Tracing a Path with Depth-First Search (DFS)
Problem Setup
Scenario:Find a path from the start node S to the goal node G by following the search order produced by the given graph.
Given:A directed graph with S as the start node and G as the goal node. The graph contains multiple branches from S, including routes through A, B, C, D, and E.
Task:Trace the search from S to G using the stack order shown in the solution, record the node selected at each step, and determine the resulting path and number of edges from S to G.
Need a quick refresher? Review Depth-First Search theory
Search Graph
Arrows show the permitted direction of travel. Several branches can reach G, but DFS follows the stack order rather than comparing which route has the fewest edges.
Step-by-Step LIFO Stack Resolution
| Step | Open List (Stack) | Evaluating Node | Closed List (Visited) |
|---|---|---|---|
1 | [ S ] | S | [ ] |
2 | [ B, A ] | A | [ S ] |
3 | [ B, D, C ] | C | [ S, A ] |
4 | [ B, D, E ] | E | [ S, A, C ] |
5 | [ B, D, G ] | G | [ S, A, C, E ] |
6 | [ B, D ] | Goal Reached | [ S, A, C, E, G ] |
Final Answer
The DFS traversal reaches the goal along the path produced by the stack-based search:
Path reached:S → A → C → E → G
Total path steps:4
DFS reaches G through the deeper branch starting at A, leaving B and D unexplored when the search stops. Although this path contains 4 edges, the graph also has a shorter route through B with only 2 edges, showing why DFS does not guarantee the shortest path.
Exam Takeaway
- Stack order determines which branch is explored first: DFS processes the most recently pushed node. When neighbors must be explored alphabetically, push them in reverse alphabetical order so the intended first neighbor is on top.
- A node should not be discovered repeatedly: Track discovered nodes so that cycles and converging paths do not cause the same node to be added to the stack multiple times. In this implementation, nodes are marked when pushed.
- Reaching the goal does not mean finding the shortest path: DFS may reach a goal through a deeper branch while shorter alternatives remain unexplored. Reconstruct the returned path using parent links and count its edges rather than assuming it is optimal.