Village (Minimum)
Contest: Baltic Olympiad in Informatics 2020, Day 2 (IOI, Unofficial Mirror Contest, Unrated)
Difficulty rating: 2100
Tags: *special, dp, greedy, trees

This problem is split into two tasks. In this task, you are required to find the minimum possible answer. In the task Village (Maximum) you are required to find the maximum possible answer. Each task is worth $$$50$$$ points.

There are $$$N$$$ houses in a certain village. A single villager lives in each of the houses. The houses are connected by roads. Each road connects two houses and is exactly $$$1$$$ kilometer long. From each house it is possible to reach any other using one or several consecutive roads. In total there are $$$N-1$$$ roads in the village.

One day all villagers decided to move to different houses — that is, after moving each house should again have a single villager living in it, but no villager should be living in the same house as before. We would like to know the smallest possible total length in kilometers of the shortest paths between the old and the new houses for all villagers.

Example village with seven houses

For example, if there are seven houses connected by roads as shown on the figure, the smallest total length is $$$8$$$ km (this can be achieved by moving $$$1 \to 6$$$, $$$2 \to 4$$$, $$$3 \to 1$$$, $$$4 \to 2$$$, $$$5 \to 7$$$, $$$6 \to 3$$$, $$$7 \to 5$$$).

Write a program that finds the smallest total length of the shortest paths in kilometers and an example assignment of the new houses to the villagers.

Input format:
The first line contains an integer $$$N$$$ ($$$1 < N \le 10^5$$$). Houses are numbered by consecutive integers $$$1, 2, \ldots, N$$$.

Then $$$N-1$$$ lines follow that describe the roads. Each line contains two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le N$$$, $$$a \neq b$$$) denoting that there is a road connecting houses $$$a$$$ and $$$b$$$.

Output format:
In the first line output the smallest total length of the shortest paths in kilometers.

In the second line describe one valid assignment of the new houses with the smallest total length: $$$N$$$ space-separated distinct integers $$$v_1, v_2, \ldots, v_N$$$. For each $$$i$$$, $$$v_i$$$ is the house number where the villager from the house $$$i$$$ should move ($$$v_i \neq i$$$). If there are several valid assignments, output any of those.

Constraints:
Time limit: 0.75 s
Memory limit: 256.0 MB

Note:
Scoring:
Subtasks:

1. (6 points) $$$N \leq 10$$$
2. (19 points) $$$N \leq 1\,000$$$
3. (25 points) No further constraints

Example input:
4
1 2
2 3
3 4

Example output:
4
2 1 4 3

Example input:
7
4 2
5 7
3 4
6 3
1 3
4 5

Example output:
8
3 4 6 2 7 1 5

Editorial:
Subtask 1
We can try each permutation of villagers, claculate the total distance and find the answer.
Complexity: $$$O(N!)$$$.
Subtask 2
See the Subtask 3 solution; here we can store the tree in slower data structures and work with the tree slower.
Complexity: $$$O(N^2)$$$.
Subtask 3
Each villager needs to move to a new place so we can process the tree from leaves (greedy): if a villager that currently is in a leaf has been there from the very beginning (still needs to move to other place), change places with its (only) neighbour node villager, add $$$2$$$ to the answer and mark the leaf node as processed (or remove it from the tree so that new nodes can become leaves). If the last villager is not moved then it can change with any of it neighbors.
Complexity: $$$O(N)$$$.