Prerequisites: Graphs, BFS.
Problem: Chef needs to send a message to Chefina through a cloud provider’s network that spans across N locations with M connections between them. The task is to determine whether Chef can send a message to Chefina and, if possible, find the minimum number of servers on such a route. Servers are numbered from 1 to N, with Chef’s server being 1 and Chefina’s server being N.
Solution Approach: We can use BFS to solve the problem. If the node N is in same connected component as node 1, then the minimum no. of servers would be the ones on the shortest path from 1 to N else answer would be -1. By modeling the network as a graph and using Breadth-First Search (BFS), we can efficiently explore the graph in a level-by-level manner. The inherent nature of BFS makes it suitable for finding the shortest path in an unweighted graph, ensuring that the first occurrence of Chefina’s server is reached using the minimum number of servers.
Key Points:
- Level-by-Level Exploration:
- BFS traverses the graph level by level, starting from the source node (Chef’s server) and moving outward.
- At each level, all vertices at the same distance from the source are explored before moving to the next level.
- Shortest Path Guarantee:
- In an unweighted graph, BFS guarantees that the first occurrence of a target node (Chefina’s server) is reached with the minimum number of edges.
- As BFS explores the graph level by level, the first time Chefina’s server is encountered, it will be through the shortest possible path.
Since the shortest distance between two nodes stores no. of edges, we add 1 to the shortest distance as we need no. of nodes (i.e., no. of servers).
Time complexity: The time complexity is O(N + M), where N is the number of servers, and M is the number of connections. Each server and connection is processed once during the BFS traversal.
Space complexity: O(N) as the space complexity of BFS is O(N).