SPOJ-FTOUR2 Free tour II
After the success of 2nd anniversary (take a look at problem FTOUR for more details), this 3rd year, Travel Agent SPOJ goes on with another discount tour.
The tour will be held on ICPC island, a miraculous one on the Pacific Ocean. We list N places (indexed from 1 to N) where the visitors can have a trip. Each road connecting them has an interest value, and this value can be negative (if there is nothing interesting to view there). Simply, these N places along with the roads connecting them form a tree structure. We will choose two places as the departure and destination of the tour.
Since September is the festival season of local inhabitants, some places are extremely crowded (we call them crowded places). Therefore, the organizer of the excursion hopes the tour will visit at most K crowded places (too tiring to visit many of them) and of course, the total number of interesting value should be maximum.
Briefly, you are given a map of N places, an integer K, and M id numbers of crowded place. Please help us to find the optimal tour. Note that we can visit each place only once (or our customers easily feel bored), also the departure and destination places don't need to be different.
Input
There is exactly one case. First one line, containing 3 integers N K M, with 1 <= N <= 200000, 0 <= K <= M, 0 <= M <= N.
Next M lines, each line includes an id number of a crowded place.
The last (N - 1) lines describe (N - 1) two-way roads connected N places, form a b i, with a, b is the id of 2 places, and i is its interest value (-10000 <= i <= 10000).
Output
Only one number, the maximum total interest value we can obtain.
Example
Input:
8 2 3
3
5
7
1 3 1
2 3 10
3 4 -2
4 5 -1
5 7 6
5 6 5
4 8 3
Output:
12
Explanation
We choose 2 and 6 as the departure and destination place, so the tour will be 2 -> 3 -> 4 -> 5 -> 6, total interest value = 10 + (-2) + (-1) + 5 = 12
* Added some unofficial cases
题目大意
n个节点带权树，每个点有黑白两种颜色。询问一条路径，使得黑色点数小于等于k且路径长度最大。\(n\leq 200000\)
题解
点分，对于每一层：
\(f[i]\)表示颜色数小于等于i的最大路径，\(num[i]\)表示点i到根的黑点个数（不包括根，如果根是黑色，k-1即可），当遇到节点\(u\)时，用\(f[k-num[i]]+dis[i]\)更新。
下面考虑如何维护\(f[i]\)，如果每一次都从\(num[x]\)搞到\(k\)，肯定爆炸，不如考虑每棵子树记录一个\(ma[i]\)，表示子树\(num[x]\)值的最大值，然后从\(1\)更新到\(ma[i]\)，然后做点\(u\)的时候，如果\(num[u] + ma[i] <= k\)，拿\(f[ma[i]] + dis[u]\)更新一下答案就行了。
不难发现，拿来额外更新的值应该取所有已经遍历过的子树的\(ma[i]\)值的最大值。
但是这样做最坏情况还是\(n^2\)的
排个序，从小到大。
变成了各个子树\(ma[i]\)相加，一定是小于n的。
复杂度变为\(O(n)\),排序可以O(n)做但我偷懒用了\(sort\)
于是总复杂度\(O(n\log n)\)(但我偷懒让复杂度退化成上界很难取到的\(O(n\log ^2n))\)
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <map>
#include <cmath>
inline int max(int a, int b){return a > b ? a : b;}
inline int min(int a, int b){return a < b ? a : b;}
inline void swap(int &x, int &y){int tmp = x;x = y;y = tmp;}
inline void read(int &x)
{
x = 0;char ch = getchar(), c = ch;
while(ch < '0' || ch > '9') c = ch, ch = getchar();
while(ch <= '9' && ch >= '0') x = x * 10 + ch - '0', ch = getchar();
if(c == '-') x = -x;
}
const int INF = 0x3f3f3f3f;
const int MAXN = 200000 + 10;
struct Edge
{
int u,v,w,nxt;
Edge(int _u, int _v, int _w, int _nxt){u = _u, v = _v, w = _w, nxt = _nxt;}
Edge(){}
}edge[MAXN << 1];
int head[MAXN], cnt, root, sum, w[MAXN], hao[MAXN], a[MAXN], ma[MAXN], ans, tot, vis[MAXN], size[MAXN], f[MAXN], dis[MAXN], num[MAXN], dp[MAXN], tmp, tag[MAXN], n, k, m;
inline void insert(int a, int b, int c)
{
edge[++ cnt] = Edge(a, b, c, head[a]), head[a] = cnt;
}
int cmp(int c, int b)
{
return ma[a[c]] < ma[a[b]];
}
void dfs_dp(int x, int pre)
{
size[x] = 1, dp[x] = 0;
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(pre == v || vis[v]) continue;
dfs_dp(v, x), size[x] += size[v], dp[x] = max(dp[x], size[v]);
}
dp[x] = max(dp[x], sum - size[x]);
if(dp[x] < dp[root]) root = x;
}
//f[i]表示黑点数小于等于i的最大dis
//num[i]表示i到根黑点个数
//ma[i]表示i子树一条根到子树内某个节点，最大的黑点个数
//dis[i]表示i子树一条根到子树内的某个节点，路径长度
void dfs_num(int x, int pre, int r)
{
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(vis[v] || v == pre) continue;
num[v] = num[x] + tag[v], ma[r] = max(ma[r], num[v]), dfs_num(v, x, r);
}
}
void dfs_dis(int x, int pre, int tmp)
{
if(num[x] <= k) ans = max(ans, f[k - num[x]] + dis[x]);
if(num[x] <= k && num[x] + tmp <= k) ans = max(ans, f[tmp] + dis[x]);
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(vis[v] || v == pre) continue;
dis[v] = dis[x] + edge[pos].w, dfs_dis(v, x, tmp);
}
}
void dfs_f(int x, int flag, int pre)
{
if(num[x] <= k)
{
if(flag) f[num[x]] = max(f[num[x]], dis[x]);
else f[num[x]] = 0;
}
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(vis[v] || v == pre) continue;
dfs_f(v, flag, x);
}
}
void solve(int x)
{
root = 0, dfs_dp(x, -1), x = root, vis[x] = 1, tot = 0;
if(tag[x]) -- k;
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(vis[v]) continue;
num[v] = ma[v] = tag[v], dfs_num(v, -1, v), a[++ tot] = v, w[tot] = edge[pos].w;
}
for(int i = 1;i <= tot;++ i) hao[i] = i;
std::sort(hao + 1, hao + 1 + tot, cmp);
for(int i = 1;i <= tot;++ i)
{
dis[a[hao[i]]] = w[hao[i]], dfs_dis(a[hao[i]], -1, ma[a[hao[i - 1]]]), dfs_f(a[hao[i]], 1, -1);
int tmp = 0, r = ma[a[hao[i]]];
for(int j = 1;j <= r;++ j)
f[j] = max(f[j], tmp), tmp = max(tmp, f[j]);
}
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(vis[v]) continue;
dfs_f(v, 0, -1);
}
if(tag[x]) ++ k;
for(int pos = head[x];pos;pos = edge[pos].nxt)
{
int v = edge[pos].v;
if(vis[v]) continue;
sum = size[v], solve(v);
}
}
int main()
{
read(n), read(k), read(m);
for(int i = 1;i <= m;++ i) read(tmp), tag[tmp] = 1;
for(int i = 1;i < n;++ i)
{
int tmp1,tmp2,tmp3;read(tmp1), read(tmp2), read(tmp3);
insert(tmp1, tmp2, tmp3), insert(tmp2, tmp1, tmp3);
}
ans = 0, sum = n, dp[0] = INF, solve(1);
printf("%d", ans);
return 0;
}
