Using the adjacency matrix representation of a graph, how long does it take (number of checks) to iterate over the vertices adjacent to a vertex v in a graph with V vertices and E edges?

Question

BlackTom AI · Accepted Answer

When using an adjacency matrix to examine the neighbors of a vertex v, you typically inspect the entire row corresponding to v to see which entries indicate an edge to another vertex. This means you perform a number of checks proportional to the number of vertices in the graph, which is V, regardless of how many edges actually connect to v.

Option a: a. V^2. This would be the cost of scanning the whole matrix in a dense uniform way, but time to list the neighbors of a single vertex specifically requires scanning only the row for that vertex, not the entire matrix. So V^2 is overestimating the needed work.

Option b: b. V. This matches the operation of checking each possible neighbor indicator in the v-th row of the matrix. You perform one check per potential neighbor, across V possible vertices, which aligns with the adjacency matrix approach.

Option c: c. E. While E is the total number of edges in the graph, listing neighbors from the v-th row of the matrix does not directly scale with E; it scales with V because you must inspect each possible neighbor slot in that row, regardless of how many edges exist from v.

Option d: d. degree(v). The degree of v counts how many neighbors v actually has, i.e., how many 1s in that row (or how many adjacent edges). However, to determine which of those neighbors are present, you still need to examine all V entries in the row to identify them, so the cost is not equal to degree(v) but rather V checks.

Using the adjacency matrix representation of a graph, how long does it take (number of checks) to iterate over the vertices adjacent to a vertex v in a graph with V vertices and E edges?单项选择题

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