danihao123's Blog

在紧张刺激的等待之后终于肝掉了这道题……

本题的DP方程长成这样（其中\(a\)指\(v\)的某个满足距离限制的祖先，\(d_v\)指\(v\)到根的路径长）：

\[f_v = min(f_a + p_v(d_v - d_a) + q_v)\]

化简之后发现：

\[f_v = q_v + p_v d_v + min(f_a - p_v d_a)\]

利用\(min\)中那一块很容易发现是截距式……但是问题在于，我们的转移来源是树上的连续一段祖先，怎样维护他们的凸包？

答案很狂暴啊……用树链剖分套上向量集那题的线段树套凸包，然后完了……

（注意一点细节：本题因为数据范围过大，故可能存在两个向量叉乘爆long long，所以在求凸包时如果直接用叉积判断是否需要删点会炸掉，建议用斜率判断）

代码：

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cctype>
#include <algorithm>
#include <utility>
#include <vector>
#include <queue>
#include <deque>
#include <cmath>
#include <set>
#include <climits>
using ll = long long;
using T = ll;
using R = long double;
const R eps = 1e-8;
int sign(R x) {
  if(fabsl(x) < eps) {
    return 0;
  } else {
    if(x > (R(0.00))) {
      return 1;
    } else {
      return -1;
    }
  }
}

struct Point {
  T x, y;
  Point(T qx = 0LL, T qy = 0LL) {
    x = qx; y = qy;
  }
};
using Vector = Point;
Vector operator +(const Vector &a, const Vector &b) {
  return Vector(a.x + b.x, a.y + b.y);
}
Vector operator -(const Point &a, const Point &b) {
  return Vector(a.x - b.x, a.y - b.y);
}
Vector operator *(const Vector &a, T lam) {
  return Vector(a.x * lam, a.y * lam);
}
Vector operator *(T lam, const Vector &a) {
  return Vector(a.x * lam, a.y * lam);
}
inline T dot(const Vector &a, const Vector &b) {
  return (a.x * b.x + a.y * b.y);
}
inline T times(const Vector &a, const Vector &b) {
  return (a.x * b.y - a.y * b.x);
}
inline bool cmp(const Point &a, const Point &b) {
  if(a.x == b.x) {
    return a.y < b.y;
  } else {
    return a.x < b.x;
  }
}
inline R slope(const Vector &a) {
  R dx = a.x, dy = a.y;
  return (dy / dx);
}

inline void andrew(Point *P, int L, std::vector<Point> &bot, std::vector<Point> &top) {
  std::sort(P + 1, P + 1 + L, cmp);
  for(int i = 1; i <= L; i ++) {
    if(i != 1 && (P[i].x == P[i - 1].x)) continue;
    while(bot.size() > 1 && sign(slope(P[i] - bot.back()) - slope(bot.back() - bot[bot.size() - 2])) <= 0) {
      bot.pop_back();
    }
    bot.push_back(P[i]);
  }
  for(int i = L; i >= 1; i --) {
    if(i != L && (P[i].x == P[i + 1].x)) continue;
    while(top.size() > 1 && sign(slope(P[i] - top.back()) - slope(top.back() - top[top.size() - 2])) <= 0) {
      top.pop_back();
    }
    top.push_back(P[i]);
  }
  std::reverse(top.begin(), top.end());
}

const int maxn = 200005;
const int maxno = maxn << 2;
const int N = 200000;
bool zen[maxno];
std::vector<Point> bot[maxno], top[maxno];
Point P[maxn];
inline void maintain(int o, int L, int R) {
  static Point tmp[maxn];
  const int lc = o << 1, rc = o << 1 | 1;
  const bool used = zen[o];
  zen[o] = (zen[lc] && zen[rc]);
  if(zen[o] != used) {
    std::copy(P + L, P + R + 1, tmp + 1);
    int len = R - L + 1;
    andrew(tmp, len, bot[o], top[o]);
  }
}
void modify(int o, int L, int R, const int &p, const Point &v) {
  if(L == R) {
    zen[o] = true;
    P[L] = v;
    bot[o].push_back(v); top[o].push_back(v);
  } else {
    const int M = (L + R) / 2;
    if(p <= M) {
      modify(o << 1, L, M, p, v);
    } else {
      modify(o << 1 | 1, M + 1, R, p, v);
    }
    maintain(o, L, R);
  }
}
inline T calc_ans(T k, const Point &v) {
  return v.y - k * v.x;
}
inline T search(const std::vector<Point> &vec, const T &k) {
  int l = 0, r = vec.size() - 1;
  while(r - l > 2) {
    int lm = (l * 2 + r) / 3, rm = (2 * r + l) / 3;
    if((calc_ans(k, vec[lm]) > calc_ans(k, vec[rm]))) {
      l = lm;
    } else {
      r = rm;
    }
  }
  T ans = LLONG_MAX;
  for(int i = l; i <= r; i ++) {
    ans = std::min(ans, calc_ans(k, vec[i]));
  }
  return ans;
}
T query(int o, int L, int R, const int &ql, const int &qr, const T &k) {
  if(ql <= L && R <= qr) {
    return search(bot[o], k);
  } else {
    int M = (L + R) / 2;
    T ans = LLONG_MAX;
    if(ql <= M) {
      ans = std::min(ans, query(o << 1, L, M, ql, qr, k));
    }
    if(qr > M) {
      ans = std::min(ans, query(o << 1 | 1, M + 1, R, ql, qr, k));
    }
    return ans;
  }
}

int first[maxn];
int next[maxn << 1], to[maxn << 1];
ll dist[maxn << 1];
void add_edge(int u, int v, ll d) {
  static int cnt = 0;
  cnt ++;
  next[cnt] = first[u];
  first[u] = cnt;
  to[cnt] = v;
  dist[cnt] = d;
}

int fa[maxn], dep[maxn], hson[maxn];
ll d[maxn];
int siz[maxn];
int bs[maxn][18];
void dfs_1(int x, int f = -1, int depth = 1) {
  fa[x] = bs[x][0] = f; dep[x] = depth;
  siz[x] = 1;
  int max_siz = 0;
  for(int i = first[x]; i; i = next[i]) {
    int v = to[i];
    if(v != f) {
      d[v] = d[x] + dist[i];
      dfs_1(v, x, depth + 1);
      siz[x] += siz[v];
      if(siz[v] > max_siz) {
        hson[x] = v; max_siz = siz[v];
      }
    }
  }
}
int dfn[maxn], tid[maxn], up[maxn];
void dfs_2(int x, int a = 1, int f = 0) {
  static int cnt = 0;
  dfn[x] = ++ cnt; tid[cnt] = x;
  up[x] = a;
  if(hson[x]) {
    dfs_2(hson[x], a, x);
  } else {
    return;
  }
  for(int i = first[x]; i; i = next[i]) {
    int v = to[i];
    if(v != f && v != hson[x]) {
      dfs_2(v, v, x);
    }
  }
}
int k_anc(int x, ll k) {
  int yx = x;
  for(int j = 17; j >= 0; j --) {
    int a = bs[x][j];
    if(a != -1 && d[yx] - d[a] <= k) {
      x = a;
    }
  }
#ifdef LOCAL
  printf("%d's %lld-th anc : %d\n", yx, k, x);
#endif
  return x;
}
int n;
ll get_up(int x, int anc, ll k) {
  ll ans = LLONG_MAX;
  while(up[x] != up[anc]) {
    ans = std::min(ans, query(1, 1, n, dfn[up[x]], dfn[x], k));
    x = fa[up[x]];
  }
  return std::min(ans, query(1, 1, n, dfn[anc], dfn[x], k));
}

ll p[maxn], q[maxn], l[maxn];
ll f[maxn];
void dp(int x) {
#ifdef LOCAL
  printf("processing %d...\n", x);
  printf("d : %lld\n", d[x]);
#endif
  if(x != 1) {
#ifdef LOCAL
    printf("b : %lld\n", get_up(fa[x], k_anc(x, l[x]), p[x]));
#endif
    f[x] = get_up(fa[x], k_anc(x, l[x]), p[x]) + d[x] * p[x] + q[x];
  } else {
    f[x] = 0;
  }
#ifdef LOCAL
  printf("ans : %lld\n", f[x]);
#endif
  modify(1, 1, n, dfn[x], Point(d[x], f[x]));
  for(int i = first[x]; i; i = next[i]) {
    int v = to[i];
    dp(v);
  }
}

int main() {
  int t; scanf("%d%d", &n, &t);
  for(int i = 2; i <= n; i ++) {
    int father; T s;
    scanf("%d%lld%lld%lld%lld", &father, &s, &p[i], &q[i], &l[i]);
    add_edge(father, i, s);
  }
  memset(bs, -1, sizeof(bs));
  dfs_1(1); dfs_2(1);
  for(int j = 1; (1 << j) < n; j ++) {
    for(int i = 1; i <= n; i ++) {
      int a = bs[i][j - 1];
      if(a != -1) {
        bs[i][j] = bs[a][j - 1];
      }
    }
  }
  dp(1);
  for(int i = 2; i <= n; i ++) {
    printf("%lld\n", f[i]);
  }
  return 0;
}

emmm做了一下这道神题……（我可能是少有的用动态凸包苟的？）

首先DP方程长这样：

\[f_i = max(f_{i - 1}, f_j\cdot\frac{A_iR_j+B_i}{A_jR_j+B_j})\]

然后这个方程炒鸡复杂……首先\(f_{i - 1}\)不要管了，然后设\(a_i = \frac{f_i}{A_iR_i + B_i}\)。在xjb推了一番之后我们终于得到了截距式……

\[-a_j R_j \frac{A_i}{B_i} + \frac{f_i}{B_i} = a_j\]

但是这玩意太毒瘤了……斜率不可能单调的，这还好，在凸壳上二分/三分一下即可。但问题在于，横坐标也不单调……

这个时候就需要动态维护凸包了（其实是我不会CDQ），我直接把我向量集那题的二进制分组线段树搬了过来……（逃

代码：

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cctype>
#include <algorithm>
#include <utility>
#include <vector>
#include <cmath>
#include <climits>
#include <deque>
#include <cassert>
using R = double;
const R eps = 1e-8;
int sign(R x) {
  if(fabs(x) < eps) {
    return 0;
  } else {
    if(x > 0.00) {
      return 1;
    } else {
      return -1;
    }
  }
}

struct Point {
  R x, y;
  Point(R qx = 0, R qy = 0) {
    x = qx; y = qy;
  }
};
using Vector = Point;
Vector operator +(const Vector &a, const Vector &b) {
  return Vector(a.x + b.x, a.y + b.y);
}
Vector operator -(const Point &a, const Point &b) {
  return Vector(b.x - a.x, b.y - a.y);
}
Vector operator *(const Vector &a, R lam) {
  return Vector(a.x * lam, a.y * lam);
}
Vector operator *(R lam, const Vector &a) {
  return Vector(a.x * lam, a.y * lam);
}
inline R dot(const Vector &a, const Vector &b) {
  return (a.x * b.x + a.y * b.y);
}
inline R times(const Vector &a, const Vector &b) {
  return (a.x * b.y - a.y * b.x);
}
inline bool cmp(const Point &a, const Point &b) {
  if(sign(a.x - b.x) == 0) {
    return a.y < b.y;
  } else {
    return a.x < b.x;
  }
}
inline void andrew(Point *P, int L, std::vector<Point> &bot, std::vector<Point> &top) {
  std::sort(P + 1, P + 1 + L, cmp);
  for(int i = 1; i <= L; i ++) {
    if(i != 1 && sign(P[i].x - P[i - 1].x) == 0) continue;
    while(bot.size() > 1 && sign(times(P[i] - bot.back(), bot.back() - bot[bot.size() - 2])) >= 0) {
      bot.pop_back();
    }
    bot.push_back(P[i]);
  }
  for(int i = L; i >= 1; i --) {
    if(i != L && sign(P[i].x - P[i + 1].x) == 0) continue;
    while(top.size() > 1 && sign(times(P[i] - top.back(), top.back() - top[top.size() - 2])) >= 0) {
      top.pop_back();
    }
    top.push_back(P[i]);
  }
  std::reverse(top.begin(), top.end());
}

const int maxn = 1000005;
const int N = 1000000;
const int maxno = maxn << 2;
bool zen[maxno];
std::vector<Point> bot[maxno], top[maxno];
Point P[maxn];
inline void maintain(int o, int L, int R) {
  static Point tmp[maxn];
  const int lc = o << 1, rc = o << 1 | 1;
  const bool used = zen[o];
  zen[o] = (zen[lc] && zen[rc]);
  if(zen[o] != used) {
    std::copy(P + L, P + R + 1, tmp + 1);
    int len = R - L + 1;
    andrew(tmp, len, bot[o], top[o]);
  }
}
void modify(int o, int L, int R, const int &p, const Point &v) {
  if(L == R) {
    zen[o] = true;
    P[L] = v;
    bot[o].push_back(v); top[o].push_back(v);
  } else {
    const int M = (L + R) / 2;
    if(p <= M) {
      modify(o << 1, L, M, p, v);
    } else {
      modify(o << 1 | 1, M + 1, R, p, v);
    }
    maintain(o, L, R);
  }
}
inline R calc_ans(R k, const Point &v) {
  return v.y - k * v.x;
}
inline R search(const std::vector<Point> &vec, const R &k) {
  int l = 0, r = vec.size() - 1;
  while(r - l > 2) {
    int lm = (l * 2 + r) / 3, rm = (2 * r + l) / 3;
    if(sign(calc_ans(k, vec[lm]) - calc_ans(k, vec[rm])) == 1) {
      r = rm;
    } else {
      l = lm;
    }
  }
  R ans = INT_MIN;
  for(int i = l; i <= r; i ++) {
    ans = std::max(ans, calc_ans(k, vec[i]));
  }
  return ans;
}
R query(int o, int L, int R, const int &ql, const int &qr, const double &k) {
  if(ql <= L && R <= qr) {
    return search(top[o], k);
  } else {
    int M = (L + R) / 2;
    double ans = INT_MIN;
    if(ql <= M) {
      ans = std::max(ans, query(o << 1, L, M, ql, qr, k));
    }
    if(qr > M) {
      ans = std::max(ans, query(o << 1 | 1, M + 1, R, ql, qr, k));
    }
    return ans;
  }
}

int n, s;
R A[maxn], B[maxn], Rate[maxn];
R f[maxn];
R dp() {
  static double a[maxn];
  f[0] = s; f[1] = s; a[1] = f[1] / (A[1] * Rate[1] + B[1]);
  modify(1, 1, n, 1, Point(a[1] * Rate[1], a[1]));
  for(int i = 2; i <= n; i ++) {
    f[i] = query(1, 1, n, 1, i - 1, -A[i] / B[i]) * B[i];
    f[i] = std::max(f[i], f[i - 1]);
    a[i] = f[i] / (A[i] * Rate[i] + B[i]);
    if(i < n) modify(1, 1, n, i, Point(a[i] * Rate[i], a[i]));
  }
  return f[n];
}

int main() {
  scanf("%d%d", &n, &s);
  for(int i = 1; i <= n; i ++) {
    scanf("%lf%lf%lf", &A[i], &B[i], &Rate[i]);
  }
  printf("%.3lf\n", dp());
  return 0;
}

xjb写了写……我评测时候心脏跳得贼快（逃

考虑如果知道了那一段区间的凸包那么怎么做。首先如果向量是往上指的话，一定在上凸壳上找点比较好，反之则在下凸壳上找点比较好（放到坐标系里脑补一下？）。然后我们观察一点，在上凸壳上的最优解往两边的点会越来越劣，所以这玩意是个上凸函数，可以三分答案（我才学的整数三分啊）。

但区间凸包求起来复杂度很爆炸啊……考虑用线段树搞？观察到一点，我们区间查询所使用的线段树节点一定是只包含了已经加进来的点。所以说，一个线段树节点的凸包需要被求的情况只有一种，那就是这个节点完全已加入点被覆盖了。那每次修改之后看是否一个节点完全被已加入点覆盖，如果被完全覆盖的话才去求它的凸包。

这样一来，线段树上每个节点之多会被求一次凸包。线段树有\(\log n\)层，每一层所有节点的大小加起来是\(n\)，所以求凸包耗费的总复杂度是\(n\log^2 n\)级别的。

其实这就是用线段树模拟二进制分组？

代码：

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cctype>
#include <algorithm>
#include <utility>
#include <vector>
#include <climits>
#include <cassert>
using ll = long long;
using T = ll;
struct Point {
  T x, y;
  Point(T qx = 0LL, T qy = 0LL) {
    x = qx; y = qy;
  }
};
using Vector = Point;
Vector operator +(const Vector &a, const Vector &b) {
  return Vector(a.x + b.x, a.y + b.y);
}
Vector operator -(const Point &a, const Point &b) {
  return Vector(a.x - b.x, a.y - b.y);
}
Vector operator *(const Vector &a, T lam) {
  return Vector(a.x * lam, a.y * lam);
}
Vector operator *(T lam, const Vector &a) {
  return Vector(a.x * lam, a.y * lam);
}
inline T dot(const Vector &a, const Vector &b) {
  return (a.x * b.x + a.y * b.y);
}
inline T times(const Vector &a, const Vector &b) {
  return (a.x * b.y - a.y * b.x);
}
inline bool cmp(const Point &a, const Point &b) {
  if((a.x - b.x) == 0LL) {
    return a.y < b.y;
  } else {
    return a.x < b.x;
  }
}
inline void andrew(Point *P, int L, std::vector<Point> &bot, std::vector<Point> &top) {
  std::sort(P + 1, P + 1 + L, cmp);
  for(int i = 1; i <= L; i ++) {
    if(i != 1 && (P[i].x - P[i - 1].x) == 0LL) continue;
    while(bot.size() > 1 && (times(P[i] - bot.back(), bot.back() - bot[bot.size() - 2])) >= 0LL) {
      bot.pop_back();
    }
    bot.push_back(P[i]);
  }
  for(int i = L; i >= 1; i --) {
    if(i != L && (P[i].x - P[i + 1].x) == 0LL) continue;
    while(top.size() > 1 && (times(P[i] - top.back(), top.back() - top[top.size() - 2])) >= 0LL) {
      top.pop_back();
    }
    top.push_back(P[i]);
  }
  std::reverse(top.begin(), top.end());
}

const int maxn = 400005;
const int maxno = maxn << 2;
const int N = 400000;
bool zen[maxno];
std::vector<Point> bot[maxno], top[maxno];
Point P[maxn];
inline void maintain(int o, int L, int R) {
  static Point tmp[maxn];
  const int lc = o << 1, rc = o << 1 | 1;
  const bool used = zen[o];
  zen[o] = (zen[lc] && zen[rc]);
  if(zen[o] != used) {
    std::copy(P + L, P + R + 1, tmp + 1);
    int len = R - L + 1;
    andrew(tmp, len, bot[o], top[o]);
  }
}
void modify(int o, int L, int R, const int &p, const Point &v) {
  if(L == R) {
    zen[o] = true;
    P[L] = v;
    bot[o].push_back(v); top[o].push_back(v);
  } else {
    const int M = (L + R) / 2;
    if(p <= M) {
      modify(o << 1, L, M, p, v);
    } else {
      modify(o << 1 | 1, M + 1, R, p, v);
    }
    maintain(o, L, R);
  }
}
inline T search(const std::vector<Point> &vec, const Point &p) {
  int l = 0, r = vec.size() - 1;
  while(r - l > 2) {
    int lm = (l * 2 + r) / 3, rm = (2 * r + l) / 3;
    if(dot(p, vec[lm]) > dot(p, vec[rm])) {
      r = rm;
    } else {
      l = lm;
    }
  }
  T ans = LLONG_MIN;
  for(int i = l; i <= r; i ++) {
    ans = std::max(ans, dot(p, vec[i]));
  }
  return ans;
}
T query(int o, int L, int R, const int &ql, const int &qr, const Point &p) {
  if(ql <= L && R <= qr) {
    if(p.y > 0LL) {
      return search(top[o], p);
    } else {
      return search(bot[o], p);
    }
  } else {
    int M = (L + R) / 2;
    T ans = LLONG_MIN;
    if(ql <= M) {
      ans = std::max(ans, query(o << 1, L, M, ql, qr, p));
    }
    if(qr > M) {
      ans = std::max(ans, query(o << 1 | 1, M + 1, R, ql, qr, p));
    }
    return ans;
  }
}

inline int decode(int x, long long lastans) {
  return x ^ (lastans & 0x7fffffff);
}
int main() {
  int q; char buf[4]; scanf("%d%s", &q, buf);
  bool typ_E = (buf[0] == 'E' && buf[1] == char(0));
  T las = 0LL;
  int tot = 0;
  while(q --) {
    char op[4]; scanf("%s", op);
    if(op[0] == 'A') {
      T x, y; scanf("%lld%lld", &x, &y);
      if(!typ_E) {
        x = decode(x, las); y = decode(y, las);
      }
      tot ++;
      modify(1, 1, N, tot, Point(x, y));
    } else {
      T x, y, l, r; scanf("%lld%lld%lld%lld", &x, &y, &l, &r);
      if(!typ_E) {
        x = decode(x, las); y = decode(y, las);
        l = decode(l, las); r = decode(r, las);
      }
      las = query(1, 1, N, l, r, Point(x, y));
      printf("%lld\n", las);
    }
  }
  return 0;
}