Bzoj3197/洛谷3296 [SDOI2013]刺客信條assassin(樹的重心+樹Hash+樹形DP+KM)
阿新 • • 發佈:2019-01-11
題面
題解
首先固定一棵樹,列舉另一棵樹,顯然另一棵樹只有與這棵樹同構才有可能產生貢獻 如果固定的樹以重心為根,那麼另一棵樹最多就只有重心為根才有可能同構了(可能有兩個) 然後就是求改動次數最小值,設$f[x][y]$表示以第一棵樹$x$為根的子樹內和第二棵樹內$y$為根的子樹內,達到目標最少需要改動的次數 我們發現只有同構的子樹需要決策,我們把同構的子樹分別拿出來,我們要做的就是做一個匹配,跑一遍$KM$或者費用流就好了。因為要最小化$f[x][y]$,所以是跑最小完美匹配。 $f[x][y]$要記憶化一下,判斷同構用樹雜湊即可
#include <cstdio> #include <vector> #include <cstring> #include <algorithm> using std::sort; using std::vector; typedef long long ll; template<typename T> void read(T &x) { int flag = 1; x = 0; char ch = getchar(); while(ch < '0' || ch > '9') { if(ch == '-') flag = -flag; ch = getchar(); } while(ch >= '0' && ch <= '9') x = x * 10 + ch - '0', ch = getchar(); x *= flag; } const int N = 1.4e3 + 10, Inf = 1e9 + 7; void upt0(int &x, int y) { if(x < y) x = y; } void upt1(int &x, int y) { if(x > y) x = y; } namespace KM { int n, w[N][N], match[N], ret, lx[N], ly[N]; bool visx[N], visy[N]; bool Hungary(int x) { visx[x] = 1; for(int y = 1; y <= n; ++y) if(!visy[y] && lx[x] + ly[y] == w[x][y]) { visy[y] = true; if(!match[y] || Hungary(match[y])) { match[y] = x; return 1; } } return 0; } int main(int opt) { for(int i = 1; i <= n; ++i) for(int j = 1; j <= n; ++j) w[i][j] = opt * w[i][j]; for(int i = 1; i <= n; ++i) { lx[i] = -Inf, ly[i] = 0; for(int j = 1; j <= n; ++j) upt0(lx[i], w[i][j]); } memset(match, 0, sizeof match); for(int x = 1; x <= n; ++x) while(1) { memset(visx, 0, sizeof visx), memset(visy, 0, sizeof visy); if(Hungary(x)) break; int inc = Inf; for(int i = 1; i <= n; ++i) if(visx[i]) for(int j = 1; j <= n; ++j) if(!visy[j]) upt1(inc, lx[i] + ly[j] - w[i][j]); for(int i = 1; i <= n; ++i) { if(visx[i]) lx[i] -= inc; if(visy[i]) ly[i] += inc; } } for(int i = 1; i <= n; ++i) if(match[i]) ret += w[match[i]][i]; return opt * ret; } }//KM模板 int n, rt, fir[N], sec[N], f[N][N], c[N][N]; ll hash[N]; int from[N], cnt, to[N << 1], nxt[N << 1]; inline void addEdge(int u, int v) { to[++cnt] = v, nxt[cnt] = from[u], from[u] = cnt; } int tmp, siz[N]; vector<int> v1[N], v2[N]; inline bool cmp(const int &i, const int &j) { return hash[i] < hash[j]; } void getrt(int u, int fa) { int max_part = 0; siz[u] = 1; for(int i = from[u]; i; i = nxt[i]) { int v = to[i]; if(v == fa) continue; getrt(v, u), siz[u] += siz[v], upt0(max_part, siz[v]); } upt0(max_part, n - siz[u]); if(max_part < tmp) tmp = max_part, rt = u; }//尋找樹的重心 void dfs(int u, int fa, vector<int> *V) { siz[u] = 1, hash[u] = 0, V[u].clear(); for(int i = from[u]; i; i = nxt[i]) { int v = to[i]; if(v == fa) continue; dfs(v, u, V), siz[u] += siz[v], V[u].push_back(v); } sort(V[u].begin(), V[u].end(), cmp); for(int i = V[u].size() - 1; ~i; --i) hash[u] = hash[u] * N + hash[V[u][i]]; hash[u] = hash[u] * N + siz[u]; }//處理各子樹hash值以及兒子(將兒子放進一個vector裡面) int dp(int x, int y) { if(f[x][y] != -1) return f[x][y]; f[x][y] = fir[x] ^ sec[y]; int lim = v1[x].size() - 1; for(int i = 0; i <= lim; ++i) { int j = i; while(j < lim && hash[v1[x][j + 1]] == hash[v1[x][i]]) ++j; for(int k = i; k <= j; ++k) for(int l = i; l <= j; ++l) dp(v1[x][k], v2[y][l]); for(int k = i; k <= j; ++k) for(int l = i; l <= j; ++l) KM::w[k - i + 1][l - i + 1] = dp(v1[x][k], v2[y][l]); //初始化邊權 KM::ret = 0, KM::n = j - i + 1, f[x][y] += KM::main(-1), i = j; //最小化f[x][y] } return f[x][y]; } int main () { read(n); for(int i = 1, u, v; i < n; ++i) read(u), read(v), addEdge(u, v), addEdge(v, u); for(int i = 1; i <= n; ++i) read(fir[i]); for(int i = 1; i <= n; ++i) read(sec[i]); tmp = Inf, getrt(1, 0), dfs(rt, 0, v2); ll tmp = hash[rt]; int ans = Inf; for(int i = 1; i <= n; ++i) {//暴力列舉重心 dfs(i, 0, v1); if(hash[i] == tmp) memset(f, -1, sizeof f), upt1(ans, dp(i, rt)); } printf("%d\n", ans); return 0; }