codeforces#P1706E. Qpwoeirut and Vertices
Qpwoeirut and Vertices
Description
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.
Input
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 $2\cdot 10^5$.
Output
For each test case, print $q$ integers — the answers to the queries.
Samples
3
2 1 2
1 2
1 1
1 2
5 5 5
1 2
1 3
2 4
3 4
3 5
1 4
3 4
2 2
2 5
3 5
3 2 1
1 3
2 3
1 3
0 1
3 3 0 5 5
2
Note
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.