GraphADT_AMtrx_2

Time Complexity – Edge Operations
Suppose your graph is stored as an adjacency matrix of size V × V. You're asked to implement the function are_adjacent(u, v) that checks if there's an edge from vertex u to vertex v.
Assume the matrix is implemented as a standard 2D array
What is the time complexity of this operation?

Question

GraphADT_AMtrx_2

Time Complexity – Edge Operations
Suppose your graph is stored as an adjacency matrix of size V × V. You're asked to implement the function are_adjacent(u, v) that checks if there's an edge from vertex u to vertex v.
Assume the matrix is implemented as a standard 2D array
What is the time complexity of this operation?

BlackTom AI · Accepted Answer

Let’s examine what the operation are_adjacent(u, v) would require when using an adjacency matrix of size V × V.
Option O(V): This would imply a dependence on the number of vertices, such as scanning a list of possible neighbors or iterating over a row or column. With a matrix, we access a specific entry directly, so this is not the right characterization of the cost.
Option O(log V): A logarithmic time bound would come from data structures that support binary search or hierarchical pruning. An array-based matrix lookup does not involve any searching or hierarchical steps, so this option does not reflect the constant-time access pattern of a 2D array.
Option O(1): This is the hallmark of a matrix-based adjacency check. With a direct 2D array, you can index into matrix[u][v] in constant time, assuming standard memory access costs, because you jump straight to the exact entry representing the presence or absence of an edge.
Option O(E): This would suggest touching every edge to determine adjacency, which is far more work than needed for a single pair (u, v). In a matrix, adjacency is determined by a single entry, not by traversing all edges.
In summary, the adjacency check in a V × V adjacency matrix corresponds to a direct, single-entry access, which is constant time, while the other scales (V, log V, E) are not representative of a direct matrix lookup.

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