2020-2021 ACM-ICPC Brazil Subregional Programming Contest

A Sticker Album

概率遞推, f[i] 表示得到 i 張貼紙要開包數的期望

則當 a > 0 時, 1~a, f[i] = 1(開一包至少 a 張貼紙)

i > a, 則 \(f[i]=\frac{\sum_{j=i-b}^{i-a}f[j]}{b-a+1}+1\) (在得到[i-b,i-a]張貼紙的基礎上, 隨便開一包得至少 a 張貼紙)

當 a == 0, 直接就是 \(f[i]=\frac{\sum_{j=i-b}^{i-a}f[j]}{b-a+1} + 1\)

帶入 a = 0, 兩邊都有f[i]移項 解得 \(f[i]=\frac{\sum_{j=i-b}^{i-1} + b + 1}{b}\)

維護個sum 即可 表示 f[i-b]~f[i - (a ? a : 1)] 即可

int main() {
    IOS; cin >> n >> a >> b;
    rep (i, 1, n) {
        if (a) f[i] = s / (b - a + 1) + 1;
        else f[i] = (s + b + 1) / b;
        s += (a ? f[max(i + 1 - a, 0ll)] : f[i]) - f[max(i - b, 0ll)];
    }
    cout << precision(5) << f[n];
    return 0;
}

B Battleship

模擬

bool v[11][11];

int main() {
    IOS; cin >> n; bool f = 1;
    rep (i, 1, n) {
        int d, l, r, c; cin >> d >> l >> r >> c;
        if (!d) for (int j = 0; j < l; ++j)
            if (c + j > 10 || v[r][c + j]) f = 0;
            else v[r][c + j] = 1;
        else for (int j = 0; j < l; ++j)
            if (r + j > 10 || v[r + j][c]) f = 0;
            else v[r + j][c] = 1;
    }
    cout << (f ? 'Y' : 'N');
    return 0;
}

C Concatenating Teams

說白了就是如果存在 $a^, = a+S, b^,=T+b, T == S \ \ \ $ 則 \(a, a^,,b,b^,\) 都不是

不存在相同字串, 直接全hash一邊, 並存下所有的 S 和 T 即可, 對於 \(b^,=T+b\) reverse一下 形式就如同 \(a^, = a+S\) 了 只不過 T 變成原來的反串了

怕卡hash, 用了雙hash

const int N = 1e5 + 5, P[2] = { 131, 13331 };
bool va[N], vb[N];
string a[N], b[N];
ull p[2][N * 10] = { {1}, {1} }, has[2][N * 10], rhas[2][N * 10];
unordered_map<ull, int> x[2], y[2];
unordered_set<ull> sta[2], stb[2];
 
void hashstr(string& a) {
    rep(i, 1, a.size()) rep(j, 0, 1) has[j][i] = has[j][i - 1] * P[j] + a[i - 1];
}
 
void rhashstr(string& a) {
    rep(i, 1, a.size()) rep(j, 0, 1) rhas[j][i] = rhas[j][i - 1] * P[j] + a[a.size() - i];
}
 
void init(int n, string* a, unordered_map<ull, int>* x, unordered_set<ull>* sta) {
    sort(a + 1, a + 1 + n, [](string& a, string& b) { return a.size() < b.size(); });
    rep(i, 1, n) {
        hashstr(a[i]); x[0][has[0][a[i].size()]] = x[1][has[1][a[i].size()]] = i;
        rep(j, 1, a[i].size() - 1) if (x[0].count(has[0][j]) && x[1].count(has[1][j]))
            sta[0].insert(has[0][a[i].size()] - has[0][j] * p[0][a[i].size() - j]),
            sta[1].insert(has[1][a[i].size()] - has[1][j] * p[1][a[i].size() - j]);
    }
}
 
int work(int n, string* a, unordered_map<ull, int>* x, unordered_set<ull>* stb, bool* va) {
    rep(i, 1, n) {
        hashstr(a[i]); rhashstr(a[i]);
        rep(j, 1, a[i].size() - 1) {
            if (x[0].count(has[0][j]) && x[1].count(has[1][j]) &&
                stb[0].count(rhas[0][a[i].size() - j]) && stb[1].count(rhas[1][a[i].size() - j]))
                va[i] = va[x[0][has[0][j]]] = 1;
        }
    }
    int ans = 0;
    rep(i, 1, n) if (va[i] == 0) ++ans;
    return ans;
}

D

E Party Company

朴素的想, 領導關系構成一棵樹(每個人最多有一個領導(父親節點)), 且年齡也滿足樹形結構(領導比手下年齡大)

任何一個party的年齡 l, r, 要找參與者就要找這個party的所有者 \(O_i\) 的手下和其領導的手下, 領導的領導的手下...

不如干脆直接把party的所有者改成 \(O_i\) 的某個祖先(領導的領導的領導...) 正好使得這個祖先的年齡 <= r (樹上倍增LCA)

然后就是 dfs 了, 每個點都會影響其孩子, 那就 dfs完當前點x的所有孩子, 再把當前x 的貢獻消掉就好, 線段樹/樹狀 都行

struct STFrom {
    static const int N = 1e5 + 5;

    int f[N][20], dep[N], lg[N], t;
    vector<int> *h;

    void init(int n, vector<int>* H) {
        t = log2(n - 1) + 1; h = H;
        rep(i, 1, n) dep[i] = 0;
	rep(i, 1, n) lg[i] = lg[i >> 1] + 1;
    }

    void bfs(int s) {
        queue<int> q; q.push(s); dep[s] = 1;
        rep(i, 0, t) f[s][i] = 0;
        while (!q.empty()) {
            int x = q.front(); q.pop();
            for (auto y : h[x]) {
                if (dep[y]) continue;
                dep[y] = dep[x] + 1; f[y][0] = x;
                for (int j = 1; j <= t; ++j) f[y][j] = f[f[y][j - 1]][j - 1];
                q.push(y);
            }
        }
    }

    int find(int x, int r, int* a) {
        per (i, lg[dep[x]], 0) if (f[x][i] && a[f[x][i]] <= r) x = f[x][i];
        return x;
    }
} ST;

const int N = 1e5 + 5;

int n, m, _, k, cas;
int a[N], c[N], fa[N], b[N];
VI h[N], g[N];

void add(int x, int k) { for (; x <= N; x += -x & x) c[x] += k; }

int ask(int x) { int ans = 0; for (; x; x -= -x & x) ans += c[x]; return ans; }

void dfs(int x) {
    for (auto i : g[x]) add(i, 1);
    b[x] = ask(a[x]);
    for (auto y : h[x]) dfs(y);
    for (auto i : g[x]) add(i, -1);
}

int main() {
    IOS; cin >> n >> m >> a[1] >> fa[1];
    rep (i, 2, n) cin >> a[i] >> fa[i], h[fa[i]].pb(i);
    ST.init(n, h); ST.bfs(1);
    rep (i, 1, m) {
        int id, l, r; cin >> id >> l >> r;
        id = ST.find(id, r, a); g[id].pb(l);
    }
    dfs(1); rep (i, 1, n) cout << b[i] << ' ';
    return 0;
}

F Fastminton

模擬

int a[2], b[2];
bool f = 0;

int main() {
    IOS; string s; cin >> s;
    for (auto c : s) {
        if (a[0] + a[1] < 3 && abs(a[0] - a[1]) < 2) {
            if (c == 'S') ++b[f];
            else if (c == 'R') ++b[f ^= 1];
            if ((b[f] >= 5 && b[f] - b[f ^ 1] >= 2) || b[f] == 10)
                b[f] = b[f ^ 1] = 0, ++a[f];
        }
        if (c != 'Q') continue;
        if (a[0] + a[1] == 3 || abs(a[0] - a[1]) == 2)
            cout << a[0] << ' ' << (a[0] > a[1] ? "(winner) - " : "- ") << a[1] << (a[1] > a[0] ? " (winner)\n" : "\n");
        else cout << a[0] << " (" << b[0] << (f ? ") - " : "*) - ") << a[1] << " (" << b[1] << (f ? "*)\n" : ")\n");
    }
    return 0;
}

G Game Show!

貪心, 不斷取a[i], 再不斷取max

int main() {
    IOS; cin >> n; k = m = 100;
    rep (i, 1, n) cin >> _, umax(m, k += _);
    cout << m;
    return 0;
}

H SBC's Hangar

組合數學 不超過 r 的方案數量 - 不超過 l - 1 的方案數量就是答案

不超過 x 的方案數量怎么求呢?

保證任意 較大的行李都比較小的行李的兩倍 重

那不就是排好序之后 \(a[i] > \sum_{j=1}^{i-1} a[i]\) 嗎

不選 a[i] 那前面的 1~i-1 個物品隨便選夠 k 減即可, 那不就是組合數學嗎?

ll c[N][N], a[N], l, r;
 
void init(int n){
    rep (i, 0, n) {
        c[i][0] = 1;
        rep (j, 1, i) c[i][j] = c[i - 1][j] + c[i - 1][j - 1];
    }
}
 
ll work(ll v, int k) {
    ll ans = 0;
    per (i, n, 1) if (k && a[i] <= v)
        ans += c[i - 1][k], v -= a[i], --k;
    return ans + (k == 0);
}
 
int main() {
    IOS; cin >> n >> k; init(n);
    rep (i, 1, n) cin >> a[i]; sort(a + 1, a + 1 + n);
    cin >> l >> r; cout << work(r, k) - work(l - 1, k);
    return 0;
}

I

J

K

L Lavaspar

暴力模擬題意, 用位壓表示這個格子有哪些串

int a[41][41], c[21][26], b[41][41], d[4][2] = {{1, 0}, {0, 1}, {1, 1}, {1, -1}};
char s[45], str[21][20];
 
void work(int k, int x, int y, int de) {
    rep (i, 0, 25) c[0][i] = c[k][i];
    for (int i = 1; str[k][i] && x <= n && y <= m && y; ++i)
        --c[0][a[x][y]], x += d[de][0], y += d[de][1];
    rep (i, 0, 25) if (c[0][i]) return;
    for (int i = 1; str[k][i]; ++i) b[x -= d[de][0]][y -= d[de][1]] |= 1 << k; 
}
 
int main() {
    IOS; cin >> n >> m;
    rep (i, 1, n) {
        cin >> s + 1;
        rep (j, 1, m) a[i][j] = s[j] ^ 'A';
    }
    cin >> cas;
    rep (i, 1, cas) {
        cin >> str[i] + 1;
        for (int j = 1; str[i][j]; ++j) ++c[i][str[i][j] ^ 'A'];
    }
    rep (k, 1, cas) rep (i, 1, n) rep (j, 1, m) rep (t, 0, 3) work(k, i, j, t);
    rep (i, 1, n) rep (j, 1, m) if (__builtin_popcount(b[i][j]) > 1) ++_;
    cout << _;
    return 0;
}

M

N Number Multiplication

因為質數有序, 直接報以算就好了

算第 i 個質數就從 pri[i - 1] 枚舉到 sqrt(w) 就好了

算出來 pri[i] 再把其對其他權值的貢獻的影響除掉

ll a[N], b[N] = {2};
vector<PII> h[N];

int main() {
    IOS; cin >> m >> n >> k; unordered_set<ll> st;
    rep (i, 1, n) cin >> a[i];
    rep (i, 1, k) {
        int u, v, c; cin >> u >> v >> c;
        h[u].pb({v, c});
    }
    rep (i, 1, m) {
        ll x = a[h[i][0].fi]; b[i] = x;
        rep (j, b[i - 1], x / j) if (x % j == 0) { b[i] = j; break; }
        for (auto &y : h[i]) rep (j, 1, y.se) a[y.fi] /= b[i];
    }
    rep (i, 1, m) cout << b[i] << ' ';
    return 0;
}

P (HDU - 3605)

10個點, 那么每個人的狀態用位壓表示就 1024 種, 轉換完之后直接 網絡流板子即可

int h[N], ne[M], to[M], co[M], tot, now[N];
int d[N], s = 1025, t = 1026, maxflow, a[N], b[11];

void add(int u, int v, int c) {
    ne[++tot] = h[u]; to[h[u] = tot] = v; co[tot] = c;
    ne[++tot] = h[v]; to[h[v] = tot] = u; co[tot] = 0;
}

bool bfs() {
    memset(d, 0, sizeof d); memcpy(now, h, sizeof h);
    queue<int> q; q.push(s); d[s] = 1;
    while (!q.empty()) {
        int x = q.front(); q.pop();
        for (int i = h[x]; i; i = ne[i]) {
            if (!co[i]) continue;
            int y = to[i];
            if (d[y]) continue;
            d[y] = d[x] + 1; q.push(y);
            if (y == t) return 1;
        }
    }
    return 0;
}

int dinic(int x, int flow) {
    if (x == t) return flow;
    int rest = flow, k;
    for (int& i = now[x]; i && rest; i = ne[i]) {
        if (!co[i]) continue;
        int y = to[i];
        if (d[y] != d[x] + 1) continue;
        k = dinic(y, min(rest, co[i]));
        if (!k) d[y] = 0;
        co[i] -= k, co[i ^ 1] += k; rest -= k;
    }
    return flow - rest;
}

int main() {
    IOS; 
    while(cin >> k >> m) {
        n = (1 << m) - 1; tot = 1;
        memset(h, 0, sizeof h); memset(a, 0, sizeof a);
        rep (i, 1, k) {
            int cur = 0;
            rep (j, 1, m) if (cin >> _, _) cur ^= 1 << j - 1;
            ++a[cur];
        }
        rep (i, 1, n) if (a[i]) {
            rep (j, 1, m) if (i >> j - 1 & 1) add(i, t + j, inf);
            add(s, i, a[i]);
        }
        rep (i, 1, m) cin >> b[i], add(i + t, t, b[i]);
        int flow = maxflow = 0;
        while (bfs()) while (flow = dinic(s, inf)) maxflow += flow;
        cout << (maxflow == k ? "YES\n" : "NO\n");
    }
    return 0;
}