Output lexicographically largest DFS traversal
Problem Description
You are given an undirected graph with n nodes, labeled 1 to n, and m edges.
Your task is to find and output a node visitation sequence of length n that is produced by a Depth-First Search (DFS) traversal, visiting every node exactly once. The sequence must be the lexicographically largest among all possible valid sequences.
The rules for generating a valid DFS visitation sequence are as follows:
- Starting Node: You may choose any node as the starting point for the initial DFS traversal.
- Multiple Components: If, after a DFS traversal completes (meaning an entire connected component has been explored), there are still unvisited nodes remaining in the graph, you may initiate a new DFS from any unvisited node. This process continues until all nodes have been visited.
- Neighbor Exploration Order: Within any active DFS traversal, when a node has multiple unvisited neighbors, you have the freedom to choose the order in which these neighbors are explored.
Your final output should be a sequence of n integers representing the lexicographically largest node visitation order.
Input
n: An integer representing the number of nodes (labeled1ton).edge list: A list of pairs of integers, where each pair(u, v)denotes an edge between nodeuand nodev.
Output
- A sequence of
nintegers, representing the lexicographically largest DFS traversal.
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Core dilemma
The central engineering conflict is making a series of locally greedy choices (which node to visit next, which neighbor to explore) that collectively guarantee a globally optimal solution: the lexicographically largest DFS traversal sequence. This involves strategic decision-making at multiple levels: selecting the initial DFS start node, ordering neighbors within a DFS, and choosing subsequent DFS start nodes for disconnected components.
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Output lexicographically largest DFS traversal
Problem Description
You are given an undirected graph with n nodes, labeled 1 to n, and m edges.
Your task is to find and output a node visitation sequence of length n that is produced by a Depth-First Search (DFS) traversal, visiting every node exactly once. The sequence must be the lexicographically largest among all possible valid sequences.
The rules for generating a valid DFS visitation sequence are as follows:
- Starting Node: You may choose any node as the starting point for the initial DFS traversal.
- Multiple Components: If, after a DFS traversal completes (meaning an entire connected component has been explored), there are still unvisited nodes remaining in the graph, you may initiate a new DFS from any unvisited node. This process continues until all nodes have been visited.
- Neighbor Exploration Order: Within any active DFS traversal, when a node has multiple unvisited neighbors, you have the freedom to choose the order in which these neighbors are explored.
Your final output should be a sequence of n integers representing the lexicographically largest node visitation order.
Input
n: An integer representing the number of nodes (labeled1ton).edge list: A list of pairs of integers, where each pair(u, v)denotes an edge between nodeuand nodev.
Output
- A sequence of
nintegers, representing the lexicographically largest DFS traversal.