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+16 votes

Let $G$ be a graph with $n$ vertices and $m$ edges.What is the tightest upper bound on the running time of Depth First Search on $G$, when $G$ is represented as an adjacency matrix?

- $\Theta(n)$
- $\Theta(n+m)$
- $\Theta(n^2)$
- $\Theta(m^2)$

+8 votes

Best answer

Depth First Search of a graph takes $O(m+n)$ time when the graph is represented using adjacency list. In adjacency matrix representation, graph is represented as an $n * n$ matrix. To do DFS, for every vertex, we traverse the row corresponding to that vertex to find all adjacent vertices (In adjacency list representation we traverse only the adjacent vertices of the vertex). Therefore time complexity becomes $O(n^2)$.

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