Challenge your understanding of key graph algorithms including Breadth-First Search (BFS), Depth-First Search (DFS), and common shortest path techniques. Assess your ability to select and apply the right traversal or path-finding strategy in typical graph problem scenarios.
Which graph traversal algorithm would be most suitable for finding the shortest path between two nodes in an unweighted, undirected graph, such as a maze or network of friends?
Explanation: Breadth-First Search (BFS) is specifically designed to find the shortest path in unweighted graphs by visiting nodes level by level, ensuring the first time a node is reached is via the shortest path. DFS is not guaranteed to find the shortest path, as it traverses as deep as possible before backtracking. Dijkstra's and Bellman-Ford algorithms are meant for weighted graphs; while they can work with unweighted graphs, BFS is simpler and more efficient in such cases. Thus, BFS is the best choice for the scenario described.
Given the following directed acyclic graph: 1 → 2, 1 → 3, 2 → 4, 3 → 5, if you start a DFS from node 1, which of these could be a valid visiting order?
Explanation: In a Depth-First Search, the algorithm explores as far as possible along each branch before backtracking. Starting at node 1, it could visit 2 first, then 4, then backtrack and visit 3 and finally 5, making '1, 2, 4, 3, 5' valid. '1, 2, 3, 4, 5' does not explore a full branch before moving to the next. '1, 3, 2, 4, 5' visits 3 before finishing node 2’s descendants, and '1, 4, 2, 3, 5' visits 4 before 2, which cannot happen since 4 is only reachable via 2.
When dealing with a positively weighted directed graph with no negative cycles, such as a road network with varying distances, which algorithm efficiently computes the shortest path from a single source to all other nodes?
Explanation: Dijkstra's Algorithm is designed for finding the shortest paths from a single source to all vertices in a graph with non-negative edge weights. DFS doesn't consider edge weights and can't guarantee shortest paths. Prim's Algorithm is used for finding minimum spanning trees, not shortest paths. BFS only finds shortest paths in unweighted graphs. For weighted graphs like road networks, Dijkstra’s is most suitable.
Which algorithm can be used to detect cycles in a directed graph by maintaining a recursion stack during traversal?
Explanation: DFS can detect cycles in directed graphs by tracking the recursion stack; encountering a node already in the stack indicates a cycle. BFS does not use a recursion stack and thus isn't suited for this method. Floyd-Warshall is an all-pairs shortest path algorithm and doesn't serve for cycle detection in the described way. Topological Sort detects cycles by failing to include all nodes but isn't itself an algorithm for cycle detection using recursion stacks.
Suppose a graph has negative edge weights but no negative cycles, such as certain currency exchange graphs. Which algorithm is suitable for finding the shortest paths from a single source to all other vertices?
Explanation: The Bellman-Ford Algorithm handles graphs with negative edge weights and is capable of finding shortest paths so long as no negative cycles are present. DFS and BFS can’t accurately compute shortest paths in weighted graphs, particularly with negative weights. Kruskal's Algorithm is used for finding minimum spanning trees and doesn't resolve shortest path queries. Therefore, Bellman-Ford is the correct choice for this scenario.