I got the BFS part down pretty fast, the real stumble was when they pushed toward weighted edges and Dijkstra.
Use BFS for unweighted graphs to compute distances from s and count shortest paths by summing counts from predecessors. For weighted graphs, use Dijkstra's algorithm with a modified relaxation step that accumulates path counts when equal distances are found. Clearly state assumptions (e.g., non-negative weights) and discuss handling of zero-weight edges or negative cycles if applicable.
Pro tip: Mention that path counts can grow exponentially, so use modular arithmetic or big integers, and clarify whether paths are considered distinct by node sequence or edge sequence. Also, note that for weighted graphs with zero-weight edges, Dijkstra may need a tweak to avoid infinite loops.
Ask about graph properties: directed/undirected, weighted/unweighted, presence of negative weights, and whether paths are simple. Confirm if the count should be modulo a number.
For unweighted graphs, BFS is optimal. For weighted graphs with non-negative weights, Dijkstra's algorithm is suitable. If negative weights exist, Bellman-Ford or Floyd-Warshall may be needed.
During traversal, maintain an array of distances and an array of path counts. When relaxing an edge, if a shorter distance is found, update distance and set count to predecessor's count; if equal distance, add predecessor's count.
Discuss handling of zero-weight edges, disconnected nodes, and large counts. Analyze time and space complexity: O(V+E) for BFS, O((V+E) log V) for Dijkstra.
Walk through a small example to verify correctness. Consider edge cases like multiple shortest paths, no path, or s == t.
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First, clarify the problem: counting all shortest paths from a source to all nodes or a target. Then, analyze BFS for unweighted graphs and Dijkstra for weighted graphs, deriving time and space complexities based on the data structures used. Compare and contrast the two approaches, noting when each is appropriate.
Pro tip: Mention that for Dijkstra, using a Fibonacci heap can improve time complexity to O(E + V log V), but in practice binary heaps are often preferred due to lower constant factors. Also, note that the space complexity for both is O(V + E) to store the graph and auxiliary arrays.
Confirm that the problem involves counting the number of shortest paths from a source to all other nodes (or a specific target) in a graph. Specify whether the graph is unweighted (BFS) or weighted with non-negative weights (Dijkstra).
For unweighted graphs, BFS explores nodes level by level. Time complexity is O(V + E) with adjacency list, and space complexity is O(V) for the queue and distance/path count arrays.
For weighted graphs, Dijkstra with a binary heap runs in O((V + E) log V) time. Space complexity is O(V + E) for the graph and O(V) for auxiliary arrays. Mention that using a Fibonacci heap gives O(E + V log V) time.
Highlight that BFS is a special case of Dijkstra for unweighted graphs, offering better time complexity. Discuss trade-offs: BFS is simpler and faster for unweighted graphs, while Dijkstra handles weights but with higher time complexity.
State the final time and space complexities clearly: BFS: O(V+E) time, O(V) space; Dijkstra (binary heap): O((V+E) log V) time, O(V+E) space. Note that path counting adds only constant overhead per node.
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Cycles tripped me up for a second because I panicked thinking they'd cause infinite loops, but BFS visited-node tracking handles that naturally.
First, clarify the problem context and the specific algorithm being used, then systematically address each edge case: no path, cycles, and any other relevant cases. Explain how your solution detects and handles these cases, emphasizing correctness and efficiency.
Pro tip: Demonstrate awareness of trade-offs: for example, in a DFS-based solution, cycles are handled via visited states, but recursion depth might be an issue; mention iterative alternatives or cycle detection techniques like coloring. Also, discuss how you would test these edge cases.
Restate the problem and specify the algorithm (e.g., BFS, DFS, Dijkstra) and data structures used. This sets the context for edge case handling.
Explain how your algorithm detects that no path exists (e.g., BFS/DFS exhausts reachable nodes without finding t) and what it returns (e.g., null, empty list, or a specific value).
Describe how your algorithm avoids infinite loops in cyclic graphs, such as using a visited set or marking nodes as processed.
Mention additional edge cases like s == t, disconnected graphs, self-loops, or negative weights (if applicable) and how they are handled.
Conclude by reiterating that the solution correctly handles these cases without compromising time/space complexity, and mention testing strategies.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.