Minimum height trees¶
Time: O(N); Space: O(N); medium
For a undirected graph with tree characteristics, we can choose any node as the root. The result graph is then a rooted tree. Among all possible rooted trees, those with minimum height are called Minimum Height Trees (MHTs). Given such a graph, write a function to find all the MHTs and return a list of their root labels.
Format The graph contains N nodes which are labeled from 0 to n - 1. You will be given the number N and a list of undirected edges (each edge is a pair of labels).
You can assume that no duplicate edges will appear in edges. Since all edges are undirected, [0, 1] is the same as [1, 0] and thus will not appear together in edges.
Example 1:
Input: n = 4, edges = [[1, 0], [1, 2], [1, 3]]
0
|
1
/ \
2 3
Output: [1]
Example 2:
Input: n = 6, edges = [[0, 3], [1, 3], [2, 3], [4, 3], [5, 4]]
0 1 2
\ | /
3
|
4
|
5
Output: [3, 4]
Hint:
How many MHTs can a graph have at most?
Notes:
According to the definition of tree on Wikipedia: “a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any connected graph without simple cycles is a tree.”
The height of a rooted tree is the number of edges on the longest downward path between the root and a leaf.
[1]:
from collections import defaultdict
class Solution1(object):
def findMinHeightTrees(self, n, edges):
"""
:type n: int
:type edges: List[List[int]]
:rtype: List[int]
"""
if n == 1:
return [0]
neighbors = defaultdict(set)
for u, v in edges:
neighbors[u].add(v)
neighbors[v].add(u)
pre_level, unvisited = [], set()
for i in range(n):
if len(neighbors[i]) == 1: # A leaf.
pre_level.append(i)
unvisited.add(i)
# A graph can have 2 MHTs at most.
# BFS from the leaves until the number
# of the unvisited nodes is less than 3.
while len(unvisited) > 2:
cur_level = []
for u in pre_level:
unvisited.remove(u)
for v in neighbors[u]:
if v in unvisited:
neighbors[v].remove(u)
if len(neighbors[v]) == 1:
cur_level.append(v)
pre_level = cur_level
return list(unvisited)
[2]:
s = Solution1()
n = 4
edges = [[1, 0], [1, 2], [1, 3]]
assert s.findMinHeightTrees(n, edges) == [1]
n = 6
edges = [[0, 3], [1, 3], [2, 3], [4, 3], [5, 4]]
assert s.findMinHeightTrees(n, edges) == [3, 4]