All Paths From Source to Target
Given a DAG with nodes 0 to n-1, find all paths from node 0 to node n-1.
Input: adjacency list, one line per node (comma-separated neighbors, or 'none'). Output: one path per line.
Example:
1,2 3 3 none
0 1 3 0 2 3
- The input represents a Directed Acyclic Graph (DAG) as an adjacency list, where each line corresponds to a node and its neighbors.
- The given input translates to the following graph structure: node 0 has neighbors 1 and 2, node 1 has a neighbor 3, node 2 has a neighbor 3, and node 3 has no neighbors.
- We perform a depth-first search (DFS) or similar traversal from node 0 to find all possible paths to node 3, which is the last node (n-1).
- The DFS yields two paths: 0 → 1 → 3 and 0 → 2 → 3, corresponding to the output lines.
- The final output lists each path on a separate line, resulting in
0 1 3and0 2 3.
Constraints:
- 2 <= n <= 15
- 0 <= edges <= n * (n-1) / 2
Background Knowledge
The problem involves finding all paths from a source node to a target node in a Directed Acyclic Graph (DAG). A DAG is a type of graph where edges have direction and there are no cycles, meaning it's not possible to start at a node and follow edges to return to the same node. This is in contrast to other types of graphs, such as undirected graphs or cyclic graphs. Understanding the properties of DAGs is crucial, as they guarantee that there are no infinite loops, making it possible to traverse the graph using depth-first search (DFS) or breadth-first search (BFS) algorithms.
In the context of this problem, the graph is represented as an adjacency list, where each node is associated with a list of its neighboring nodes. This representation is useful for efficiently traversing the graph, as it allows for quick lookup of neighboring nodes. The problem requires finding all possible paths from the source node (node 0) to the target node (node n-1), which involves exploring all possible branches of the graph.
To solve this problem, it's essential to understand the concepts of graph traversal, recursion, and backtracking. Graph traversal refers to the process of visiting nodes in a graph, while recursion and backtracking are techniques used to explore all possible paths in the graph. Recursion involves breaking down the problem into smaller sub-problems, while backtracking involves exploring all possible solutions and reverting to a previous state when a dead end is reached.
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