help me solve with these rules: For questions where you are asked to write an algorithm:

$$ You are required to write pseudocode. Do not write code in any specific language such as

Java, Python, C$/$C$++,$ etc.

$$ You should include a brief explanation of it and justification or a proof $($when appropriate$)$

for why it works. Part of your grade depends on this.

$$ Derive the runtime of the designed algorithm, part of the grade depends on the efficiency of

your algorithm$(25$ points$)$ Given a connected undirected graph, write an algorithm to determine the existence

of simple cycle in the graph. A simple cycle is defined as a sequence of vertices $v_1,v_2,$dots,$v_k,$

where all vertices are distinct and AAiin$[1,k-1],(v_i,v_{i+1})inE$ and $(v_k,v_1)inE.$

$(25$ points$)$ You are a city planner working for a metropolitan area with a complex road

network. The city has recently expanded, and there is a need to establish a centralized

emergency service station $($such as a fire station, hospital, or police station$)$ that can be

quickly accessed from any intersection in the city.

The city has a mix of one$-$way and two$-$way streets, all of equal length. To ensure that

emergency response times are minimized, you need to identify a strategic location for this

emergency service station. Write an algorithm to determine if there exists an intersection as

emergency service station in the city that can be reached from every other intersection, or to

conclude that no such intersection exists.

$(25$ points$)$ The city planning commission are also trying to implement a new network of

electric buses to enhance the public transportation. Each bus is routed to travel along the

shortest path from the starting intersection to its destination intersection. To ensure the elec$-$

tric buses are used efficiently and to alleviate range anxiety among drivers, the city planning

commission wants to determine the longest bus route within the city. Write an algorithm to

output the length of the longest shortest path.

$(25$ points$)$Given an undirected graph $G$ and node $s,$ Write an algorithm that outputs an

array ShortestCount in which ShortestCount $[$i$]$ is the number of shortest paths from $s$ to

vertex i$.$