A15192. 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 $1, 2, \ldots, k$ ).

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 $2\cdot 10^5$ .

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

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.