\(A\) \(Falling\) \(Asleep\)
\(description:\)
Give you a song list \ (, \) \ (n \) song name \ (s_i \) and playback duration \ (t_i \) \ (, \)
Reading a song name \ (st, \) requires calculating the total playing time of all songs after this song \ (. \) in the song list
\(solution:\)
Violence \ (. \) time complexity \ (O(n + \sum|s_i| \)
\(code:\)
#include <bits/stdc++.h>
using namespace std;
inline int read(){
int x = 0; char c = getchar();
while (!isdigit(c)) c = getchar();
while (isdigit(c)) x = x * 10 + c - '0',c = getchar();
return x;
}
template <typename T> void read(T &x){
x = 0; int f = 1; char ch = getchar();
while (!isdigit(ch)) {if (ch == '-') f = -1; ch = getchar();}
while (isdigit(ch)) {x = x * 10 + ch - '0'; ch = getchar();}
x *= f;
}
inline void write(int x){if (x > 9) write(x/10); putchar(x%10+'0'); }
int n; string st,s[500]; int t[500];
int main(){
cin >> n;
for (int i = 1; i<= n; ++i) cin >> s[i] >> t[i];
cin >> st;
int id = -1; long long ans = 0;
for (int i = 1; i <= n; ++i) if (s[i] == st) id = i;
for (int i = id+1; i <= n; ++i) ans += t[i];
cout << ans << endl;
return 0;
}
\(B\) \(Fusing\) \(Slimes\)
\(description :\)
There are \ (n(n ≤ 10 ^ 5) \) stones \ (, \) each stone has its initial coordinates \ (x_i. \)
Each time, you will randomly select a pile \ (, \) that is not on the far right, move the pile to the position \ (, \) of the pile of stones closest to it on its right, and combine the two piles of stones into a pile \ (. \)
Find the sum of the moving distance of \ (, \) stones in all cases \ (, \) times \ ((n-1)! \)\ (.\)
Answer pair \ (P = 1e9 + 7 \) modulo \ (. \)
\(solution :\)
Consider a distance \ (d_i = x_{i+1} - x_i \)
Definition $dp[n] = $the expected number of times that a \ (n \) stone \ (, \) passes a distance after the \ (n \) stone is moved
It is not difficult to get \ (dp[i] = dp[i-1] + 1/i \)
You can directly calculate the distance of each pair of answers (\.)
Time complexity \ (O(n). \)
\(code :\)
#include <bits/stdc++.h>
using namespace std;
inline int read(){
int x = 0; char c = getchar();
while (!isdigit(c)) c = getchar();
while (isdigit(c)) x = x * 10 + c - '0',c = getchar();
return x;
}
template <typename T> void read(T &x){
x = 0; int f = 1; char ch = getchar();
while (!isdigit(ch)) {if (ch == '-') f = -1; ch = getchar();}
while (isdigit(ch)) {x = x * 10 + ch - '0'; ch = getchar();}
x *= f;
}
inline void write(int x){if (x > 9) write(x/10); putchar(x%10+'0'); }
const int P = 1e9 + 7,N = 500050;
int n,fac[N],inv[N];
int x[N];
int f[N];
long long ans,now;
int main(){
int i;
read(n);
for (i = 1; i <= n; ++i) read(x[i]);
for (fac[0] = i = 1; i <= n; ++i) fac[i] = 1ll*fac[i-1]*i%P;
for (inv[0] = inv[1] = 1,i = 2; i <= n; ++i) inv[i] = 1ll*inv[P%i]*(P-P/i)%P;
for (i = 1; i <= n; ++i){
int prob; prob = inv[i];
f[i] = 1ll*prob*(f[i-1]+1) % P;
f[i] += 1ll*(P+1-prob)*f[i-1] % P;
f[i] %= P;
}
ans = 0;
for (i = 1; i <= n-1; ++i){
int dist = x[i+1] - x[i];
ans = (ans + 1ll*dist*f[i]%P)%P;
}
cout << 1ll*fac[n-1]*ans%P << endl;
return 0;
\(C\) \(Cookie\) \(Distribution\)
$description : $
There are \ (n \) individuals \ (, \) you want to send them candy \ (. \) in \ (k \) days
Read in \ (a_1.. a_k, \), where \ (a_i \) means that on the \ (I \) day \ (a_i \) individuals \ (, \) will be evenly and randomly selected from \ (n \) individuals, and each of them will be given a sugar \ (. \)
Record \ (c_i \) as the number of candy sent by the \ (I \) \ (, \) and find the expectation \ (. \) of \ (\ Pi c_i \)
\(n<=10^3,k<=20\)
\(solution:\)
Seeking \ (\ prod c_i \) is equivalent to seeking the number of schemes to choose a sugar from each of the \ (n \) individuals
Consider enumerating \ (x_i \) to indicate when the candy selected from the \ (I \) person came from \ (. \)
\It doesn't matter what (x_i \) is \ (, \) what matters is that \ (cnt2_i: \) indicates how many \ (x_j\) \(=\) \(i \)
What is the contribution of this choice to the answer
Then use a \ (O(nk^2) dp \) to solve this problem \ (. \)
\(code :\)
#include <bits/stdc++.h>
using namespace std;
inline int read(){
int x = 0; char c = getchar();
while (!isdigit(c)) c = getchar();
while (isdigit(c)) x = x * 10 + c - '0',c = getchar();
return x;
}
template <typename T> void read(T &x){
x = 0; int f = 1; char ch = getchar();
while (!isdigit(ch)) {if (ch == '-') f = -1; ch = getchar();}
while (isdigit(ch)) {x = x * 10 + ch - '0'; ch = getchar();}
x *= f;
}
inline void write(int x){if (x > 9) write(x/10); putchar(x%10+'0'); }
const int N = 1050,K = 20 + 5,P = 1e9 + 7;
int n,k,a[K];
int fac[N],nfac[N],inv[N];
inline int C(int n,int m){ return (n<m||n<0||m<0) ? 0 : 1ll*fac[n]*nfac[m]%P*nfac[n-m]%P; }
int t[K][N];
int dp[K][N];
int main(){
int i,j,e;
read(n),read(k); for (i = 1; i <= k; ++i) read(a[i]);
inv[0] = fac[0] = nfac[0] = inv[1] = fac[1] = nfac[1] = 1;
for (i = 2; i <= n; ++i){
fac[i] = 1ll*fac[i-1]*i%P;
inv[i] = 1ll*(P-P/i)*inv[P%i]%P;
nfac[i] = 1ll*nfac[i-1]*inv[i]%P;
}
for (i = 1; i <= k; ++i)
for (j = 0; j <= n; ++j) t[i][j] = 1ll*C(n-j,a[i]-j)*nfac[j]%P;
dp[0][0] = 1;
for (i = 1; i <= k; ++i)
for (j = 0; j <= n; ++j){
dp[i][j] = 0;
for (e = 0; e <= j; ++e) dp[i][j] = (dp[i][j] + 1ll * t[i][e] * dp[i-1][j-e]) % P;
}
int ans = 1ll * dp[k][n] * fac[n] % P;
cout << ans << endl;
return 0;
}
\(D\) \(Arrangement\)
\(description :\)
Give you a picture \ (G,\) \(G \) has \ (n \) points \ (, \) \ (n*(n-2) \) directed edges \ (. \)
The complementary graph of graph \ (G \) is a directed graph \ (. \) with all point out degrees \ (= 1 \) \ ((\) may have self ring \ () \)
Find a Hamiltonian path with the smallest dictionary order \ (, \) and output \ (- 1. \) if it does not exist
\(n<=10^5\)
\(solution :\)
First of all, if the scale is small enough \ ((n < = 8) \), it can be directly violent \ (O(n!) \)
Then we consider dealing with the answer \ (. \) for the larger case of \ (n \)
Record $cnt_i = $how many \ (, \) are in the current figure (a_j = i \)
If there is a point \ (P \) that satisfies \ (cnt_p = n-1 \), then we must choose this point as the first point of the path \ (. \)
If there is no such point \ (, \), we can select the point with the lowest number \ (. \) among the current points
Then the problem becomes a subproblem with the scale of \ (n-1 \) \ (, \) but there is an additional limit \ (. \) that cannot start with a certain number
Then we can write a data structure that supports query \ (cnt \) maximum value \ (, \) \ (cnt \) single point modification and query the lowest number of the current point set \ (, \) delete a point \ (, \) and write another \ (O(n!) \) Violence is enough \ (. \)
Complexity \ (O(8! +nlogn) \)
\(code:\)
#include <bits/stdc++.h>
using namespace std;
inline int read(){
int x = 0; char c = getchar();
while (!isdigit(c)) c = getchar();
while (isdigit(c)) x = x * 10 + c - '0',c = getchar();
return x;
}
template <typename T> void read(T &x){
x = 0; int f = 1; char ch = getchar();
while (!isdigit(ch)) {if (ch == '-') f = -1; ch = getchar();}
while (isdigit(ch)) {x = x * 10 + ch - '0'; ch = getchar();}
x *= f;
}
inline void write(int x){if (x > 9) write(x/10); putchar(x%10+'0'); }
const int N = 100050;
int n,a[N];
namespace subtask0{
int ans[100],vis[100]; bool ok;
int p[100];
inline void chk(){
for (int i = 2; i <= n; ++i) if (a[ans[i-1]] == ans[i]) return;
ok = 1; for (int i = 1; i <= n; ++i) p[i] = ans[i];
return;
}
inline void dfs(int dep){
if (ok) return;
if (dep > n){ chk(); return; }
for (int i = 1; i <= n; ++i) if (!vis[i]){
vis[i] = 1,ans[dep] = i;
dfs(dep+1);
vis[i] = 0;
}
}
inline void solve(){
ok = 0; memset(vis,0,sizeof(vis)); memset(ans,0,sizeof(ans)); dfs(1);
if (!ok){ puts("-1"); return; }
for (int i = 1; i <= n; ++i) cout << p[i] << ((i<n) ? (' '):('\n'));
}
}
int mx[N<<2],mxi[N<<2],data[N<<2],cnt[N];
inline void Build(int o,int l,int r){
if (l^r){
int mid = l+r>>1; Build(o<<1,l,mid); Build(o<<1|1,mid+1,r);
mx[o] = max(mx[o<<1],mx[o<<1|1]); data[o] = min(data[o<<1],data[o<<1|1]);
mxi[o] = mxi[ (mx[o<<1]>mx[o<<1|1]) ? (o<<1) : (o<<1|1) ];
return;
}
mx[o] = cnt[l]; data[o] = (mx[o] >= 0) ? (l) : (n+1); mxi[o] = l;
}
inline int Ask(int o,int l,int r,int p){
if (l==r) return mx[o];
int mid = l+r>>1; return (p<=mid) ? Ask(o<<1,l,mid,p) : Ask(o<<1|1,mid+1,r,p);
}
inline void Add(int o,int l,int r,int p,int v){
if (l==r){ mx[o]=v; data[o] = (mx[o] >= 0) ? (l) : (n+1); return; }
int mid = l+r>>1; if (p<=mid) Add(o<<1,l,mid,p,v); else Add(o<<1|1,mid+1,r,p,v);
mx[o] = max(mx[o<<1],mx[o<<1|1]); data[o] = min(data[o<<1],data[o<<1|1]);
mxi[o] = mxi[ (mx[o<<1]>mx[o<<1|1]) ? (o<<1) : (o<<1|1) ];
}
inline int Nowv(){ return data[1]; }
inline int Query(int pos){ if (pos==0) return -1; return Ask(1,1,n,pos); }
inline void Modify(int x,int v){ cnt[x] = v; Add(1,1,n,x,v); }
int ans[N];
int vis[N],nowc[N],lc;
int qwq;
inline void chk(){
for (int i = qwq; i <= n; ++i) if (ans[i]==a[ans[i-1]]) return;
for (int i = 1; i <= n; ++i) write(ans[i]),putchar((i<n)?(' '):('\n'));
exit(0);
}
inline void dfs(int dep){
if (dep>n){ chk(); return; }
for (int i = 1; i <= lc; ++i) if (!vis[i]){
vis[i] = 1;
ans[dep] = nowc[i];
dfs(dep+1);
vis[i] = 0;
}
}
inline void solve_force(int l,int r){
qwq = l;
for (int i = 1; i <= n; ++i) if (Query(i) >= 0) nowc[++lc] = i,vis[lc] = 0;
dfs(l);
}
inline bool unproperty(){
for (int i = 1; i <= n; ++i) cnt[i] = 0;
for (int i = 1; i <= n; ++i) ++cnt[ans[i]];
for (int i = 1; i <= n; ++i) if (cnt[i] != 1) return 1;
for (int i = 2; i <= n; ++i) if (ans[i] == a[ans[i-1]]) return 1;
return 0;
}
int main(){
int i,banp,ret,siz,p;
int pp,val;
read(n);
for (i = 1; i <= n; ++i) read(a[i]);
for (i = 1; i <= n; ++i) if (a[i]==i) a[i]=0;
for (i = 1; i <= n; ++i) ++cnt[a[i]];
// if (n==2){ puts("-1"); return 0; }
if (n<=8){ subtask0::solve(); return 0; }
Build(1,1,n);
for (banp = 0,siz = n,i = 1; i <= n; ++i,--siz){
if (siz <= 6){
solve_force(i,n);
}
banp = a[ans[i-1]];
if (banp != 0 && (ret=Query(banp)) >= 0){
Modify(banp,-1);
if (mx[1] == siz-1){
p = mxi[1];
ans[i] = p;
Modify(ans[i],-1);
}
else{
p = data[1];
ans[i] = p;
Modify(ans[i],-1);
}
pp = a[ans[i]];
if ((val=Query(pp))>=0){ --val; Modify(pp,val); }
Modify(banp,ret);
}
else{
if (mx[1] == siz-1){
p = mxi[1];
ans[i] = p;
Modify(ans[i],-1);
}
else{
p = data[1];
ans[i] = p;
Modify(ans[i],-1);
}
pp = a[ans[i]];
if ((val=Query(pp))>=0){ --val; Modify(pp,val); }
}
}
if (unproperty()){ puts("-1"); return 0; }
for (i = 1; i <= n; ++i) write(ans[i]),putchar((i<n)?(' '):('\n'));
}
\(E\) \(Span\) \(Covering\)
\(description:\)
There are \ (n \) segments with length \ (l_i \) \ (. \)
You want to put these segments on a coordinate axis with a length of \ (X \) \ (\) \ (n < = 100 \) \ (X < = 500 \) \ () \)
And if a condition \ (: \) is met, it cannot cover the outside \ (, \) and there cannot be a place that is not covered by any line segment \ (. \)
Number of solutions \ (. \)
\(solution:\)
First, sort \ (l_i \) from large to small \ (. \)
Consider a \ (dp: \)
$f[i][j][k] = \ (I \) intervals before $constitute \ (j \) adjacent open intervals \ (, \) and the number of schemes with a total length of \ (K \) \ (. \)
There are three types of transfers \ (: \)
\(1. \) open a new interval \ (; (J - > j + 1) \)
\(2. \) merge the current interval and the interval I added into one interval \ (; (J - > J) \)
\(3. \) merge two adjacent intervals into one interval \ ((J - > J-1) \)
The length of the \ (1 \) transfer is determined \ (, \) is \ (k + len \)
The length of the \ (2 \) \ (3 \) transfer is \ (. \) that needs to be enumerated
It seems that the complexity is \ (O(n^2X^2), \), but when enumerating to \ (I \), only the state of \ (k * l_i < = J \) is useful \ (. \)
Therefore, the complexity is \ (O(?? \) can pass \ () \ (\) as if \ (O(nX^2)? \)\ ()\)
\(code:\)
#include <bits/stdc++.h>
using namespace std;
inline int read(){
int x = 0; char c = getchar();
while (!isdigit(c)) c = getchar();
while (isdigit(c)) x = x * 10 + c - '0',c = getchar();
return x;
}
template <typename T> void read(T &x){
x = 0; int f = 1; char ch = getchar();
while (!isdigit(ch)) {if (ch == '-') f = -1; ch = getchar();}
while (isdigit(ch)) {x = x * 10 + ch - '0'; ch = getchar();}
x *= f;
}
inline void write(int x){if (x > 9) write(x/10); putchar(x%10+'0'); }
const int N = 100 + 5,M = 1050,P = 1e9 + 7;
int n,m;
int f[N][M],g[N][M];
inline void upd(int &x,int y){ x+=y; x>=P?x-=P:0; x<0?x+=P:0; }
inline void DP(int L){
int i,j,k;
// cout <<"DP " << endl;
// for (i = 1; i <= n; ++i,cout << endl)
// for (j = 1; j <= m; ++j) cout <<f[i][j] << ' ';
for (i = 1; i <= n; ++i)
for (j = 1; j <= m; ++j) g[i][j] = f[i][j],f[i][j] = 0;
for (i = 1; i <= n; ++i)
for (j = 1; j <= m; ++j) if (g[i][j]>0){
upd(f[i+1][j+L],1ll*(i+1)*g[i][j]%P);
upd(f[i][j],1ll*g[i][j]*(P+j-i*(L-1))%P);
for (k = 1; k < L; ++k) upd(f[i][j+k],2ll*g[i][j]%P*i%P);
for (k = 0; k < L-1; ++k) upd(f[i-1][j+k],1ll*g[i][j]*(L-k-1)%P*(i-1)%P);
}
}
int a[N];
int main(){
int i,j;
int rm;
read(n),read(m); for (i = 1; i <= n; ++i) read(a[i]),++a[i]; rm = m; m += 1;
sort(a+1,a+n+1); reverse(a+1,a+n+1);
memset(f,0,sizeof(f)); f[1][a[1]] = 1;
for (i = 2; i <= n; ++i) DP(a[i]);
// cout <<"DP " << endl;
// for (i = 1; i <= n; ++i,cout << endl)
// for (j = 1; j <= m; ++j) cout <<f[i][j] << ' ';
int ans = 0;
for (i = 1; i <= 1; ++i) ans += f[i][rm+i],ans %= P;
cout << ans << endl;
return 0;
}