road Addition Queries II
/**
 * [3244] Shortest Distance After Road Addition Queries II
 */
pub struct Solution {}

// submission codes start here

impl Solution {
    pub fn shortest_distance_after_queries(n: i32, queries: Vec<Vec<i32>>) -> Vec<i32> {
        let n = n as usize;

        let mut roads = vec![None; n];
        for i in 0..n {
            roads[i] = Some(i + 1);
        }

        let mut result = Vec::with_capacity(queries.len());
        let mut distance = n - 1;

        for query in queries {
            let (start, end) = (query[0] as usize, query[1] as usize);
            let mut k = roads[start];
            roads[start] = Some(end);

            while let Some(now) = k {
                if now >= end {
                    break;
                }

                k = roads[now];
                roads[now] = None;
                distance -= 1;
            }

            result.push(distance);
        }

        result.into_iter().map(|x| x as i32).collect()
    }
}

// submission codes end

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_3244() {
        assert_eq!(
            vec![3, 2, 1],
            Solution::shortest_distance_after_queries(5, vec![vec![2, 4], vec![0, 2], vec![0, 4]])
        );
        assert_eq!(
            vec![1, 1],
            Solution::shortest_distance_after_queries(4, vec![vec![0, 3], vec![0, 2]])
        );
    }
}

// Accepted solution for LeetCode #3244: Shortest Distance After Road Addition Queries II
function shortestDistanceAfterQueries(n: number, queries: number[][]): number[] {
    const nxt: number[] = Array.from({ length: n - 1 }, (_, i) => i + 1);
    const ans: number[] = [];
    let cnt = n - 1;
    for (const [u, v] of queries) {
        if (nxt[u] && nxt[u] < v) {
            let i = nxt[u];
            while (i < v) {
                --cnt;
                [nxt[i], i] = [0, nxt[i]];
            }
            nxt[u] = v;
        }
        ans.push(cnt);
    }
    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 + q)

Space

O(n)

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.

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.