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T1 奥术神杖

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

#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
[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

[p]用通项公式，扩域计算。Θ(k2log⁡r)\Theta(k^2 \log r)Θ(k2logr)

#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+nm)log⁡nm)\Theta((q+nm)\log nm)Θ((q+nm)lognm)

#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;
}