# 332 Reconstruct Itinerary

Given a list of airline tickets represented by pairs of departure and arrival airports`[from, to]`, reconstruct the itinerary in order. All of the tickets belong to a man who departs from`JFK`. Thus, the itinerary must begin with`JFK`.

**Note:**

1. If there are multiple valid itineraries, you should return the itinerary that has the smallest lexical order when read as a single string. For example, the itinerary `["JFK", "LGA"]` has a smaller lexical order than `["JFK", "LGB"]`.
2. All airports are represented by three capital letters (IATA code).
3. You may assume all tickets form at least one valid itinerary.

**Example 1:**\
`tickets`=`[["MUC", "LHR"], ["JFK", "MUC"], ["SFO", "SJC"], ["LHR", "SFO"]]`\
Return`["JFK", "MUC", "LHR", "SFO", "SJC"]`.

**Example 2:**\
`tickets`=`[["JFK","SFO"],["JFK","ATL"],["SFO","ATL"],["ATL","JFK"],["ATL","SFO"]]`\
Return`["JFK","ATL","JFK","SFO","ATL","SFO"]`.\
Another possible reconstruction is`["JFK","SFO","ATL","JFK","ATL","SFO"]`. But it is larger in lexical order.

**The Idea:** This is a eularian path/cycle problem. Our goal is to visit every edge of the graph once, and return to our original position (in our case, which is '`JFK`'). The algorithm that I have used to implement this idea is called *Hierholzer's Algorithm*. More details and discussion here: <https://maksimdan.gitbooks.io/ecs122a-algorithm-design-lecture-notes/content/eularian-cycle.html>

**Complexity:** O(|E|) time and O(|E|) space

```python
from collections import defaultdict


class Solution:
    def findItinerary(self, tickets):
        """
        :type tickets: List[List[str]]
        :rtype: List[str]
        """

        if not tickets or not tickets[0]:
            return []

        # build representative graph
        g = defaultdict(list)
        for _to, _from in tickets:
            g[_to].append(_from)

        # now sort to create smallest
        # lexicographical itinerary
        for _to, _ in g.items():
            g[_to].sort()

        # maintain an index where the eularian path
        # left off
        left_off = {}
        for _to, _from in tickets:
            left_off[_to] = 0
            left_off[_from] = 0

        itinerary = []
        s = ['JFK']

        # now find the eularian path
        while s:
            top = s[-1]
            g_index = left_off[top]
            g_list = g[top]
            if g_index < len(g_list):
                s.append(g_list[g_index])
                left_off[top] = g_index + 1
            else:
                itinerary.append(top)
                s.pop()

        return list(reversed(itinerary))
```