BZOJ 1758 / Luogu P4292 [WC2010]重建计划 (分数规划(二分/迭代) + 长链剖分/点分治)

题意

自己看.

分析

求这个平均值的最大值就是分数规划,二分一下就变成了求一条长度在[L,R]内路径的权值和最大.有淀粉质的做法但是我没写,感觉常数会很大.这道题可以用长链剖分做.

先对树长链剖分. 我们像做dsu on tree一样先做重儿子，用线段树继承重儿子的全部信息，然后做其他轻儿子

查询的时候枚举一下路径的长度len，一边单点$O(1)$查询长度为len的最大权值，一边线段树$O(logn)$查询长度为[L-len,R-len]的区间即可

时间复杂度我不会证(后面有)…反正枚举轻儿子的深度是总共$O(n)$的.所以加上外面的二分,总时间复杂度为$O(nlog^2n)$.

$Upd:$ 枚举轻儿子的深度的时间复杂度证明如下

时间复杂度$O(n)$。
分析如下：
每个点x只会暴力统计其所有轻儿子的信息，而每个轻儿子的信息大小为该轻儿子所在长链长度。
而当递归到x的父节点fa(x)时，若x不是fa(x)的重儿子，则fa(x)会暴力统计大小为x长链长度的信息。
故，每个长链只会对转移的复杂度做一次大小为其长度的贡献。

超强的长链剖分a!!! 证明摘自下面的博客.

学习长链剖分的看这里…博客传送门

CODE

#include<bits/stdc++.h>
using namespace std;
char cb[1<<15],*cs=cb,*ct=cb;
#define getc() (cs==ct&&(ct=(cs=cb)+fread(cb,1,1<<15,stdin),cs==ct)?0:*cs++)
template<class T>inline void read(T &res) {
    char ch; int flg = 1; for(;!isdigit(ch=getc());)if(ch=='-')flg=-flg;
    for(res=ch-'0';isdigit(ch=getc());res=res*10+ch-'0'); res*=flg;
}
template<class T>inline void chkmax(T &x, T y) { if(x < y) x = y; }
const int MAXN = 100005;
const double INF = 1e16;
int n, L, R, cnt, fir[MAXN], init_len[MAXN<<1];
struct edge { int to, nxt; double w; }e[MAXN<<1];
inline void add(int u, int v, int wt) {
    e[cnt] = (edge) { v, fir[u], 0 }, init_len[cnt] = wt, fir[u] = cnt++;
    e[cnt] = (edge) { u, fir[v], 0 }, init_len[cnt] = wt, fir[v] = cnt++;
}
int mxd[MAXN], dep[MAXN], son[MAXN], Eson[MAXN], dfn[MAXN], tmr;
void dfs1(int u, int ff) {
    mxd[u] = dep[u] = dep[ff] + 1;
    for(int i = fir[u], v; ~i; i = e[i].nxt)
        if((v=e[i].to) != ff) {
            dfs1(v, u); chkmax(mxd[u], mxd[v]);
            if(mxd[v] > mxd[son[u]]) son[u] = v, Eson[u] = i;
        }
}
void dfs2(int u, int ff) {
    dfn[u] = ++tmr;
    if(son[u]) dfs2(son[u], u);
    for(int i = fir[u], v; ~i; i = e[i].nxt)
        if((v=e[i].to) != ff && v != son[u])
            dfs2(v, u);
}
int num[MAXN];
double dis[MAXN], mx[MAXN<<2];
void build(int i, int l, int r) {
    mx[i] = -INF;
    if(l == r) { num[l] = i; return; }
    int mid = (l + r) >> 1;
    build(i<<1, l, mid);
    build(i<<1|1, mid+1, r);
}
void modify(int i, int l, int r, int x, double val) {
    chkmax(mx[i], val);
    if(l == r) return;
    int mid = (l + r) >> 1;
    if(x <= mid) modify(i<<1, l, mid, x, val);
    else modify(i<<1|1, mid+1, r, x, val);
}
double query(int i, int l, int r, int x, int y) {
    if(x > y) return -INF;
    if(l == x && r == y) return mx[i];
    int mid = (l + r) >> 1;
    if(y <= mid) return query(i<<1, l, mid, x, y);
    else if(x > mid) return query(i<<1|1, mid+1, r, x, y);
    else return max(query(i<<1, l, mid, x, mid), query(i<<1|1, mid+1, r, mid+1, y));
}
double tmp[MAXN];
bool solve(int u, int ff) {
    modify(1, 1, n, dfn[u], dis[u]);
    if(son[u]) {
		dis[son[u]] = dis[u] + e[Eson[u]].w;
		if(solve(son[u], u)) return 1;
	}
    for(int i = fir[u], v; ~i; i = e[i].nxt)
        if((v=e[i].to) != ff && v != son[u]) {
            dis[v] = dis[u] + e[i].w;
			if(solve(v, u)) return 1;
            for(int j = 1; j <= mxd[v]-dep[u]; ++j) {
                tmp[j] = mx[num[dfn[v]+j-1]];
                if(j <= R) {
                    double temp = query(1, 1, n, dfn[u] + max(L-j, 0), dfn[u] + min(R-j, mxd[u]-dep[u]));
					if(tmp[j] + temp - 2 * dis[u] >= 0) return 1;
                }
            }
            for(int j = 1; j <= mxd[v]-dep[u]; ++j)
                modify(1, 1, n, dfn[u]+j, tmp[j]);
        }
    return query(1, 1, n, dfn[u]+L, dfn[u]+min(R, mxd[u]-dep[u]))-dis[u] >= 0;
}
inline bool judge(double mid) {
    for(int i = 0; i < cnt; ++i)
        e[i].w = init_len[i] - mid;
    build(1, 1, n);
    return solve(1, 0);
}
int main() {
    memset(fir, -1, sizeof fir);
    read(n), read(L), read(R);
    for(int i = 1, x, y, z; i < n; ++i)
        read(x), read(y), read(z), add(x, y, z);
    dfs1(1, 0), dfs2(1, 0);
    double l = 0, r = 1e6, mid;
    while(r - l > (1e-4)) {
        mid = (l + r) / 2;
        if(judge(mid)) l = mid;
        else r = mid;
    }
    printf("%.3f
", l);
}

CODE2

把二分转成迭代后直接洛谷rank 1了…

迭代就是随便取一个值作为mid,带进去算后得到更优的答案,然后把更优的答案作为mid继续迭代下去…直白点就是向答案逼近.

然后初值取一个适中的数就跑的超快了…感觉相当于没有了二分的那一个log

那分数规划都可以用迭代咯…

#include<bits/stdc++.h>
using namespace std;
char cb[1<<15],*cs=cb,*ct=cb;
#define getc() (cs==ct&&(ct=(cs=cb)+fread(cb,1,1<<15,stdin),cs==ct)?0:*cs++)
template<class T>inline void read(T &res) {
    char ch; int flg = 1; for(;!isdigit(ch=getchar());)if(ch=='-')flg=-flg;
    for(res=ch-'0';isdigit(ch=getchar());res=res*10+ch-'0'); res*=flg;
}
template<class T>inline void chkmax(T &x, T y) { if(x < y) x = y; }
const int MAXN = 100005;
const double INF = 1e16;
int n, L, R, cnt, fir[MAXN], init_len[MAXN<<1];
struct edge { int to, nxt; double w; }e[MAXN<<1];
inline void add(int u, int v, int wt) {
    e[cnt] = (edge) { v, fir[u], 0 }, init_len[cnt] = wt, fir[u] = cnt++;
    e[cnt] = (edge) { u, fir[v], 0 }, init_len[cnt] = wt, fir[v] = cnt++;
}
int mxd[MAXN], dep[MAXN], son[MAXN], Eson[MAXN], dfn[MAXN], tmr;
void dfs1(int u, int ff) {
    mxd[u] = dep[u] = dep[ff] + 1;
    for(int i = fir[u], v; ~i; i = e[i].nxt)
        if((v=e[i].to) != ff) {
            dfs1(v, u); chkmax(mxd[u], mxd[v]);
            if(mxd[v] > mxd[son[u]]) son[u] = v, Eson[u] = i;
        }
}
void dfs2(int u, int ff) {
    dfn[u] = ++tmr;
    if(son[u]) dfs2(son[u], u);
    for(int i = fir[u], v; ~i; i = e[i].nxt)
        if((v=e[i].to) != ff && v != son[u])
            dfs2(v, u);
}
int num[MAXN];
double dis[MAXN];
struct node {
    double x; int y; //x:最大路径和   y:边数
    node(){}
    node(double xx, int yy):x(xx), y(yy){}
    inline bool operator <(const node &o)const { return x < o.x; }
}Ans, tmp[MAXN], mx[MAXN<<2];
void build(int i, int l, int r) {
    mx[i] = node(-INF, 0);
    if(l == r) { num[l] = i; return; }
    int mid = (l + r) >> 1;
    build(i<<1, l, mid);
    build(i<<1|1, mid+1, r);
}
void modify(int i, int l, int r, int x, node val) {
    chkmax(mx[i], val);
    if(l == r) return;
    int mid = (l + r) >> 1;
    if(x <= mid) modify(i<<1, l, mid, x, val);
    else modify(i<<1|1, mid+1, r, x, val);
}
node query(int i, int l, int r, int x, int y) {
    if(x > y) return node(-INF, 0);
    if(l == x && r == y) return mx[i];
    int mid = (l + r) >> 1;
    if(y <= mid) return query(i<<1, l, mid, x, y);
    else if(x > mid) return query(i<<1|1, mid+1, r, x, y);
    else return max(query(i<<1, l, mid, x, mid), query(i<<1|1, mid+1, r, mid+1, y));
}

void solve(int u, int ff) {
    modify(1, 1, n, dfn[u], node(dis[u], dep[u]));
    if(son[u]) dis[son[u]] = dis[u] + e[Eson[u]].w, solve(son[u], u);
    for(int i = fir[u], v; ~i; i = e[i].nxt)
        if((v=e[i].to) != ff && v != son[u]) {
            dis[v] = dis[u] + e[i].w, solve(v, u);
            for(int j = 1; j <= mxd[v]-dep[u]; ++j) {
                tmp[j] = mx[num[dfn[v]+j-1]];
                if(j <= R) {
                    node temp = query(1, 1, n, dfn[u] + max(L-j, 0), dfn[u] + min(R-j, mxd[u]-dep[u]));
                    chkmax(Ans, node(tmp[j].x + temp.x - 2 * dis[u], tmp[j].y + temp.y - 2*dep[u]));
                }
            }
            for(int j = 1; j <= mxd[v]-dep[u]; ++j)
                modify(1, 1, n, dfn[u]+j, tmp[j]);
        }
    node temp = query(1, 1, n, dfn[u]+L, dfn[u]+min(R, mxd[u]-dep[u]));
    temp.x -= dis[u]; temp.y -= dep[u];
    chkmax(Ans, temp);
}
inline node judge(double mid) {
    for(int i = 0; i < cnt; ++i)
        e[i].w = init_len[i] - mid;
    build(1, 1, n);
    Ans = node(-INF, 0);
    solve(1, 0);
    return Ans;
}
int main() {
    memset(fir, -1, sizeof fir);
    read(n), read(L), read(R);
    for(int i = 1, x, y, z; i < n; ++i)
        read(x), read(y), read(z), add(x, y, z);
    dep[0] = -1; dfs1(1, 0), dfs2(1, 0);
    double mid = 600000;
    while(1) { //高级操作
        node now = judge(mid);
        double ans = (now.x + mid*now.y) / now.y; //把y条边减去的mid加回来,再算一次平均值
        if(fabs(ans-mid) < 1e-3) break;
        mid = ans;
    }
    printf("%.3f
", mid);
}