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I knew it was a graph problem right away and jumped to DFS, which was the right instinct.
Model the tickets as a directed graph and use Hierholzer's algorithm to find an Eulerian path starting from JFK. To ensure the lexicographically smallest itinerary, sort each airport's destinations in reverse order and use a stack-based DFS, building the itinerary in reverse.
Pro tip: Clarify that the problem guarantees a valid itinerary exists; if not, discuss handling edge cases. Also, mention that using a min-heap for destinations can achieve the same lexicographic order, but sorting once is more efficient.
Recognize that the tickets form a directed graph where each airport is a node and each ticket is a directed edge. The goal is to find an Eulerian path starting from JFK that uses all edges exactly once, and among all such paths, return the lexicographically smallest one.
Use Hierholzer's algorithm to find an Eulerian path. This algorithm is efficient (O(E log E) due to sorting) and naturally handles the requirement to use all edges.
Sort the destinations for each airport in reverse lexicographic order (or use a min-heap) so that when we explore, we visit the smallest lexical destination first. This ensures the final itinerary is lexicographically smallest.
Perform an iterative DFS using a stack, starting from JFK. At each step, pop the next destination from the current airport's list (which is sorted in reverse) and push it onto the stack. When no destinations remain, add the airport to the itinerary. Finally, reverse the itinerary to get the correct order.
Discuss time complexity: O(E log E) due to sorting, where E is the number of tickets. Space complexity: O(E). Mention edge cases like multiple tickets between same airports, and the guarantee of a valid itinerary.
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