time = 11 minutes.

Solution

To find the second minimum time required to travel from vertex 1 to vertex n in a graph with traffic signals, we need to account for both path traversal and the waiting times caused by the traffic signals' cycle. The key insights are:

Each edge traversal takes exactly time minutes.
Each signal alternates every change minutes between green and red.
You can only leave a vertex when the signal is green.

Method 1 - Dijkstra Algorithm

We can solve this problem using a modified Dijkstra's or shortest path algorithm. This will help us account for the different paths' travel times and keep track of the second minimum time.

Here are the steps we can take:

Build the graph as adjacency list to representation.
Setup the dijkstra algorithm
- We use Priority Queue to store states represented as (current_time, current_vertex), and it is a min heap based on time.
- Add node 1 with time 0
- Time tracking: We use distance[n + 1][2] matrix to not just minimum time to reach each neighboring vertex, but also the second minimum time as well. For dist[1][0] minimum time will be 0. We use distance[n + 1] as nodes start with 1, not 0.
Start Dijkstra Algorithm
- Poll out the node from priority queue based on time
- Check if it's a red light when we are at current node, if it is, calculate the waiting time until the next green light. Our

Related Tags