# Group Quiz 2

```Group Quiz 2
1
Question 1
Give an algorithm to detect whether a
given undirected graph contains a cycle.
If the graph contains a cycle, then your
algorithm should output one. It should not
output all the cycles in the graph, just one
of them. The running time of your
algorithm should be O(m + n) for a graph
with n nodes and m edges.
2
Question 2
Let’s consider a long country road with houses
scattered very sparsely along it. We can picture
the road as a long line segment, with an eastern
endpoint and a western endpoint. Further,
suppose the residents are avid cell phone users.
You want to place cell phone tower stations at
certain points along the road, so that every
house is within 4 miles of one of them.
Give an efficient algorithm that achieves this
goal, using as few towers as possible.
3
Question 3
A binary tree is a rooted tree in which
each node has at most two children.
Show by induction that in any binary tree
the number of nodes with two children is
exactly one less than the number of
leaves.
4
```