Given a graph-G that has |V| vertices and |E| edges, that is:Connected. Directed. Weighted What is the worst case time complexity to run Breadth-First Search (BFS), if G is implemented using adjacency matrix?

Question

BlackTom AI · Accepted Answer

Using an adjacency matrix, scanning the neighbor list of any vertex requires checking an entire row of length |V|, so per-vertex neighbor scanning costs Θ(|V|).

1) Relevant knowledge points and concepts
- BFS explores each vertex once and examines all its outgoing adjacency entries.
- An adjacency matrix for |V| vertices stores |V|^2 entries; checking whether an edge exists to each candidate neighbor is O(1) per entry.
- Whether the graph is directed or weighted does not change the per-entry check cost for an adjacency matrix representation.

2) Step-by-step reasoning
- BFS will dequeue each vertex at most once (Θ(|V|) dequeues).
- For each dequeued vertex, with an adjacency matrix you must inspect every column in that vertex’s row to find neighbors; there are |V| columns.
- Thus the total number of adjacency checks = |V| vertices × |V| checks per vertex = |V|^2 checks.

3) Intermediate calculation steps
- Per-vertex neighbor checks: Θ(|V|)
- Number of vertices processed: Θ(|V|)
- Total adjacency checks: |V| × |V| = |V|^2
- Each check and any constant-time queue operations give overall Θ(|V|^2).

4) Why the correct answer is correct
- The adjacency matrix forces examination of all |V| possible neighbor slots for each vertex regardless of how many actual edges exist, so BFS runtime is Θ(|V|^2) in the worst (and typical) case for matrix representation.

5) Why other options are incorrect
- Θ(1), Θ(|V|): far too small; they ignore scanning neighbors.
- Θ(|E|), Θ(|V|+|E|): these are the runtimes for adjacency-list implementations; with a matrix you don’t gain by having few edges.
- Θ(|E|^2), Θ(|V| log |V|), Θ(|E| log |E|): unrelated complexities that do not reflect the matrix neighbor-scan pattern.

6) Conclusion
- Therefore the worst-case time complexity for BFS on a graph implemented with an adjacency matrix is Θ(|V|^2), which matches the provided answer.

Given a graph-G that has |V| vertices and |E| edges, that is:Connected. Directed. Weighted What is the worst case time complexity to run Breadth-First Search (BFS), if G is implemented using adjacency matrix?单项选择题

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