310 Minimum Height Trees
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:
Given n = 4, edges = [[1, 0], [1, 2], [1, 3]]
0
|
1
/ \
2 3
return [1]
Example 2:
Given n = 6, edges = [[0, 3], [1, 3], [2, 3], [4, 3], [5, 4]]
0 1 2
\ | /
3
|
4
|
5
return [3, 4]
Note:
(1) 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.”
(2) The height of a rooted tree is the number of edges on the longest downward path between the root and a leaf.
The Idea: For every node in the tree, find the height of that node. Store these into a dictionary. Sort by height and return the smallest homogenious group. The program below is a bit unconventional using this idea because I was initially trying to find an optimal solution.
Complexity: O(n^2) time and O(n) space
import queue
import operator
class Solution:
def findMinHeightTrees(self, n, edges):
"""
:type n: int
:type edges: List[List[int]]
:rtype: List[int]
"""
if not n:
return []
g = {i:set() for i in range(0, n)}
for s, e in edges:
g[s].add(e)
g[e].add(s)
g_heights = {}
def BFS_heights(root):
visited = set()
q = queue.Queue()
q.put((root, 0))
while not q.empty():
node, height = q.get()
g_heights[node] = height
visited.add(node)
for neighbor in g[node]:
if neighbor not in visited:
q.put((neighbor, height + 1))
leafs = []
def id_leafs(root, visited):
visited.add(root)
if len(g[root]) == 1:
leafs.append(root)
for neighbor in g[root]:
if neighbor not in visited:
id_leafs(neighbor, visited)
id_leafs(0, set())
MHTs = {node:float('-inf') for node in range(0, n)}
for node in range(0, n):
BFS_heights(node)
for leaf in leafs:
MHTs[node] = max(MHTs[node], abs(g_heights[leaf] - g_heights[node]))