樹狀結構學習(1)、最大-最小劃分樹
有這樣一類題:
給一棵n個點的樹,要求將它分成k個子樹,使權值和最小的子樹權值和儘量大。
解法一:二分答案
我們知道,如果已知權值下限,我們可以從下往上搜以O(n)的複雜度判斷是否可以分成k個權值大於權值下限的子樹。
缺點:沒有真正解出這道題,如果權值是實數或很大將會使這種方法失效。
#include <iostream>
#include <cmath>
#include <cstring>
#include <cstdio>
#include <algorithm>
#include <vector>
using namespace std;
const int MAXN = 1e5 + 10;
int n, k;
int size[MAXN];
int bri[MAXN];
int fa[MAXN];
int w[MAXN];
vector<int> map[MAXN];
void dfs(int x, int father) {
size[x] = w[x];
for (int i = 0; i < map[x].size(); ++i) {
int y = map[x][i];
if (y == father) continue;
fa[y] = x;
dfs(y, x);
size[x] += size[y];
}
}
int tot;
void dfs(int x, int father, int val) {
int delta = w[x];
for (int i = 0; i < map[x].size(); ++i) {
int y = map[x][i];
if (y == father) continue;
dfs(y, x, val);
delta += size[y];
}
size[x] = delta;
if (size[x] >= val) {
++tot;
size[x] = 0;
}
}
bool check(int x) {
tot = 0;
for (int i = 1; i <= n; ++i)
bri[i] = size[i];
dfs(1, -1, x);
for (int i = 1; i <= n; ++i)
size[i] = bri[i];
if (tot >= k) return true;
else return false;
}
int main()
{
freopen("in.txt", "r", stdin);
freopen("out1.txt", "w", stdout);
int a, b;
scanf("%d%d", &n, &k);
for (int i = 1; i < n; ++i) {
scanf("%d%d", &a, &b);
map[a].push_back(b);
map[b].push_back(a);
}
int l = 1, r = 0;
for (int i = 1; i <= n; ++i)
scanf("%d", &w[i]), r += w[i];
r /= k;
dfs(1, -1);
while (l < r) {
int mid = (l + r + 1) >> 1;
if (check(mid))
`
l = mid;
else
r = mid - 1;
}
printf("%d", l);
return 0;
}
解法二:
移動割法
算法流程:
(1)選擇入度爲1的結點作爲起始,將k-
1個割都放在與根相連的唯一的邊上。
(2)計算當前劃分的最小值Wmin
(3)移動割,並求出所有移動方法最大的子樹權值和Wmax
(4)如果Wmax > Wmin,那麼移動,並跳到(2)
(5)結束。Wmin爲所求。當前割的位置即爲一個可行的最優方案。
算法證明之後傳。
#include <iostream>
#include <cstdio>
#include <cmath>
#include <cstring>
#include <algorithm>
#include <vector>
using namespace std;
const int MAXN = 100000 + 10;
const int INF = 0x7fffffff;
int n, k, Wmin, start;
int in[MAXN]; //入度
int size[MAXN];
int fa[MAXN]; //father
int vec[MAXN]; //每一個割現在的容量
int have[MAXN]; //結點x是否有一個割
int pos[MAXN]; //割的位置
int w[MAXN]; //權重
int totSize = 0;
vector<int> map[MAXN];
void dfs(int x, int father) {
size[x] = w[x];
totSize += size[x];
for (int i = 0; i < map[x].size(); ++i) {
int y = map[x][i];
if (y == father) continue;
fa[y] = x;
dfs(y, x);
size[x] += size[y];
}
}
bool down() {
int Wmax = 0, next = 0, id = 0;
for (int i = 1; i <= k; ++i) {
int x = pos[i];
for (int j = 0; j < map[x].size(); ++j) {
int y = map[x][j];
if (y == fa[x] || have[y]) continue;
if (size[y] >= Wmin && size[y] > Wmax) {
Wmax = size[y];
next = y;
id = i;
}
}
}
if (!next) return false;
--have[pos[id]];
int tmp = size[pos[id]] - size[next], i = fa[pos[id]], flag = 1;
while (i && flag) {
if (have[i]) flag = 0;
size[i] += tmp;
i = fa[i];
}
size[pos[id]] -= size[next];
pos[id] = next;
++have[next];
return true;
}
int query() {
/* cout<<"Status of have[]"<<endl;
for (int i = 1; i <= n; ++i)
printf("(%d, %d), ", i, have[i]);
cout<<endl;
cout<<"Status of size[]"<<endl;
for (int i = 1; i <= k; ++i)
cout<<size[pos[i]]<<" ";*/
int res = INF, sum = 0;
for (int i = 1; i <= k; ++i)
if (pos[i] != pos[i-1]) {
res = min(res, size[pos[i]]);
sum += size[pos[i]];
}
// cout<<totSize - sum<<endl;
res = min(res, totSize - sum);
return res;
}
int move(int start) {
Wmin = w[start];
while (true) {
if (!down()) break;
Wmin = query();
}
return Wmin;
}
int main()
{
freopen("in.txt", "r", stdin);
freopen("out2.txt", "w", stdout);
int a, b, Wstart = INF;
scanf("%d%d", &n, &k);
--k;
for (int i = 1; i < n; ++i) {
scanf("%d%d", &a, &b);
map[a].push_back(b);
map[b].push_back(a);
++in[a];
++in[b];
}
for (int i = 1; i <= n; ++i) {
scanf("%d", &w[i]);
if ((in[i] == 0 || in[i] == 1) && w[i] < Wstart)
start = i, Wstart = w[i];
}
dfs(start, -1);
if (k == 0) printf("%d", totSize);
else {
for (int i = 1; i <= k; ++i)
pos[i] = map[start][0];
have[map[start][0]] = k;
printf("%d", move(start));
}
return 0;
}
附:隨機數生成程序
#include <iostream>
#include <cstdio>
#include <cmath>
#include <cstring>
#include <algorithm>
#include <ctime>
using namespace std;
int main()
{
srand(time(0));
freopen("in.txt", "w", stdout);
int n = rand()%10000 + 1, k = rand()%n / 2 + 1;
cout<<n<<" "<<k<<endl;
for (int i = 2; i <= n; ++i)
cout<<rand()%(i-1)+1<<" "<<i<<endl;;
for (int i = 1; i <= n; ++i)
cout<<rand()%100000 + 1<<endl;
return 0;
}