BJOI2019 Day1 简要题解

其他 2020-06-23 10:38:00 阅读次数: 0

T1 奥术神杖

二分，AC 自动机。 Θ ( n s ∣ Σ ∣ log ⁡ log ⁡ v ϵ ) \Theta \left(ns|\Sigma|\log \frac{\log v}{\epsilon}\right) 

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>

#include <algorithm>
#include <numeric>
#include <queue>
#include <functional>
#include <limits>

#define LOG(FMT...) fprintf(stderr, FMT)

using namespace std;

typedef long long ll;

const int N = 1510, A = 10;
double EPS;

int n, m, k;
int trie[N][A], fail[N], pos[N];
vector<int> fch[N];
double x[N], v[N];
double dp[N][N];
pair<int, int> sol[N][N];
char s[N], t[N];

void buildac() {
    queue<int> q;
    q.push(1);
    fail[1] = 1;
    while (!q.empty()) {
        int p = q.front();
        q.pop();
        if (p == 1) {
            for (int i = 0; i < A; ++i)
                if (trie[p][i]) {
                    fail[trie[p][i]] = p;
                    q.push(trie[p][i]);
                } else
                    trie[p][i] = p;
        } else {
            for (int i = 0; i < A; ++i)
                if (trie[p][i]) {
                    fail[trie[p][i]] = trie[fail[p]][i];
                    q.push(trie[p][i]);
                } else
                    trie[p][i] = trie[fail[p]][i];
        }
    }
    for (int i = 2; i <= k; ++i)
        fch[fail[i]].push_back(i);
}

void dfs(int u) {
    for (int i = 0; i < fch[u].size(); ++i) {
        v[fch[u][i]] += v[u];
        dfs(fch[u][i]);
    }
}

double pred(double x) {
    memset(v, 0, sizeof(v));
    for (int i = 1; i <= m; ++i)
        v[pos[i]] += ::x[i] - x;
    dfs(1);
    for (int i = 0; i <= n; ++i)
        fill(dp[i] + 1, dp[i] + k + 1, -1e9);
    dp[0][1] = 0;
    for (int i = 0; i < n; ++i) {
        for (int j = 1; j <= k; ++j) {
            if (s[i + 1] == '.') {
                for (int c = 0; c < A; ++c)
                    dp[i + 1][trie[j][c]] = max(dp[i + 1][trie[j][c]], dp[i][j]);
            } else {
                int c = s[i + 1] - '0';
                dp[i + 1][trie[j][c]] = max(dp[i + 1][trie[j][c]], dp[i][j]);
            }
        }
        for (int j = 1; j <= k; ++j)
            dp[i + 1][j] += v[j];
    }
    return *max_element(dp[n] + 1, dp[n] + k + 1);
}

int main() {
    scanf("%d%d%s", &n, &m, s + 1);
    ++k;
    if (n <= 501)
        EPS = 1e-6;
    else
        EPS = 3e-5;
    for (int i = 1; i <= m; ++i) {
        int v;
        scanf("%s%d", t + 1, &v);
        x[i] = log(v);
        int p = 1;
        for (char* c = t + 1; *c; ++c) {
            int cc = *c - '0';
            if (trie[p][cc])
                p = trie[p][cc];
            else
                p = trie[p][cc] = ++k;
        }
        pos[i] = p;
    }
    buildac();
    double l = 0, r = 22;
    while (r - l > EPS) {
        double mid = (l + r) / 2;
        if (pred(mid) > 0)
            l = mid;
        else
            r = mid;
    }
    
    memset(v, 0, sizeof(v));
    for (int i = 1; i <= m; ++i)
        v[pos[i]] += x[i] - l;
    dfs(1);
    for (int i = 0; i <= n; ++i)
        fill(dp[i] + 1, dp[i] + k + 1, -1e9);
    dp[0][1] = 0;
    for (int i = 0; i < n; ++i) {
        for (int j = 1; j <= k; ++j)
            if (s[i + 1] == '.') {
                for (int c = 0; c < A; ++c) {
                    if (dp[i + 1][trie[j][c]] < dp[i][j]) {
                        dp[i + 1][trie[j][c]] = dp[i][j];
                        sol[i + 1][trie[j][c]] = make_pair(j, c);
                    }
                }
            } else {
                int c = s[i + 1] - '0';
                if (dp[i + 1][trie[j][c]] < dp[i][j]) {
                    dp[i + 1][trie[j][c]] = dp[i][j];
                    sol[i + 1][trie[j][c]] = make_pair(j, c);
                }
            }
        for (int j = 1; j <= k; ++j)
            dp[i + 1][j] += v[j];
    }
    int cur = max_element(dp[n] + 1, dp[n] + k + 1) - dp[n];
    for (int i = n; i; --i) {
        t[i] = '0' + sol[i][cur].second;
        cur = sol[i][cur].first;
    }
    puts(t + 1);
    
    return 0;
}

T2 勘破神机

用通项公式，扩域计算。 Θ ( k 2 log ⁡ r ) \Theta(k^2 \log r) 

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>

#include <algorithm>
#include <numeric>
#include <functional>
#include <limits>

#define LOG(FMT...) fprintf(stderr, FMT)

using namespace std;

typedef long long ll;

const int P = 998244353, K = 510;

int norm(int x) { return x >= P ? x - P : x; }

int mpow(int x, int k) {
    int ret = 1;
    while (k) {
        if (k & 1)
            ret = ret * (ll)x % P;
        k >>= 1;
        x = x * (ll)x % P;
    }
    return ret;
}

#define DEFZ(W) \
struct Z##W {\
    int a, b;\
    \
    Z##W(int a = 0, int b = 0) : a(a), b(b) {}\
    \
    Z##W operator+(const Z##W& rhs) const { return Z##W(norm(a + rhs.a), norm(b + rhs.b)); }\
    Z##W operator-() const { return Z##W(norm(P - a), norm(P - b)); }\
    Z##W operator-(const Z##W& rhs) const { return *this + -rhs; }\
    Z##W operator*(const Z##W& rhs) const { return Z##W((a * (ll)rhs.a + b * (ll)rhs.b * W) % P, (a * (ll)rhs.b + b * (ll)rhs.a) % P); }\
    Z##W inv() const { return Z##W(a, norm(P - b)) * mpow(norm((a * (ll)a - b * (ll)b * W) % P + P), P - 2); }\
    Z##W operator/(const Z##W& rhs) const { return *this * rhs.inv(); }\
    \
    bool operator==(const Z##W& rhs) const { return a == rhs.a && b == rhs.b; }\
};

DEFZ(3)
DEFZ(5)

#define DEFPOW(W) \
Z##W mpow(Z##W x, ll k) {\
    Z##W ret(1, 0);\
    while (k) {\
        if (k & 1)\
            ret = ret * x;\
        k >>= 1;\
        x = x * x;\
    }\
    return ret;\
}

DEFPOW(5)
DEFPOW(3)

int fac[K], ifac[K];
int fall[K][K];

int binom(int n, int m) { return fac[n] * (ll)ifac[m] % P * ifac[n - m] % P; }

void prepare(int k) {
    fac[0] = 1;
    for (int i = 1; i <= k; ++i)
        fac[i] = fac[i - 1] * (ll)i % P;
    ifac[1] = 1;
    for (int i = 2; i <= k; ++i)
        ifac[i] = -(P / i) * (ll)ifac[P % i] % P + P;
    ifac[0] = 1;
    for (int i = 1; i <= k; ++i)
        ifac[i] = ifac[i - 1] * (ll)ifac[i] % P;
    
    fall[0][0] = 1;
    for (int i = 1; i <= k; ++i) {
        int v = norm(P - (i - 1));
        fall[i][0] = fall[i - 1][0] * (ll)v % P;
        for (int j = 1; j <= i; ++j)
            fall[i][j] = (fall[i - 1][j - 1] + fall[i - 1][j] * (ll)v) % P;
    }
}

int g2(ll l, ll r, int k) {
    int I2 = mpow(2, P - 2);
    Z5 a = Z5(I2, I2), b = Z5(I2, P - I2);
    Z5 res = 0;
    for (int j = 0; j <= k; ++j) {
        Z5 cur = mpow(a, j) * mpow(b, k - j), con = (cur == 1) ? ((r - l + 1) % P) : ((mpow(cur, r + 2) - mpow(cur, l + 1)) / (cur - 1));
        if ((k - j) & 1) {
            res = res - con * binom(k, j);
        } else
            res = res + con * binom(k, j);
    }
    return (res / mpow(Z5(0, 1), k)).a;
}

int g3(ll l, ll r, int k) {
    int I2 = mpow(2, P - 2), I6 = mpow(6, P - 2);
    Z3 a = Z3(2, 1), b = Z3(2, P - 1), x = Z3(I2, I6), y = Z3(I2, P - I6);
    Z3 res = 0;
    for (int j = 0; j <= k; ++j) {
        Z3 cur = mpow(a, j) * mpow(b, k - j), con = (cur == 1) ? ((r - l + 1) % P) : ((mpow(cur, r + 1) - mpow(cur, l)) / (cur - 1));
        res = res + con * mpow(x, j) * mpow(y, k - j) * binom(k, j);
    }
    return res.a;
}

int main() {
    prepare(505);
    int t, m;
    scanf("%d%d", &t, &m);
    while (t--) {
        ll l, r;
        int k;
        scanf("%lld%lld%d", &l, &r, &k);
        int in = mpow((r - l + 1) % P, P - 2);
        if (m == 3) {
            r /= 2;
            l = (l + 1) / 2;
            if (l > r) {
                puts("0");
                continue;
            }
        }
        int ans = 0;
        for (int i = 0; i <= k; ++i)
            ans = (ans + fall[k][i] * (ll)(m == 2 ? g2(l, r, i) : g3(l, r, i))) % P;
        ans = ans * (ll)ifac[k] % P;
        ans = ans * (ll)in % P;
        printf("%d\n", ans);
    }
    
    return 0;
}

T3 送别

平衡树维护若干个环路。 Θ ( ( q + n m ) log ⁡ n m ) \Theta((q+nm)\log nm) 

#include <cstdio>
#include <cassert>
#include <cstring>
#include <cstdlib>
#include <cmath>

#include <algorithm>
#include <numeric>
#include <functional>
#include <limits>

#define LOG(FMT...) fprintf(stderr, FMT)

using namespace std;

typedef long long ll;

struct Node {
    int v;
    bool cyc;
    Node* prt;
    union {
        struct {
            Node *ls, *rs;
        };
        Node* ch[2];
    };

    Node() : v(1) {}

    bool rel() const { return this == prt->rs; }

    void upd() {
        v = 1;
        for (int i = 0; i < 2; ++i)
            if (ch[i])
                v += ch[i]->v;
    }

    void rot() {
        cyc = prt->cyc;
        prt->cyc = false;
        bool f = rel();
        Node* p = prt;
        prt = p->prt;
        if (prt)
            prt->ch[p->rel()] = this;
        p->ch[f] = ch[!f];
        if (ch[!f])
            ch[!f]->prt = p;
        ch[!f] = p;
        p->prt = this;
        p->upd();
        upd();
    }

    void spl(Node* goal = NULL) {
        while (prt != goal) {
            if (prt->prt == goal)
                return rot();
            if (rel() == prt->rel()) {
                prt->rot();
                rot();
            } else {
                rot();
                rot();
            }
        }
    }
};

void link(Node* x, Node* y) {
    x->spl();
    Node* o = y;
    while (o->prt) o = o->prt;
    bool f = o == x;
    y->spl();
    if (f) {
        y->cyc = true;
    } else {
        x->rs = y;
        y->prt = x;
        x->upd();
    }
}

void cut(Node* x) {
    x->spl();
    Node* o = NULL;
    if (x->rs) {
        o = x->rs;
        o->prt = NULL;
        x->rs = NULL;
        x->upd();
    }
    if (x->cyc) {
        x->cyc = false;
        if (o) {
            while (o->rs) o = o->rs;
            o->spl();
            o->rs = x;
            x->prt = o;
            o->upd();
        }
    }
}

void rcut(Node* x) {
    x->spl();
    Node* o = NULL;
    if (x->ls) {
        o = x->ls;
        o->prt = NULL;
        x->ls = NULL;
        x->upd();
    }
    if (x->cyc) {
        x->cyc = false;
        if (o) {
            while (o->ls) o = o->ls;
            o->spl();
            o->ls = x;
            x->prt = o;
            o->upd();
        }
    }
}

int ind(Node* x) {
    x->spl();
    return x->ls ? x->ls->v : 0;
}

int tour(Node* x, Node* y) {
    x->spl();
    Node* o = y;
    while (o->prt) o = o->prt;
    bool f = o == x;
    y->spl();
    if (!f)
        return -1;
    int dy = ind(y);
    int dx = ind(x);
    if (dy >= dx)
        return dy - dx;
    return x->v + dy - dx;
}

Node* pred(Node* x) {
    x->spl();
    if (!x->ls) {
        if (!x->rs)
            return x;
        Node* o = x->rs;
        while (o->rs) o = o->rs;
        return o->spl(), o;
    }
    Node* o = x->ls;
    while (o->rs) o = o->rs;
    return o->spl(), o;
}

Node* succ(Node* x) {
    x->spl();
    if (!x->rs) {
        if (!x->ls)
            return x;
        Node* o = x->ls;
        while (o->ls) o = o->ls;
        return o->spl(), o;
    }
    Node* o = x->rs;
    while (o->ls) o = o->ls;
    return o->spl(), o;
}

const int N = 510;

int n, m;
bool col[N][N], row[N][N];
Node o[N][N][4];

#define X first.first
#define Y first.second
#define D second

typedef pair<pair<int, int>, int> Pos;

Pos bpos(int x, int y, int d) { return make_pair(make_pair(x, y), d); }

Pos gpos(int x1, int y1, int x2, int y2, int d) {
    if (x1 == x2) {
        if (y1 > y2)
            swap(y1, y2);
        return bpos(x1, y1, 1 + (1 - d) * 2);
    } else {
        if (x1 > x2)
            swap(x1, x2);
        return bpos(x1, y1, (1 - d) << 1);
    }
}

Pos go(const Pos& p) {
    if (p.D == 0) {
        if (row[p.X][p.Y])
            return bpos(p.X, p.Y, 1);
        if (p.X > 0 && col[p.X - 1][p.Y])
            return bpos(p.X - 1, p.Y, 0);
        if (p.Y > 0 && row[p.X][p.Y - 1])
            return bpos(p.X, p.Y - 1, 3);
        return bpos(p.X, p.Y, 2);
    } else if (p.D == 1) {
        if (col[p.X][p.Y + 1])
            return bpos(p.X, p.Y + 1, 2);
        if (row[p.X][p.Y + 1])
            return bpos(p.X, p.Y + 1, 1);
        if (p.X > 0 && col[p.X - 1][p.Y + 1])
            return bpos(p.X - 1, p.Y + 1, 0);
        return bpos(p.X, p.Y, 3);
    } else if (p.D == 2) {
        if (p.Y > 0 && row[p.X + 1][p.Y - 1])
            return bpos(p.X + 1, p.Y - 1, 3);
        if (col[p.X + 1][p.Y])
            return bpos(p.X + 1, p.Y, 2);
        if (row[p.X + 1][p.Y])
            return bpos(p.X + 1, p.Y, 1);
        return bpos(p.X, p.Y, 0);
    } else {
        if (p.X > 0 && col[p.X - 1][p.Y])
            return bpos(p.X - 1, p.Y, 0);
        if (p.Y > 0 && row[p.X][p.Y - 1])
            return bpos(p.X, p.Y - 1, 3);
        if (col[p.X][p.Y])
            return bpos(p.X, p.Y, 2);
        return bpos(p.X, p.Y, 1);
    }
}

Pos rgo(const Pos& p) {
    for (int i = -1; i <= 1; ++i)
        for (int j = -1; j <= 1; ++j)
            if (p.X + i >= 0 && p.Y + j >= 0) {
                for (int d = 0; d < 4; ++d) {
                    if (((d & 1) ? row : col)[p.X + i][p.Y + j])
                        if (go(bpos(p.X + i, p.Y + j, d)) == p)
                            return bpos(p.X + i, p.Y + j, d);
                }
            }
}

Node* geto(const Pos& p) { return &o[p.X][p.Y][p.D]; }

void ins(int x, int y, int c) {
    if (c == 1)
        row[x][y] = true;
    else
        col[x][y] = true;
    Pos p1 = bpos(x, y, c), p2 = bpos(x, y, c + 2);
    Node *o1 = geto(p1), *o2 = geto(p2);
    if (go(p1) != p2) {
        rcut(geto(go(p1)));
        cut(geto(rgo(p2)));
    }
    if (go(p2) != p1) {
        rcut(geto(go(p2)));
        cut(geto(rgo(p1)));
    }
    if (go(p1) != p2) {
        link(o1, geto(go(p1)));
        link(geto(rgo(p2)), o2);
    } else
        link(o1, o2);
    if (go(p2) != p1) {
        link(o2, geto(go(p2)));
        link(geto(rgo(p1)), o1);
    } else
        link(o2, o1);
}

void rmv(int x, int y, int c) {
    Pos p1 = bpos(x, y, c), p2 = bpos(x, y, c + 2);
    Node *o1 = geto(p1), *o2 = geto(p2);
    cut(o1);
    rcut(o1);
    cut(o2);
    rcut(o2);
    if (go(p1) != p2)
        link(geto(rgo(p2)), geto(go(p1)));
    if (go(p2) != p1)
        link(geto(rgo(p1)), geto(go(p2)));
    if (c == 1)
        row[x][y] = false;
    else
        col[x][y] = false;
}

int main() {
    int q;
    scanf("%d%d%d", &n, &m, &q);
    for (int i = 0; i < m; ++i) {
        ins(0, i, 1);
        ins(n, i, 1);
    }
    for (int i = 0; i < n; ++i) {
        ins(i, 0, 0);
        ins(i, m, 0);
    }
    for (int i = 0; i < n; ++i)
        for (int j = 1; j < m; ++j) {
            int v;
            scanf("%d", &v);
            if (v)
                ins(i, j, 0);
        }
    for (int i = 1; i < n; ++i)
        for (int j = 0; j < m; ++j) {
            int v;
            scanf("%d", &v);
            if (v)
                ins(i, j, 1);
        }
    while (q--) {
        int opt, x1, y1, x2, y2;
        scanf("%d%d%d%d%d", &opt, &x1, &y1, &x2, &y2);
        if (opt == 1) {
            if (x1 == x2)
                ins(x1, min(y1, y2), 1);
            else
                ins(min(x1, x2), y1, 0);
        } else if (opt == 2) {
            if (x1 == x2)
                rmv(x1, min(y1, y2), 1);
            else
                rmv(min(x1, x2), y1, 0);
        } else {
            int d1, x3, y3, x4, y4, d2;
            scanf("%d%d%d%d%d%d", &d1, &x3, &y3, &x4, &y4, &d2);
            Pos p1 = gpos(x1, y1, x2, y2, d1), p2 = gpos(x3, y3, x4, y4, d2);
            printf("%d\n", tour(geto(p1), geto(p2)));
        }
    }
    return 0;
}

猜你喜欢

转载自blog.csdn.net/EI_Captain/article/details/89449400
今日推荐
Arc Browser for Windows 1.0 正式 GA
90后程序员开发视频搬运软件、不到一年获利超 700 万，结局很刑！
《美国对全球网络空间安全与发展的威胁和破坏》报告发布
火速冲上 GitHub 热榜 —— 开源编程语言、框架哪有这么可爱？
北京人形机器人创新中心发布全球首个纯电驱拟人奔跑的全尺寸人形机器人“天工”
周排行
rbac——界面、权限
Apache CXF + SpringMVC 整合发布WebService
so插件化
Vue.js实战系列---图标字体制作（svg格式）
PAT乙级 1007 素数对猜想(孪生素数对) (20分) ---（C语言 + 详细注释）
被IRM保护的文档，打开失败
Calendar和Date计算日期差的小问题
win10子系统ubuntu18.4安装docker
利用Wrap Shell Script定位Android Native内存泄漏
MySQL: Transaction (Part I - Basic Concept)
每日归档
更多
2024-05-03(19)
2024-05-02(0)
2024-05-01(4)
2024-04-30(1)
2024-04-29(40)
2024-04-28(0)
2024-04-27(56)
2024-04-26(39)
2024-04-25(22)
2024-04-24(36)

代码天地 © 2018 codetd.com

境外服务器   最新电视剧2022