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CF464E The Classic Problem

时间:2022-09-01 19:22:08浏览次数:80  
标签:rt Classic int CF464E rs mid Problem now sum

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思路

\(2^{100000}\) ?别想了,普通高精度肯定不行

但我们发现,求最短路的过程中,其实是用到了比较大小和加法操作

细想比较大小的过程,当长度相同的数,我们会先略过前面相同的部分,比较第一个不同的数字,时间大部分都耗在了相同部分的枚举上

我们就可以使用二分,找出第一个不同的数字

对于如何判断前缀是否相同,我们可以使用 hashLCP 的方法,在这道题中,我们可以将 hash 的底数设为 \(2\),模数设为 \(1e9+7\),可以方便输出

对于加法操作,我们注意到这是二进制加法,当加入 \(2^w\) 时,先找到 \(w\) 位往后的第一个 \(0\) 的位置(设为 \(p\)),然后将 \(w\) ~ \(p-1\) 赋值为 \(0\),将 \(p\) 改为 \(1\)

有区间赋值,我们就考虑可以用线段树;但每个数都开一棵线段树显然是不够空间的,因此我们考虑用主席树,让能够共用的区间更多

找 \(0\) 操作就是在线段树二分

hash 也放在线段树上,合并左右儿子时一起更新;比较大小改为线段树二分即可

有了这些,就可以正常跑 dij


代码

#include<iostream>
#include<fstream>
#include<algorithm>
#include<cmath>
#include<cstdlib>
#include<cstring>
#include<queue>
#include<map>
#include<set>
#include<bitset>
#define LL long long
#define maxn 100040
#define mod 1000000007
#define FOR(i, x, y) for(int i = (x); i <= (y); i++)
#define ROF(i, x, y) for(int i = (x); i >= (y); i--)
#define PFOR(i, x) for(int i = he[x]; i; i = e[i].nxt)
inline int reads()
{
    int sign = 1, re = 0; char c = getchar();
    while(c < '0' || c > '9'){if(c == '-') sign = -1; c = getchar();}
    while('0' <= c && c <= '9'){re = re * 10 + (c - '0'); c = getchar();}
    return sign * re;
}
int n, m, S, T;
struct Node
{
    int to, nxt, w;
}e[200005]; int he[100005];
inline void Edge_add(int u, int v, int w)
{
    static int cnt = 0;
    e[++cnt] = (Node){v, he[u], w};
    he[u] = cnt;
}
inline int mod_add(int x) {return x >= mod ? x - mod : x;}
int pow2[maxn + 5];
namespace Seg_Tree
{
    int ls[4000005], rs[4000005];
    int sum[4000005], hash[4000005];
    int tcnt, rt[100005];
    inline void up(int now)
    {
        sum[now] = sum[ls[now]] + sum[rs[now]];
        hash[now] = mod_add(hash[ls[now]] + hash[rs[now]]);
    }
    int build(int l, int r, int v)
    {
        int now = ++tcnt;
        if(l == r)
        {
            sum[now] = v;
            hash[now] = pow2[l] * v % mod;
            return now;
        }
        int mid = (l + r) >> 1;
        ls[now] = build(l, mid, v);
        rs[now] = build(mid + 1, r, v);
        up(now);
        return now;
    }
    bool cmp(int now1, int now2, int l, int r)
    {
        if(l == r) return sum[now1] < sum[now2];
        int mid = (l + r) >> 1;
        bool chk = sum[rs[now1]] == sum[rs[now2]] && hash[rs[now1]] == hash[rs[now2]];
        if(chk) return cmp(ls[now1], ls[now2], l, mid);
        return cmp(rs[now1], rs[now2], mid + 1, r);
    }
    int get_sum(int now, int l, int r, int L, int R)
    {
        if(L <= l && r <= R) return sum[now];
        int mid = (l + r) >> 1, re = 0;
        if(L <= mid) re += get_sum(ls[now], l, mid, L, R);
        if(mid < R) re += get_sum(rs[now], mid + 1, r, L, R);
        return re;
    }
    int Div(int now, int l, int r, int st)
    {
        if(l == r) return l;
        int mid = (l + r) >> 1;
        if(mid < st) return Div(rs[now], mid + 1, r, st);
        if(get_sum(ls[now], l, mid, st, mid) == mid - st + 1)
            return Div(rs[now], mid + 1, r, mid + 1);
        return Div(ls[now], l, mid, st);
    }
    int modify_one(int pre, int l, int r, int to)
    {
        int now = ++tcnt;
        ls[now] = ls[pre], rs[now] = rs[pre];
        if(l == r)
        {
            sum[now] = 1, hash[now] = pow2[l];
            return now;
        }
        int mid = (l + r) >> 1;
        if(to <= mid) ls[now] = modify_one(ls[pre], l, mid, to);
        else rs[now] = modify_one(rs[pre], mid + 1, r, to);
        up(now);
        return now;
    }
    int modify_zero(int now, int zero, int l, int r, int L, int R)
    {
        if(L <= l && r <= R) return zero;
        int nnow = ++tcnt; ls[nnow] = ls[now], rs[nnow] = rs[now];
        int mid = (l + r) >> 1;
        if(L <= mid) ls[nnow] = modify_zero(ls[now], ls[zero], l, mid,L,  R);
        if(mid < R) rs[nnow] = modify_zero(rs[now], rs[zero], mid + 1, r, L, R);
        up(nnow);
        return nnow;
    }
    int add(int _rt, int w)
    {
        int pos = Div(_rt, 0, maxn, w);
        int rp = modify_one(_rt, 0, maxn, pos);
        if(pos == w) return rp;
        rp = modify_zero(rp, rt[0], 0, maxn, w, pos - 1);
        return rp;
    }
} using namespace Seg_Tree;
std::bitset<100005> vis;
struct Dis
{
    int u, rt;
};
bool operator > (const Dis a, const Dis b) {return cmp(b.rt, a.rt, 0, maxn);}
std::priority_queue<Dis, std::vector<Dis>, std::greater<Dis>> q;
int pre[100005], ans[100005], acnt;
inline void Dij()
{
    int rp = build(0, maxn, 1);
    FOR(i, 1, n) rt[i] = rp;
    rt[0] = rt[S] = build(0, maxn, 0);
    q.push((Dis){S, rt[S]});
    while(!q.empty())
    {
        while(!q.empty() && vis[q.top().u])
            q.pop();
        if(q.empty()) break;
        int now = q.top().u; q.pop();
        vis[now] = 1;
        PFOR(i, now)
        {
            int to = e[i].to;
            if(vis[to]) continue;
            int rp = add(rt[now], e[i].w);
            if(cmp(rp, rt[to], 0, maxn))
            {
                rt[to] = rp, pre[to] = now;
                q.push((Dis){to, rt[to]});
            }
        }
    }

    if(rt[T] == rp) {puts("-1"); return;}
    printf("%d\n", hash[rt[T]]);
    int dd = T;
    while(dd != S) ans[++acnt] = dd, dd = pre[dd];
    ans[++acnt] = S;
    printf("%d\n", acnt);
    ROF(i, acnt, 1) printf("%d ", ans[i]);
}
signed main()
{
#ifndef ONLINE_JUDGE
    freopen("test.in", "r", stdin);
    freopen("test.out", "w", stdout);
#endif
    n = reads(), m = reads();
    FOR(i, 1, m)
    {
        int u = reads(), v = reads(), w = reads();
        Edge_add(u, v, w), Edge_add(v, u, w);
    }
    S = reads(), T = reads();
    pow2[0] = 1;
    FOR(i, 1, maxn)
        pow2[i] = pow2[i - 1] << 1,
        pow2[i] = mod_add(pow2[i]);
    Dij();
    return 0;
}

标签:rt,Classic,int,CF464E,rs,mid,Problem,now,sum
From: https://www.cnblogs.com/zuytong/p/16647596.html

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