HDU 4822 Tri-war（LCA树上倍增）（2013 Asia Regional Changchun）

题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=4822

Problem Description

Three countries, Red, Yellow, and Blue are in war. The map of battlefield is a tree, which means that there are N nodes and (N – 1) edges that connect all the nodes. Each country has a base station located in one node. All three countries will not place their station in the same node. And each country will start from its base station to occupy other nodes. For each node, country A will occupy it iff other two country's base stations have larger distances to that node compared to country A. Note that each edge is of the same length.

Given three country's base station, you task is to calculate the number of nodes each country occupies (the base station is counted).

 

Input

The input starts with a single integer T (1 ≤ T ≤ 10), the number of test cases.
Each test cases starts with a single integer N (3 ≤ N ≤ 10 ^ 5), which means there are N nodes in the tree.
Then N - 1 lines follow, each containing two integers u and v (1 ≤ u, v ≤ N, u ≠ v), which means that there is an edge between node u and node v.
Then a single integer M (1 ≤ M ≤ 10 ^ 5) follows, indicating the number of queries.
Each the next M lines contains a query of three integers a, b, c (1 ≤ a, b, c ≤ N, a, b, c are distinct), which indicates the base stations of the three countries respectively.

 

Output

For each query, you should output three integers in a single line, separated by white spaces, indicating the number of nodes that each country occupies. Note that the order is the same as the country's base station input.

 

题目大意：给一棵n个点的树，每次询问有三个点，问离每个点比另外两个点近的点有多少个。

思路：贴一下官方题解：

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本题抽象的题意是给出一棵树，有许多询问，每次询问，给出3个点，问有多少个点，到这三个点的最短距离是递增的。
首先考虑两个点的简单情况，因为是树，有特殊性，任意两点间只有唯一的一条路，找到路的中点，就可以把树分成两部分，其中一部分的点是合法解。
回到本题，问题就变成了两个子树的交集。这个考虑一个子树是否是另一子树的子树即可。用dfs序列来判断即可。
时间复杂度是O(nlogn)

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即考虑每一个点，求这个点与另两个点劈开成的两颗子树（或者是整棵树减去一棵子树），这里要用到树上倍增求第k祖先。然后求两个子树的交，这个分类讨论一下即可。

PS：用G++交居然栈溢出了。只好换C++开栈了。

 

 

代码（2781MS）：

  1 #ifdef ONLINE_JUDGE

  2 #pragma comment(linker, "/STACK:1024000000,1024000000")

  3 #endif // ONLINE_JUDGE

  4 

  5 #include <cstdio>

  6 #include <cstring>

  7 #include <algorithm>

  8 #include <iostream>

  9 using namespace std;

 10 

 11 const int MAXV = 100010;

 12 const int MAXE = 200010;

 13 const int MAX_LOG = 20;

 14 

 15 int head[MAXV], ecnt;

 16 int to[MAXE], next[MAXE];

 17 int n, m, T;

 18 

 19 void init() {

 20     memset(head + 1, -1, n * sizeof(int));

 21     ecnt = 0;

 22 }

 23 

 24 void add_edge(int u, int v) {

 25     to[ecnt] = v; next[ecnt] = head[u]; head[u] = ecnt++;

 26     to[ecnt] = u; next[ecnt] = head[v]; head[v] = ecnt++;

 27 }

 28 

 29 int fa[MAX_LOG][MAXV];

 30 int size[MAXV], dep[MAXV];

 31 

 32 void dfs(int u, int f, int depth) {

 33     fa[0][u] = f; size[u] = 1; dep[u] = depth;

 34     for(int p = head[u]; ~p; p = next[p]) {

 35         int v = to[p];

 36         if(v == f) continue;

 37         dfs(v, u, depth + 1);

 38         size[u] += size[v];

 39     }

 40 }

 41 

 42 void initfa() {

 43     dfs(1, -1, 0);

 44     for(int k = 0; k < MAX_LOG - 1; ++k) {

 45         for(int u = 1; u <= n; ++u) {

 46             if(fa[k][u] == -1) fa[k + 1][u] = 1;

 47             else fa[k + 1][u] = fa[k][fa[k][u]];

 48         }

 49     }

 50 }

 51 

 52 int upslope(int u, int p) {

 53     for(int k = 0; k < MAX_LOG; ++k) {

 54         if((p >> k) & 1) u = fa[k][u];

 55     }

 56     return u;

 57 }

 58 

 59 int lca(int u, int v) {

 60     if(dep[u] < dep[v]) swap(u, v);

 61     u = upslope(u, dep[u] - dep[v]);

 62     if(u == v) return u;

 63     for(int k = MAX_LOG - 1; k >= 0; --k) {

 64         if(fa[k][u] != fa[k][v])

 65             u = fa[k][u], v = fa[k][v];

 66     }

 67     return fa[0][u];

 68 }

 69 

 70 struct Node {

 71     int type, r;

 72     Node(int type, int r): type(type), r(r) {}

 73 };

 74 

 75 Node get_middle(int a, int b, int ab) {

 76     int len = dep[a] + dep[b] - 2 * dep[ab];

 77     if(dep[a] >= dep[b]) {

 78         return Node(1, upslope(a, (len - 1) / 2));

 79     } else {

 80         return Node(2, upslope(b, len / 2));

 81     }

 82 }

 83 

 84 int calc(int a, int b, int c, int ab, int ac) {

 85     Node bn = get_middle(a, b, ab), cn = get_middle(a, c, ac);

 86     if(bn.type == 1 && cn.type == 1) {

 87         if(dep[bn.r] < dep[cn.r]) swap(bn, cn);

 88         if(lca(bn.r, cn.r) == cn.r) return size[bn.r];

 89         else return 0;

 90     } else if(bn.type == 2 && cn.type == 2) {

 91         if(dep[bn.r] < dep[cn.r]) swap(bn, cn);

 92         if(lca(bn.r, cn.r) == cn.r) return n - size[cn.r];

 93         else return n - size[bn.r] - size[cn.r];

 94     } else {

 95         if(bn.type == 2) swap(bn, cn);

 96         int t = lca(bn.r, cn.r);

 97         if(t == cn.r) return n - size[cn.r];

 98         if(t == bn.r) return size[bn.r] - size[cn.r];

 99         return size[bn.r];

100     }

101 }

102 

103 int main() {

104     scanf("%d", &T);

105     while(T--) {

106         scanf("%d", &n);

107         init();

108         for(int i = 1, u, v; i < n; ++i) {

109             scanf("%d%d", &u, &v);

110             add_edge(u, v);

111         }

112         initfa();

113         scanf("%d", &m);

114         for(int i = 0, a, b, c; i < m; ++i) {

115             scanf("%d%d%d", &a, &b, &c);

116             int ab = lca(a, b), ac = lca(a, c), bc = lca(b, c);

117             printf("%d %d %d\n", calc(a, b, c, ab, ac), calc(b, a, c, ab, bc), calc(c, a, b, ac, bc));

118         }

119     }

120 }

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