[LeetCode] 310. Minimum Height Trees 解题思路

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]]

return [1]

Example 2:

Given n = 6, edges = [[0, 3], [1, 3], [2, 3], [4, 3], [5, 4]]

return [3, 4]

Hint:

How many MHTs can a graph have at most?

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.

问题：给定一个拥有树性质的无向图，图的每一个节点都可以视为一棵树的根节点。在所有可能的树中，找出高度最小的树，并返回他们的树根。

思路一：

求最小高度的树，实际上是一个求最优解题目。求最优解，我首先想到的是动态规划(DP)思路。这个题目也确实满足 DP 的两个主要性质：overlapping substructure & optimal substructure 。思路比较直观：

视某个节点为树根节点，求这棵树的树高。
对所有节点进行上面的求解，得到 n 个树高，其中高度最小的树的树根即为原题目的解。

对于1，如何求解节点 A 为树根的树高？

假设已知 A 的所有相邻节点分别为树根的各个子树的树高，那么 A根的树高等于已知的各个子树树高中的最大值加一。方程式表达如下，即状态转换方程：

height(A) = max(height(A.next0), height(A.next2),... height(A.nextk)) + 1

对于2，存在大量重复计算。可以借助表格，将已计算的树分支高度保存下来后续使用，避免重复计算。将当前节点以及其中一个相邻节点组合分支方向，求得该分支高度后存入 map<string, int> direc_height 中，其中 direc 有这两个节点组成作为 key ，height 表示高度。

若不使用表格优化，时间复杂度为 O(V*E)。使用表格，相当于进行了剪枝，会快不少。跑 LeetCode 的大集合 V = 1000, E =2000 ，测试耗时是 820ms，可惜没能过 submit 要求。

 private:
 
     // dirct consists of two node cur_next, height means the height of the subtree which souce node is the current node and walk forward on the cur_next direction
     map<string, int> direc_height;
 
 public:
 
     /**
      * calulate the height of a tree which parent node is p and current node is node
      *
      */
     int getHeight(gnode* node, gnode* p){
         int h = ;
         for (int i = ; i < node->neighbours.size(); i++){
             gnode* tmpn = node->neighbours[i];
             if (tmpn == p){
                 continue;
             }
 
             int tmph;
             string str = to_string(node->val) + "_" + to_string(tmpn->val);
             if(direc_height.count(str) != ){
                 tmph = direc_height[str];
             }else{
                 tmph = getHeight(tmpn, node);
                 direc_height[str] = tmph;
             }
 
             h = max(h, tmph);
         }
 
         return h + ;
     }
 
     vector<int> findMinHeightTrees(int n, vector<pair<int, int>>& edges) {
 
         map<int, gnode*> val_node;
         for(int i = ; i < n ; i++){
             gnode* node = new gnode(i);
             val_node[i] = node;
         }
 
         for (int i = ; i < edges.size(); i++){
             pair<int, int> pp = edges[i];
             val_node[pp.first]->neighbours.push_back(val_node[pp.second]);
             val_node[pp.second]->neighbours.push_back(val_node[pp.first]);
         }
 
         int minh = INT_MAX;
 
         map<int, int> node_height;
 
         map<int, gnode*>::iterator m_iter;
         for(m_iter = val_node.begin(); m_iter != val_node.end(); m_iter++){
             int h = getHeight(m_iter->second, NULL);
             node_height[m_iter->first] = h;
             minh = min(minh, h);
         }
 
         vector<int> res;
         map<int, int>::iterator mii_iter;
         for(mii_iter = node_height.begin(); mii_iter != node_height.end(); mii_iter++){
             if(mii_iter->second == minh){
                 res.push_back(mii_iter->first);
             }
         }
 
         return res;
     }

思路二：

除了 DP 方案，没有想到其他思路，在网上借鉴了其他了的想法，理解后实现通过。

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。

 class TNode{
 public:
     int val;
     unordered_set<TNode*> neighbours;
     TNode(int val){
         this->val = val;
     }
 };
 
     vector<int> findMinHeightTrees(int n, vector<pair<int, int>>& edges) {
         map<int, TNode*> val_node;
 
         for (int i =  ; i < n ; i++){
             TNode* tmp = new TNode(i);
             val_node[i] = tmp;
         }
 
         for (int i =  ; i < edges.size(); i++){
             pair<int, int> pp = edges[i];
             val_node[pp.first]->neighbours.insert(val_node[pp.second]);
             val_node[pp.second]->neighbours.insert(val_node[pp.first]);
         }
 
         map<int, TNode*>::iterator m_iter;
 
         while(val_node.size() > ){
 
             // obtain all leaves in current graph;
             list<TNode*> listg;
             for ( m_iter = val_node.begin(); m_iter != val_node.end(); m_iter++){
                 if(m_iter->second->neighbours.size() == ){
                     listg.push_back(m_iter->second);
                 }
             }
 
             // remove all leaves
             list<TNode*>::iterator l_iter;
             for(l_iter = listg.begin(); l_iter != listg.end(); l_iter++){
                 TNode* p = (*(*l_iter)->neighbours.begin());
                 p->neighbours.erase(*l_iter);
                 (*l_iter)->neighbours.erase(p);
 
                 val_node.erase((*l_iter)->val);
             }
         }
 
         vector<int> res;
         for ( m_iter = val_node.begin(); m_iter != val_node.end(); m_iter++){
             res.push_back(m_iter->first);
 
         }
 
         return res;
     }

参考资料：

C++ Solution. O(n)-Time, O(n)-Space, LeetCode OJ