Lecture Notes – Graph Problem: Minimum Time to Reach a Destination

Introduction

• Topic: Graph Problems
• Video Number: 54
• Problem Level: Medium to Hard
• Ideal for interviews, especially high-level ones.

Problem Statement

• Problem ID: LeetCode 20457
• Title: Minimum Time to Reach a Destination
• Given a bidirectional connected graph represented as a 2D matrix.
• Contains n vertices numbered from 1 to n and edges between vertices u and v.
• No vertex has an edge to itself, thus eliminating cycles.

Graph Characteristics

• Edges: Represented as a 2D matrix.
• Traversal Time: Each edge traversal takes a specific number of minutes.
• Traffic Signals: Each vertex has a traffic signal that alternates between green and red every t minutes
  • Signals change at the same time for all vertices.
  • Can only leave a vertex when the signal is green.
  • You cannot wait at a vertex when the signal turns green.

Constraints

• Minimum Time: Define as the smallest value strictly greater than the minimum value.
• Understand seconds/minutes required to reach from start to destination.

Conditions for the Problem

• Given inputs: numbers of vertices, edges, time taken to traverse each edge, time to change signals.
• You can enter a vertex at any time, but can leave only when the signal is green.
• At the start of the journey, assume all signals are green.

Example Walkthrough

• Example Input: (not provided in full detail)
• Clarification of how to determine the output based on signal timings and edge weights.
• Example demonstrated that a specific path led to a minimum time of 13 seconds to reach the destination.

Algorithm Planning

1. Dijkstra's Algorithm: The best fit for finding the shortest path with a single source to a single destination in a weighted graph.
   • Dijkstra's algorithm can be applied to derive both minimum and second minimum distances from the source to the destination.
2. Consideration of Traffic Signals: The challenge is monitoring when the signals change and ensuring you only traverse the edges when the signal is green.
   • Calculation of time will determine when the signal would turn green.

Dijkstra’s Algorithm Details

• Initialize distance vectors for both minimum and second minimum.
• Priority Queue will help in selecting the next vertex based on the shortest distance.
• Maintain a count of how many times each vertex has been visited to differentiate between the cases for d1 and d2.

Conclusion

• This problem effectively combines knowledge of graph traversal, specifically Dijkstra's algorithm, with the additional complexity of time-dependent traffic signals.
• A BFS approach can also potentially solve the problem, focusing on the shortest path and graph characteristics.

Final Thoughts

• Understanding how to incorporate altered weights based on traffic signal states could be crucial.
• The practical application of this problem in coding interviews and real-life scenarios.