310. Minimum Height Trees

内容：

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.

思路：

这个思路实际上是一个 BFS 思路。和常见的从根节点进行 BFS 不同，这里从叶子节点开始进行 BFS。

所有入度（即相连边数）为 1 的节点即是叶子节点。找高度最小的节点，即找离所有叶子节点最远的节点，也即找最中心的节点。

找最中心的节点的思路很简单：

• 每次去掉当前图的所有叶子节点，重复此操作直到只剩下最后的根。

根据这个思路可以回答题目中的 [ hint : How many MHTs can a graph have at most? ]，只能有一个或者两个最小高度树树根。证明省略。

public class Solution {  
    private int[] radius;  
    private int[][] height;  
    private int[][] htnode;  
    private List<Integer>[] graph;  
    private void find(int prev, int node) {  
        for(int next: graph[node]) {  
            if (next == prev) continue;  
            find(node, next);  
            if (height[next][0] + 1 > height[node][0]) {  
                height[node][0] = height[next][0] + 1;  
                htnode[node][0] = next;  
            }  
        }  
        for(int next: graph[node]) {  
            if (next == prev) continue;  
            if (next == htnode[node][0]) continue;  
            if (height[next][0] + 1 > height[node][1]) {  
                height[node][1] = height[next][0] + 1;  
                htnode[node][1] = next;  
            }  
        }  
    }  
      
    private void find(int prev, int node, int sum) {  
        radius[node] = Math.max(sum, height[node][0]);  
          
        for(int next: graph[node]) {  
            if (next == prev) continue;  
            if (next == htnode[node][0]) {  
                find(node, next, Math.max(sum+1, height[node][1]+1));  
            } else {  
                find(node, next, Math.max(sum+1, height[node][0]+1));  
            }  
        }  
    }  
  
    public List<Integer> findMinHeightTrees(int n, int[][] edges) {  
        radius = new int[n];  
        height = new int[n][2];  
        htnode = new int[n][2];  
        graph = new List[n];  
        for(int i=0; i<n; i++) graph[i] = new ArrayList<>();  
        for(int[] edge: edges) {  
            graph[edge[0]].add(edge[1]);  
            graph[edge[1]].add(edge[0]);  
        }  
          
        find(-1, 0);  
        find(-1, 0, 0);  
          
        int min = Integer.MAX_VALUE;  
        for(int r: radius) min = Math.min(min, r);  
        List<Integer> roots = new ArrayList<>();  
        for(int i=0; i<radius.length; i++) {  
            if (radius[i] == min) roots.add(i);  
        }  
        return roots;  
    }  
}