Problem Description

The city is modeled as a graph with n vertices, or points, which stand for different places within the city. The connections between these points are represented by bi-directional edges, which imply that one can travel in either direction between any two connected points. The set of these connections is given by an array of edge pairs, edges, where each pair [u_i, v_i] indicates a direct path between the vertices labeled u_i and v_i.

Travel time on any edge is constant, represented by the time variable. Importantly, the vertices in this city have traffic signals that toggle between green and red at fixed intervals given by the change variable. When a signal is green, one can leave the vertex, but they cannot do so during a red light, nor can they remain at the vertex waiting for it to turn green.

The task is to determine the second minimum time it takes to travel from vertex 1 to vertex n. The second minimum time is defined as the smallest value that is larger than the minimum travel time. Note that the travel is described with the following conditions:

• Multiple passes through any vertex are permitted, including the start and end points.
• The journey commences with all signals just turning green.

Given the setup of the graph (n and edges), the time taken to cross an edge (time), and the signal switching interval (change), the objective is to compute the second fastest time one can get from point 1 to point n.

Flowchart Walkthrough

Let's analyze the problem using the algorithm flowchart provided. Here's a detailed step-by-step walkthrough based on the Flowchart:

Is it a graph?

• Yes: The problem involves stations (nodes) and buses (edges) which connect these stations.

Is it a tree?

• No: There can be multiple paths between two nodes, representing different bus routes, so it's not a tree.

Is the problem related to directed acyclic graphs (DAGs)?

• No: The nature of transport systems implies possible cycles (e.g., a bus route can loop back to previously visited stations) and the problem itself doesn't specify acyclic constraints.

Is the problem related to shortest paths?

• Yes: The primary goal is to find the second minimum time to reach the destination, which is a variation of finding shortest or constrained shortest paths.

Is the graph weighted?

• Yes: Different routes can take different amounts of time, each bus's travel time acts as a weight.

Consequently, using the flowchart, the optimal solution points towards breadth-first search for both unweighted shortest paths (for understanding route connectivity quickly) and weighted scenarios when fine-tuning times or in scenarios like finding the second shortest that require further exploration beyond straightforward shortest path strategies.

Conclusion: Given this is a shortest path problem on a weighted graph, the evolved use of BFS, possibly in conjunction with priority queues (to handle weights efficiently), appears appropriate. Although the flowchart directly suggests Dijkstra's Algorithm for weighted shortest paths, BFS can be adapted for certain interpretations of the problem (e.g., when the weights are uniformly distributed or when modified to handle scenarios like "second minimum" through layers or state expansion techniques).

Intuition

The solution approach involves the use of a breadth-first search (BFS) algorithm, which is typically used for finding the shortest path in a graph. For this problem, however, we need to consider traffic signals' timings while traversing the graph, hence a modification of the standard BFS.

In this solution, a graph g is constructed from the edges list where each vertex points to its set of adjacent vertices. The dist list holds the shortest and second shortest number of edges needed to reach every vertex from the starting vertex. The BFS is initiated from vertex 1.

The primary intuition here revolves around capturing the shortest and second shortest paths to each vertex. In a typical BFS, only the shortest path is considered, but due to our second minimum value requirement, we maintain two path lengths for each vertex. The BFS explores the graph, constantly updating these path lengths.

After the BFS traversal, we compute the actual time considering traffic signals. Since we have captured the number of edges traversed on the second shortest path to vertex n, we multiply this number by time to get the base travel time. However, due to the signal switching, we may have to wait at certain points if the signal is red when we arrive. This is modeled in the code by checking if the elapsed time is in a red signal interval. If it's a red signal, we wait until it turns green before continuing our journey.

To summarize, the solution consists of conducting a modified BFS to determine the second shortest path by the number of edges crossed, then converting this path length into actual time by taking into account the traffic signal intervals.

Learn more about Breadth-First Search, Graph and Shortest Path patterns.

Solution Approach

To execute our solution, we utilize a Breadth-First Search (BFS) algorithm with a slight twist to account for the traffic signals. Here is an explanation of how the provided Python code implements the solution:

• First, we use a defaultdict to create a graph g where each vertex u has a set of its neighbors v. This represents our bi-directional graph.

• We initialize a double-ended queue (deque) called q that will keep track of vertices to visit, starting with vertex 1. Each item in this queue will be a tuple containing a vertex and the number of edges crossed to reach it.

• The dist array is initialized to hold the smallest and second smallest number of edges required to reach each vertex, starting from inf (infinity). For vertex 1, the distance is set to 0 since we start there.

• The BFS begins by iterating through the queue q. At each step:

  • We pop the leftmost item (u, d) from the queue.
  • u represents the current vertex, while d is the distance (edge count) from vertex 1.
  • We then iterate through each neighbor v of the current vertex u.
  • For each neighbor v, we check whether we've found a shorter path (fewer edges) to reach it (d + 1 < dist[v][0]). If so, we update the shortest distance dist[v][0] and enqueue (v, d + 1) for further exploration.
  • If the shortest path is already found but the second shortest isn't (dist[v][0] < d + 1 < dist[v][1]), we update the second shortest distance dist[v][1] and, unless we're at the destination vertex n, we enqueue (v, d + 1).

• Upon completion of the BFS, we compute the actual time taken to travel on the second shortest path using traffic signals consideration. The variable ans represents this time.

  • We iterate i from 0 to one less than the second shortest distance dist[n][1] (the number of edges in the second shortest path).
  • For each edge, we increase ans by time.
  • After traveling each edge, except the last one, we check if the signal would be red ((ans // change) % 2 == 1). If so, we wait for the next green (ans = (ans + change) // change * change).

• Finally, ans holds the second minimum time it takes to go from vertex 1 to vertex n considering all traffic signals, which we return.

The key data structures used include:

• A graph built as a dictionary of sets, providing efficient lookups and neighbor traversal.
• A double-ended queue facilitating BFS.
• A list of lists, dist, for tracking the shortest and second shortest paths to vertices.

This solution utilizes BFS for path exploration and leverages arithmetic to manage time spent waiting at traffic signals. The pairing of graph traversal with time management based on the graph's constraints makes it an elegant solution to the problem.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Assume we have the following setup:

• A city with n = 4 vertices.
• Connections between them: edges = [[1, 2], [2, 3], [3, 4], [1, 4]].
• Constant travel time on any edge: time = 2 seconds.
• Traffic signals toggle interval: change = 3 seconds.

The city graph will look like this:

1 -- 2
|    |
4 -- 3

Vertex 1 is our starting point, and 4 is our destination.

Initial Setup

We construct the graph g using the edges:

g = {
  1: {2, 4},
  2: {1, 3},
  3: {2, 4},
  4: {1, 3}
}

The distance array dist is initialized as:

dist = [
  [inf, inf],  # Vertex 1
  [inf, inf],  # Vertex 2
  [inf, inf],  # Vertex 3
  [inf, inf]   # Vertex 4
]

And updated for vertex 1 as dist[0] = [0, inf], meaning the shortest distance to vertex 1 is 0 edges, and the second shortest is not yet found.

BFS Exploration

We enqueue vertex 1 with a distance of 0 into the queue q.

Dequeueing 1, we observe its neighbors are 2 and 4. We visit vertex 2 first, noting a distance of 1 edge. Since dist[2][0] is inf, we update it to 1. The same is done for vertex 4.

Now our dist looks like this:

dist = [
  [0, inf],  # Vertex 1
  [1, inf],  # Vertex 2
  [inf, inf],  # Vertex 3
  [1, inf]   # Vertex 4
]

In our queue, we now have vertices 2 and 4, both with distances of 1.

Continuing BFS, we dequeue 2 and then 4. Both have 3 as a common neighbor, which we reach with 2 edges. dist[3][0] becomes 2. From 3, we can reach 4 with 3 edges, updating dist[4][1] to 3.

Now our dist looks like this:

dist = [
  [0, inf],  # Vertex 1
  [1, inf],  # Vertex 2
  [2, inf],  # Vertex 3
  [1, 3]   # Vertex 4
]

This finalizes the BFS as we've found both the shortest (1 edge) and second-shortest (3 edges) paths to vertex 4.

Actual Time Computation

The shortest path to vertex 4 takes 1 edge, which at 2 seconds per edge, would take 2 seconds without signal delays.

The second shortest path takes 3 edges, which base time is 3 * 2 = 6 seconds.

Before crossing each edge except the last one, we consider the signal's state. On the second shortest path, for each edge:

• At 0 seconds, we're at vertex 1.
• After the first edge, 2 seconds have passed. No wait time is needed.
• Arriving at vertex 3, 4 seconds have passed. The signal state alternates every 3 seconds, so we wait 2 more seconds for the next green signal.
• After the second edge, 6 + 2 = 8 seconds have passed. Again, no wait is needed because we depart at the start of a green signal cycle.
• Arriving at vertex 4, 10 seconds have passed, and no further wait is needed.

The second minimum time to go from vertex 1 to vertex 4 is 10 seconds, considering traffic signals.

Thus, by combining BFS to track paths and managing time at signals, we've efficiently solved the problem.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n * m) where n is the number of nodes in the graph and m is the number of edges. Each node gets processed at most twice (once for each of its two shortest paths since we're looking for the second minimum time). For each node, we explore all its adjacent edges. In the worst case, the total number of operations can be proportional to n times m.

The space complexity is O(n). The extra space is used for:

1. The adjacency list g which, in the worst case, could hold all nodes and edges, so O(n + m), but since m is at most n(n-1)/2, it simplifies to O(n).
2. The queue q which could, in the worst case, contain all nodes. The queue holds nodes with their distance which is O(n).
3. The dist array which contains two entries for each node, so 2n which simplifies to O(n).

Learn more about how to find time and space complexity quickly using problem constraints.