3244. Shortest Distance After Road Addition Queries II

Problem

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.

https://leetcode.cn/problems/shortest-distance-after-road-addition-queries-ii/

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].

Test Cases

1 2	class Solution: def shortestDistanceAfterQueries(self, n: int, queries: List[List[int]]) -> List[int]:

solution_test.py

import pytest

from solution import Solution


@pytest.mark.parametrize('n, queries, expected', [
    (5, [[2,4],[0,2],[0,4]], [3,2,1]),
    (4, [[0,3],[0,2]], [1,1]),
])
class Test:
    def test_solution(self, n, queries, expected):
        sol = Solution()
        assert sol.shortestDistanceAfterQueries(n, queries) == expected

Thoughts

这是 3243. Shortest Distance After Road Addition Queries I 的进阶版，增加了限制条件，任意两条道路不会交叉。

因为道路不会交叉，设有一条道路 [u, v]，那么城市 u 和 v 之间的城市（即 [u+1, u+2, ..., v-2, v-1]）就都可以忽略掉（因为对于任何 u < i < v，从城市 i 出发，不可能一步走到 v 之后）。所有的道路都加好之后，剩下的城市就是从 0 走到 n-1 的必经点，剩余城市个数减去一就是距离。

记录从城市 i 出发，一步可以走到的最远的城市为 f[i]。初始时 f[i] = i + 1。

初始的时候，集合中有所有城市。增加一条道路 [u, v] 时，如果 u 已经不在集合中或者 v <= f[u]，则这条道路对于缩短距离没有帮助，直接忽略。否则就把集合中所有 u < i < v 的城市 i 都删掉。删的时候不需要遍历 u 到 v 的所有值，因为现在集合中只有 f[u], f[f[u]], ..., f[...f[u]] < v，依次删掉这些即可。

把 f[u] 从集合中删掉，并继续看 f[f[u]]，直到走到 v 就停止，然后将 f[u] 的值更新为 v。

附加的空间复杂度 O(n)。时间复杂度 O(n + q)，因为除了遍历所有 query 之外，相当于每个城市都加入再移出集合各一次。

写代码的时候不用真的创建一个包含所有城市的集合，再逐个去掉没用的。实际上当城市 i 从集合中删除后，f[i] 就没用了，可以给 f[i] 赋一个非法值（如 n）表示被删除了。而剩余城市的数量可以在每次去掉某个城市时直接减去一（循环逻辑保证不会多次去掉同一个城市）。

Code

solution.py

class Solution:
    def shortestDistanceAfterQueries(self, n: int, queries: list[list[int]]) -> list[int]:
        f = [i+1 for i in range(n)] # f[i] == n means i was removed.
        dis = n - 1
        results = [dis] * len(queries)
        for q, (u, v) in enumerate(queries):
            if (i := f[u]) < v:
                f[u] = v
                while i < v:
                    f[i], i = n, f[i]
                    dis -= 1 # City `i` removed.

            results[q] = dis

        return results