CF1706E.Qpwoeirut and Vertices

You are given a connected undirected graph with $n$ vertices and $m$ edges. Vertices of the graph are numbered by integers from $1$ to $n$ and edges of the graph are numbered by integers from $1$ to $m$.

Your task is to answer $q$ queries, each consisting of two integers $l$ and $r$. The answer to each query is the smallest non-negative integer $k$ such that the following condition holds:

• For all pairs of integers $(a, b)$ such that $l\le a\le b\le r$, vertices $a$ and $b$ are reachable from one another using only the first $k$ edges (that is, edges $1, 2, \ldots, k$).

给你一个包含 $n$ 个顶点和 $m$ 条边的连通无向图。图的顶点用 $1$ 到 $n$ 的整数编号，图的边用 $1$ 到 $m$ 的整数编号。

你需要回答 $q$ 个查询，每个查询由两个整数 $l$ 和 $r$ 组成。每个查询的答案是最小的非负整数 $k$，使得以下条件成立：

• 对于所有满足 $l\le a\le b\le r$ 的整数对 $(a, b)$，顶点 $a$ 和 $b$ 仅通过前 $k$ 条边（即边 $1, 2, \ldots, k$）即可相互到达。

The first line contains a single integer $t$ ($1\le t\le 1000$) — the number of test cases.

The first line of each test case contains three integers $n$, $m$, and $q$ ($2\le n\le 10^5$, $1\le m, q\le 2\cdot 10^5$) — the number of vertices, edges, and queries respectively.

Each of the next $m$ lines contains two integers $u_i$ and $v_i$ ($1\le u_i, v_i\le n$) — ends of the $i$-th edge.

It is guaranteed that the graph is connected and there are no multiple edges or self-loops.

Each of the next $q$ lines contains two integers $l$ and $r$ ($1\le l\le r\le n$) — descriptions of the queries.

It is guaranteed that that the sum of $n$ over all test cases does not exceed $10^5$, the sum of $m$ over all test cases does not exceed $2\cdot 10^5$, and the sum of $q$ over all test cases does not exceed $2\cdot 10^5$.

第一行包含一个整数 $t$（$1\le t\le 1000$）—— 测试用例的数量。

每个测试用例的第一行包含三个整数 $n$、$m$ 和 $q$（$2\le n\le 10^5$，$1\le m, q\le 2\cdot 10^5$）—— 分别表示顶点数、边数和查询数。

接下来的 $m$ 行中，每行包含两个整数 $u_i$ 和 $v_i$（$1\le u_i, v_i\le n$）—— 表示第 $i$ 条边的两个端点。

保证该图是连通的，且不存在重边或自环。

接下来的 $q$ 行中，每行包含两个整数 $l$ 和 $r$（$1\le l\le r\le n$）—— 表示一次查询。

保证所有测试用例的 $n$ 之和不超过 $10^5$，所有测试用例的 $m$ 之和不超过 $2\cdot 10^5$，所有测试用例的 $q$ 之和不超过 $2\cdot 10^5$。

For each test case, print $q$ integers — the answers to the queries.

对于每个测试用例，输出 $q$ 个整数——即各查询的答案。

Graph from the first test case. The integer near the edge is its number.

In the first test case, the graph contains $2$ vertices and a single edge connecting vertices $1$ and $2$.

In the first query, $l=1$ and $r=1$. It is possible to reach any vertex from itself, so the answer to this query is $0$.

In the second query, $l=1$ and $r=2$. Vertices $1$ and $2$ are reachable from one another using only the first edge, through the path $1 \longleftrightarrow 2$. It is impossible to reach vertex $2$ from vertex $1$ using only the first $0$ edges. So, the answer to this query is $1$.

Graph from the second test case. The integer near the edge is its number.

In the second test case, the graph contains $5$ vertices and $5$ edges.

In the first query, $l=1$ and $r=4$. It is enough to use the first $3$ edges to satisfy the condition from the statement:

• Vertices $1$ and $2$ are reachable from one another through the path $1 \longleftrightarrow 2$ (edge $1$).
• Vertices $1$ and $3$ are reachable from one another through the path $1 \longleftrightarrow 3$ (edge $2$).
• Vertices $1$ and $4$ are reachable from one another through the path $1 \longleftrightarrow 2 \longleftrightarrow 4$ (edges $1$ and $3$).
• Vertices $2$ and $3$ are reachable from one another through the path $2 \longleftrightarrow 1 \longleftrightarrow 3$ (edges $1$ and $2$).
• Vertices $2$ and $4$ are reachable from one another through the path $2 \longleftrightarrow 4$ (edge $3$).
• Vertices $3$ and $4$ are reachable from one another through the path $3 \longleftrightarrow 1 \longleftrightarrow 2 \longleftrightarrow 4$ (edges $2$, $1$, and $3$).

If we use less than $3$ of the first edges, then the condition won't be satisfied. For example, it is impossible to reach vertex $4$ from vertex $1$ using only the first $2$ edges. So, the answer to this query is $3$.

In the second query, $l=3$ and $r=4$. Vertices $3$ and $4$ are reachable from one another through the path $3 \longleftrightarrow 1 \longleftrightarrow 2 \longleftrightarrow 4$ (edges $2$, $1$, and $3$). If we use any fewer of the first edges, nodes $3$ and $4$ will not be reachable from one another.

第一个测试用例中的图。边旁的整数表示该边的编号。

在第一个测试用例中，图包含 $2$ 个顶点，以及一条连接顶点 $1$ 和 $2$ 的边。

在第一个查询中，$l=1$ 且 $r=1$。任一顶点均可从自身到达，因此该查询的答案为 $0$。

在第二个查询中，$l=1$ 且 $r=2$。仅使用第 $1$ 条边，即可通过路径 $1 \longleftrightarrow 2$ 实现顶点 $1$ 与 $2$ 之间的相互可达。若仅使用前 $0$ 条边，则无法从顶点 $1$ 到达顶点 $2$。因此，该查询的答案为 $1$。

第二个测试用例中的图。边旁的整数表示该边的编号。

在第二个测试用例中，图包含 $5$ 个顶点和 $5$ 条边。

在第一个查询中，$l=1$ 且 $r=4$。仅需使用前 $3$ 条边，即可满足题目所述条件：

• 顶点 $1$ 和 $2$ 可通过路径 $1 \longleftrightarrow 2$（边 $1$）相互可达；
• 顶点 $1$ 和 $3$ 可通过路径 $1 \longleftrightarrow 3$（边 $2$）相互可达；
• 顶点 $1$ 和 $4$ 可通过路径 $1 \longleftrightarrow 2 \longleftrightarrow 4$（边 $1$ 和 $3$）相互可达；
• 顶点 $2$ 和 $3$ 可通过路径 $2 \longleftrightarrow 1 \longleftrightarrow 3$（边 $1$ 和 $2$）相互可达；
• 顶点 $2$ 和 $4$ 可通过路径 $2 \longleftrightarrow 4$（边 $3$）相互可达；
• 顶点 $3$ 和 $4$ 可通过路径 $3 \longleftrightarrow 1 \longleftrightarrow 2 \longleftrightarrow 4$（边 $2$、$1$ 和 $3$）相互可达。

若使用的前若干条边数量少于 $3$，则无法满足条件。例如，仅使用前 $2$ 条边时，无法从顶点 $1$ 到达顶点 $4$。因此，该查询的答案为 $3$。

在第二个查询中，$l=3$ 且 $r=4$。顶点 $3$ 和 $4$ 可通过路径 $3 \longleftrightarrow 1 \longleftrightarrow 2 \longleftrightarrow 4$（边 $2$、$1$ 和 $3$）相互可达。若使用的前若干条边数量更少，则顶点 $3$ 与 $4$ 将无法相互可达。