[UVA][樹形dp] 10859 - Placing Lampposts
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As a part of the mission �Beautification of Dhaka City�, the
government has decided to replace all the old lampposts with new expensive ones.
Since the new ones are quite expensive and the budget is not up to the
requirement, the government has decided to buy the minimum number of lampposts
required to light the whole city.
Dhaka city can be modeled as an undirected graph with no cycles, multi-edges or
loops. There are several roads and junctions. A lamppost can only be placed on
junctions. These lampposts can emit light in all the directions, and that means a
lamppost that is placed in a junction will light all the roads leading away from
it.
The �Dhaka City Corporation� has given you the road map of Dhaka city. You are
hired to find the minimum number of lampposts that will be required to light the
whole city. These lampposts can then be placed on the required junctions to
provide the service. There could be many combinations of placing these lampposts
that will cover all the roads. In that case, you have to place them in
such a way that the number of roads receiving light from two lampposts is
maximized.
Input
There will be several cases in the input file. The first line
of input will contain an integer T(T<=30) that will determine the number
of test cases. Each case will start with two integers N(N<=1000) and
M( M<N) that will indicate the number of junctions and roads respectively.
The junctions are numbered from 0 to N-1. Each of the next M
lines will contain two integers a and b, which implies there is a
road from junction a to b,
( 0<= a,b < N ) and a != b. There is a blank line separating two
consecutive input sets.
Output
For each line of input, there will be one line of output. Each output line will contain 3 integers, with one space separating two consecutive numbers. The first of these integers will indicate the minimum number of lampposts required to light the whole city. The second integer will be the number of roads that are receiving lights from two lampposts and the third integer will be the number of roads that are receiving light from only one lamppost.
Sample Input
| 2 4 3 0 1 1 2 2 3 5 4 0 1 0 2 0 3 0 4 |
Sample Output
| 2 1 2 1 0 4 |
Problem Setter: Sohel Hafiz.
Special thanks to Per Austrin.
題目描述：
這個國家要放置路燈於所有道路的路口處，而這個國家道路呈現樹形結構。
一個路燈放置於路口處時，相鄰的道路都會被照亮，希望能用最少路燈照亮所有道路。
在最少路燈的情況下，希望用兩個路燈照亮的道路越多越好。
分別輸出：最少路燈個數、兩個路燈的個數、剩餘使用一個路燈的個數。
題目解法：
樹狀圖上的最少點集覆蓋，將狀態劃分成 dp[i][2] // 分別表示這一點可選可不選。
dp[i][0] = 所有子節點要選的最少花費 dp[j][1]
dp[i][1] = 所有子節點 min(dp[j][0], dp[j][1])
然而要同時最大化兩個相鄰的放置個數，多記錄一個附加個數即可。
#include <stdio.h>
#include <vector>
using namespace std;
vector<int> g[1005];
int dp[1005][2], used[1005];
int dp2[1005][2];
void dfs(int nd) {
int i, u;
used[nd] = 1;
dp[nd][0] = dp[nd][1] = 0; // dp[node][place_]
dp2[nd][0] = dp2[nd][1] = 0;
for(i = 0; i < g[nd].size(); i++) {
u = g[nd][i];
if(used[u]) continue;
dfs(u);
dp[nd][1] += min(dp[u][0], dp[u][1]);
if(dp[u][0] != dp[u][1])
dp2[nd][1] += dp[u][0] < dp[u][1] ? dp2[u][0] : (dp2[u][1]+1);
else
dp2[nd][1] += max(dp2[u][0], dp2[u][1]+1);
dp[nd][0] += dp[u][1];
dp2[nd][0] += dp2[u][1];
}
dp[nd][1]++;
// printf("[%d] %d %d %d %d\n", nd, dp[nd][0], dp[nd][1], dp2[nd][0], dp2[nd][1]);
}
int main() {
int testcase;
int n, m;
int i, j, k, x, y;
scanf("%d", &testcase);
while(testcase--) {
scanf("%d %d", &n, &m);
for(i = 0; i < n; i++)
g[i].clear(), used[i] = 0;
for(i = 0; i < m; i++) {
scanf("%d %d", &x, &y);
g[x].push_back(y);
g[y].push_back(x);
}
int lamppost = 0, two = 0;
for(i = 0; i < n; i++) {
if(used[i] == 0) {
dfs(i);
lamppost += min(dp[i][0], dp[i][1]);
if(dp[i][0] != dp[i][1])
two += dp[i][0] < dp[i][1] ? dp2[i][0] : dp2[i][1];
else
two += max(dp2[i][0], dp2[i][1]);
}
}
printf("%d %d %d\n", lamppost, two, m - two);
}
return 0;
}