The 13th Chinese Northeast Collegiate Programming Contest
题解：
Code：
A. Apple Business
#include<cstdio>
#include<algorithm>
#include<vector>
using namespace std;
typedef long long ll;
const int N=100010;
int Case,len[N],n,m,i,mx,a[N],size[N],tmp[N];ll ans;
vector<ll>v[N],f[N];
struct E{int u,v,c,w;}e[N];
inline bool cmp(const E&a,const E&b){return a.w>b.w;}
void dfs(int x,int y){
if(x>n)return;
if(y>mx)mx=y;
tmp[y]=a[x];
dfs(x<<1,y<<1);
dfs(x<<1|1,y<<1|1);
}
inline void init(int o){
mx=0;
dfs(o,1);
size[o]=mx;
v[o].resize(mx+1);
f[o].resize(mx+1);
for(int i=1;i<=mx;i++)v[o][i]=f[o][i]=tmp[i];
}
inline int getid(int x,int y){
int k=1<<(len[y]-len[x]);
return (y&(k-1))|k;
}
inline void append(int A,int B,int C,int W){
int x,y,o,t,m;
for(x=A;x;x>>=1){
o=getid(x,B);
m=size[x];
ll dp=f[x][o];
for(y=o>>1;y;o=y,y>>=1){
dp+=v[x][y];
t=y<<1;
if(t<=m&&t!=o&&f[x][t]<0)dp+=f[x][t];
t=y<<1|1;
if(t<=m&&t!=o&&f[x][t]<0)dp+=f[x][t];
}
if(dp<C)C=dp;
}
if(!C)return;
ans+=1LL*C*W;
for(x=A;x;x>>=1){
o=getid(x,B);
m=size[x];
v[x][o]-=C;
f[x][o]-=C;
for(y=o>>1;y;y>>=1){
f[x][y]=v[x][y];
t=y<<1;
if(t<=m&&f[x][t]<0)f[x][y]+=f[x][t];
t=y<<1|1;
if(t<=m&&f[x][t]<0)f[x][y]+=f[x][t];
}
}
}
int main(){
for(i=1;i<N;i++)len[i]=len[i>>1]+1;
scanf("%d",&Case);
while(Case--){
scanf("%d%d",&n,&m);
for(i=1;i<=n;i++)scanf("%d",&a[i]);
for(i=1;i<=n;i++)init(i);
for(i=1;i<=m;i++)scanf("%d%d%d%d",&e[i].u,&e[i].v,&e[i].c,&e[i].w);
sort(e+1,e+m+1,cmp);
ans=0;
for(i=1;i<=m;i++)append(e[i].u,e[i].v,e[i].c,e[i].w);
printf("%lld\n",ans);
}
}
B. Balanced Diet
#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long ll;
const int N=100010;
int T,n,m,i,j,k,l[N];ll f[N],s,A,B,d;
struct E{int a,b;}e[N];
inline bool cmp(const E&a,const E&b){return a.b==b.b?a.a>b.a:a.b<b.b;}
ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
int main(){
scanf("%d",&T);
while(T--){
scanf("%d%d",&n,&m);
for(i=1;i<=m;i++)scanf("%d",&l[i]);
for(i=1;i<=n;i++)scanf("%d%d",&e[i].a,&e[i].b);
sort(e+1,e+n+1,cmp);
for(i=1;i<=n;i++)f[i]=0;
for(i=1;i<=n;i=j){
for(j=i;j<=n&&e[i].b==e[j].b;j++);
s=0;
for(k=i;k<j;k++){
s+=e[k].a;
if(k-i+1==l[e[i].b])f[k-i+1]+=s;
if(k-i+1>l[e[i].b])f[k-i+1]+=e[k].a;
}
}
for(i=1;i<=n;i++)f[i]+=f[i-1];
A=f[1],B=1;
for(i=2;i<=n;i++)if(f[i]*B>A*i)A=f[i],B=i;
d=gcd(A,B);
printf("%lld/%lld\n",A/d,B/d);
}
}
C. Line-line Intersection
#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long ll;
const int N=100010;
int Case,n,i;pair<ll,pair<ll,ll> >f[N],g[N];
inline ll myabs(ll x){return x>0?x:-x;}
ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll cal(pair<ll,pair<ll,ll> >f[]){
int i,j;
ll ret=0;
sort(f+1,f+n+1);
for(i=1;i<=n;i=j){
for(j=i;j<=n&&f[i]==f[j];j++);
ret+=1LL*(j-i)*(j-i-1);
}
return ret/2;
}
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d",&n);
for(i=1;i<=n;i++){
int x1,y1,x2,y2;
scanf("%d%d%d%d",&x1,&y1,&x2,&y2);
int dx=x2-x1,dy=y2-y1;
ll a,b,c;
if(dx==0){
a=1;
b=0;
c=-x1;
}else{
a=-dy;
b=dx;
c=1LL*dy*x1-1LL*dx*y1;
}
if(a<0)a=-a,b=-b,c=-c;
if(a==0&&b<0)b=-b,c=-c;
ll d=gcd(myabs(a),gcd(myabs(b),myabs(c)));
a/=d,b/=d,c/=d;
f[i].first=a,f[i].second.first=b,f[i].second.second=c;
d=gcd(myabs(a),myabs(b));
a/=d,b/=d;
g[i].first=a,g[i].second.first=b,g[i].second.second=0;
}
printf("%lld\n",1LL*n*(n-1)/2-cal(g)+cal(f));
}
}
D. Master of Data Structure
#include<cstdio>
#include<cstring>
#include<cstdlib>
const int N=500010,M=2010,K=M*4,inf=~0U>>1;
int Case,n,m,i,o,x,y,z,root,op[M][4];
int vip[N],g[N],v[N<<1],nxt[N<<1],ed,f[N],d[N],id[N],cnt;
int at[K],G[K],W[K],NXT[K],F[K],D[K];
int vv[K],ve[K];
inline void add(int x,int y){v[++ed]=y;nxt[ed]=g[x];g[x]=ed;}
inline void addedge(int x,int y,int z){NXT[y]=G[x];G[x]=y;W[y]=z;}
inline void umin(int&a,int b){a>b?(a=b):0;}
inline void umax(int&a,int b){a<b?(a=b):0;}
inline int abs(int x){return x>0?x:-x;}
inline void swap(int&a,int&b){int c=a;a=b;b=c;}
void dfs(int x){
int deg=0;
for(int i=g[x];i;i=nxt[i]){
int u=v[i];
if(u==f[x])continue;
f[u]=x;
d[u]=d[x]+1;
dfs(u);
if(!id[u])continue;
deg++;
id[x]^=id[u];
}
if(deg>1)vip[x]=1;
if(!vip[x])return;
id[x]=++cnt;
at[cnt]=x;
for(int i=g[x];i;i=nxt[i]){
int u=v[i];
if(u==f[x])continue;
u=id[u];
if(!u)continue;
addedge(cnt,u,d[at[u]]-d[x]-1);
}
}
void dfs2(int x,int y){
F[x]=y;
D[x]=D[y]+1;
for(int i=G[x];i;i=NXT[i])dfs2(i,x);
}
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d%d",&n,&m);
for(ed=cnt=i=0;i<=n;i++)f[i]=d[i]=id[i]=vip[i]=g[i]=0;
memset(G,0,sizeof G);
memset(W,0,sizeof W);
memset(F,0,sizeof F);
memset(D,0,sizeof D);
memset(vv,0,sizeof vv);
memset(ve,0,sizeof ve);
for(i=1;i<n;i++)scanf("%d%d",&x,&y),add(x,y),add(y,x);
for(i=1;i<=m;i++){
scanf("%d%d%d",&o,&x,&y);
vip[x]=vip[y]=1;
op[i][0]=o;
op[i][1]=x;
op[i][2]=y;
if(o==1||o==2||o==3||o==7)scanf("%d",&op[i][3]);
}
for(i=1;i<=n;i++)if(vip[i])root=i;
dfs(root);
dfs2(id[root],0);
for(i=1;i<=m;i++){
o=op[i][0];
x=id[op[i][1]];
y=id[op[i][2]];
z=op[i][3];
if(o==1){
while(x!=y){
if(D[x]<D[y])swap(x,y);
vv[x]+=z;
ve[x]+=z;
x=F[x];
}
vv[x]+=z;
}
if(o==2){
while(x!=y){
if(D[x]<D[y])swap(x,y);
vv[x]^=z;
ve[x]^=z;
x=F[x];
}
vv[x]^=z;
}
if(o==3){
while(x!=y){
if(D[x]<D[y])swap(x,y);
if(vv[x]>=z)vv[x]-=z;
if(ve[x]>=z)ve[x]-=z;
x=F[x];
}
if(vv[x]>=z)vv[x]-=z;
}
if(o==4){
long long ans=0;
while(x!=y){
if(D[x]<D[y])swap(x,y);
ans+=vv[x];
ans+=1LL*ve[x]*W[x];
x=F[x];
}
printf("%lld\n",ans+vv[x]);
}
if(o==5){
int ans=0;
while(x!=y){
if(D[x]<D[y])swap(x,y);
ans^=vv[x];
if(W[x]&1)ans^=ve[x];
x=F[x];
}
printf("%d\n",ans^vv[x]);
}
if(o==6){
int mi=inf,ma=0;
while(x!=y){
if(D[x]<D[y])swap(x,y);
umin(mi,vv[x]);
umax(ma,vv[x]);
if(W[x]){
umin(mi,ve[x]);
umax(ma,ve[x]);
}
x=F[x];
}
umin(mi,vv[x]);
umax(ma,vv[x]);
printf("%d\n",ma-mi);
}
if(o==7){
int ans=inf;
while(x!=y){
if(D[x]<D[y])swap(x,y);
umin(ans,abs(vv[x]-z));
if(W[x])umin(ans,abs(ve[x]-z));
x=F[x];
}
umin(ans,abs(vv[x]-z));
printf("%d\n",ans);
}
}
}
}
E. Minimum Spanning Tree
#include<cstdio>
#include<algorithm>
#include<vector>
using namespace std;
typedef long long ll;
const int N=100010;
int Case,n,i,j,x,y,z;vector<int>f[N];ll ans;
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d",&n);
for(i=1;i<=n;i++)f[i].clear();
for(i=1;i<n;i++)scanf("%d%d%d",&x,&y,&z),f[x].push_back(z),f[y].push_back(z);
ans=0;
for(i=1;i<=n;i++)if(f[i].size()>1){
sort(f[i].begin(),f[i].end());
for(j=1;j<f[i].size();j++)ans+=f[i][0]+f[i][j];
}
printf("%lld\n",ans);
}
}
F. Mini-game Before Contest
#include<cstdio>
const int N=100010,M=200010,K=1<<24;
int Case,n,m,i,j,k,S,A,B,x,y,z,o,deg[N],g[N],v[M],nxt[M],ed;
bool isactor[6],isa[6],ismin[6];
char tmp[9];
int f[N][6],w[N][6][3];
int q[K+5],h,t;
inline void add(int x,int y){v[++ed]=y;nxt[ed]=g[x];g[x]=ed;}
inline void ext(int x,int y,int B){
int A=f[x][y];
if(A==B)return;
t=(t+1)&(K-1);
q[t]=(x<<7)|(y<<4)|(A<<2)|B;
f[x][y]=B;
}
inline void check(int x,int y){
int ret;
if(ismin[y]){
if(w[x][y][0])ret=0;
else if(w[x][y][1])ret=1;
else ret=2;
}else{
if(w[x][y][2])ret=2;
else if(w[x][y][1])ret=1;
else ret=0;
}
ext(x,y,ret);
}
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d%d",&n,&m);
for(ed=i=0;i<=n;i++)deg[i]=g[i]=0;
while(m--)scanf("%d%d",&x,&y),deg[x]++,add(y,x);
scanf("%s",tmp);
for(i=0;i<6;i++)isa[i]=tmp[i]=='A';
scanf("%s",tmp);
for(i=0;i<6;i++)isactor[i]=tmp[i]=='1';
for(i=0;i<6;i++)ismin[i]=isa[i]^isactor[i];
for(i=1;i<=n;i++)for(j=0;j<6;j++){
f[i][j]=1;
for(k=0;k<3;k++)w[i][j][k]=0;
w[i][j][1]=deg[i];
}
h=1,t=0;
for(i=1;i<=n;i++)if(!deg[i])for(j=0;j<6;j++){
if(isa[j])ext(i,j,2);
else ext(i,j,0);
}
while(h!=((t+1)&(K-1))){
S=q[h];
h=(h+1)&(K-1);
x=S>>7,y=S>>4&7,A=S>>2&3,B=S&3;
o=(y+5)%6;
for(i=g[x];i;i=nxt[i]){
z=v[i];
w[z][o][A]--;
w[z][o][B]++;
check(z,o);
}
}
for(i=1;i<=n;i++){
if(f[i][0]==0)putchar('A');
if(f[i][0]==1)putchar('D');
if(f[i][0]==2)putchar('B');
}
puts("");
}
}
G. Radar Scanner
#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long ll;
const int N=100010;
int Case,n,m,i,j,e[N<<1];
struct P{int l,r;}a[N],b[N];
inline ll solve(P*a){
m=0;
ll ans=0;
for(i=1;i<=n;i++){
e[++m]=a[i].l;
e[++m]=a[i].r;
ans-=a[i].r-a[i].l;
}
sort(e+1,e+m+1);
for(i=1;i<=m;i++)ans+=abs(e[i]-e[n]);
return ans/2;
}
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d",&n);
for(i=1;i<=n;i++)scanf("%d%d%d%d",&a[i].l,&b[i].l,&a[i].r,&b[i].r);
printf("%lld\n",solve(a)+solve(b));
}
}
H. Skyscraper
#include<cstdio>
typedef long long ll;
const int N=100010;
int Case,n,m,i,op,x,y;ll z,a[N],b[N],f[N],g[N];
inline void ins(ll*f,int x,ll p){for(;x<=n;x+=x&-x)f[x]+=p;}
inline ll ask(ll*f,int x){ll t=0;for(;x;x-=x&-x)t+=f[x];return t;}
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d%d",&n,&m);
for(i=1;i<=n;i++)scanf("%lld",&a[i]);
for(i=1;i<=n;i++)b[i]=a[i]-a[i-1];
for(i=1;i<=n;i++)f[i]=g[i]=0;
for(i=1;i<=n;i++){
ins(f,i,b[i]);
if(b[i]>0)ins(g,i,b[i]);
}
while(m--){
scanf("%d%d%d",&op,&x,&y);
if(op==1){
y++;
scanf("%lld",&z);
ins(f,x,z);
ins(f,y,-z);
if(b[x]>0)ins(g,x,-b[x]);
if(b[y]>0)ins(g,y,-b[y]);
b[x]+=z;
b[y]-=z;
if(b[x]>0)ins(g,x,b[x]);
if(b[y]>0)ins(g,y,b[y]);
}else{
printf("%lld\n",ask(f,x)+ask(g,y)-ask(g,x));
}
}
}
}
I. Temperature Survey
#include<cstdio>
#include<algorithm>
#include<map>
using namespace std;
const int N=524288,M=200010,K=18,P=998244353,G=3;
typedef unsigned int uint32;
typedef long long int64;
typedef unsigned long long uint64;
typedef uint32 word;
typedef uint64 dword;
typedef int sword;
const int word_bits=sizeof(word)*8;
word mod,Modinv,r2;
struct UnsafeMod{
word x;
UnsafeMod(): x(0) {}
UnsafeMod(word _x): x(init(_x)) {}
UnsafeMod& operator += (const UnsafeMod& rhs) {
(x += rhs.x) >= mod && (x -= mod);
return *this;
}
UnsafeMod& operator -= (const UnsafeMod& rhs) {
sword(x -= rhs.x) < 0 && (x += mod);
return *this;
}
UnsafeMod& operator *= (const UnsafeMod& rhs) {
x = reduce(dword(x) * rhs.x);
return *this;
}
UnsafeMod operator + (const UnsafeMod &rhs) const {
return UnsafeMod(*this) += rhs;
}
UnsafeMod operator - (const UnsafeMod &rhs) const {
return UnsafeMod(*this) -= rhs;
}
UnsafeMod operator * (const UnsafeMod &rhs) const {
return UnsafeMod(*this) *= rhs;
}
UnsafeMod pow(uint64 e) const {
UnsafeMod ret(1);
for (UnsafeMod base = *this; e; e >>= 1, base *= base) {
if (e & 1) ret *= base;
}
return ret;
}
word get() const {
return reduce(x);
}
static word modulus() {
return mod;
}
static word init(word w) {
return reduce(dword(w) * r2);
}
static void set_mod(word m) {
mod = m;
Modinv = mul_inv(mod);
r2 = -dword(mod) % mod;
}
static word reduce(dword x) {
word y = word(x >> word_bits) - word((dword(word(x) * Modinv) * mod) >> word_bits);
return sword(y) < 0 ? y + mod : y;
}
static word mul_inv(word n, int e = 6, word x = 1) {
return !e ? x : mul_inv(n, e - 1, x * (2 - x * n));
}
};
int Case,n,i,j,lim[M];
UnsafeMod a[M],b[M],c[M],d[M];
UnsafeMod ta[N+5],tb[N+5];
UnsafeMod g[K+1],ng[K+10],gw[N+10],ngw[N+10];
int pos[N+10],inv[N+10];
UnsafeMod fac[N+5],rev[N+5];
map<int,UnsafeMod>f[M];
struct E{int xl,xr,yl,yr;}e[M];
inline UnsafeMod C(int n,int m){return n<m?0:fac[n]*rev[m]*rev[n-m];}
inline void NTT(UnsafeMod*a,int n,int t){
int j=__builtin_ctz(n)-1;
for(int i=0;i<n;i++)pos[i]=pos[i>>1]>>1|((i&1)<<j);
for(int i=1;i<n;i++)if(i<pos[i])std::swap(a[i],a[pos[i]]);
for(int d=0;(1<<d)<n;d++){
int m=1<<d,m2=m<<1;
UnsafeMod*_w=t==1?gw:ngw;
_w+=m;
for(int i=0;i<n;i+=m2){
UnsafeMod*w=_w;
for(int j=i;j<m+i;j++,w++){
UnsafeMod t=*w*a[j+m];
a[j+m]=a[j]-t;
a[j]+=t;
}
}
}
if(t==-1){
UnsafeMod j=inv[n];
for(int i=0;i<n;i++)a[i]*=j;
}
}
void build(int l,int r,int yl){
if(l>r)return;
int mid=(l+r)>>1;
e[mid].xl=mid;
e[mid].xr=r;
e[mid].yl=yl;
e[mid].yr=lim[mid];
build(l,mid-1,yl);
build(mid+1,r,lim[mid]+1);
}
inline UnsafeMod cal(int x,int y){
UnsafeMod t=0;
if(x>1)t=f[x-1][y];
if(y>1)t+=f[x][y-1];
return f[x][y]=t;
}
inline void work0(int n,int m,UnsafeMod*a,UnsafeMod*b){
int i,k;
for(k=1;k<=(n-1)*2;k<<=1);
for(i=0;i<k;i++)ta[i]=tb[i]=0;
for(i=0;i<n;i++){
ta[i]=a[i]*rev[m];
tb[i]=fac[i+m]*rev[i];
}
NTT(ta,k,1),NTT(tb,k,1);
for(i=0;i<k;i++)ta[i]*=tb[i];
NTT(ta,k,-1);
for(i=0;i<n;i++)b[i]+=ta[i];
}
inline void work1(int n,int m,UnsafeMod*a,UnsafeMod*b){
int i,k;
for(k=1;k<=n+n+m-2;k<<=1);
for(i=0;i<k;i++)ta[i]=tb[i]=0;
for(i=0;i<n;i++)ta[i]=a[i]*rev[n-i];
for(i=0;i<n+m;i++)tb[i]=fac[i];
NTT(ta,k,1),NTT(tb,k,1);
for(i=0;i<k;i++)ta[i]*=tb[i];
NTT(ta,k,-1);
for(i=0;i<m;i++)b[i]+=ta[i+n]*rev[i];
}
inline void solve(int o){
int xl=e[o].xl,xr=e[o].xr,yl=e[o].yl,yr=e[o].yr;
if(yl>yr)return;
int i,j,k,ca,cb;
if(xl==1){
for(i=xl;i<=xr;i++)for(j=yl;j<=yr;j++)f[i][j]=C(i+j-2,i-1);
return;
}
cal(xl,yl);
for(i=yl+1,k=0;i<=yr;i++,k++)a[k]=cal(xl,i);
for(i=xl+1,k=0;i<=xr;i++,k++)b[k]=cal(i,yl);
ca=yr-yl-1;
cb=xr-xl-1;
if(ca<=0||cb<=0){
for(i=xl;i<=xr;i++)for(j=yl;j<=yr;j++){
if(i==xl||j==yl)continue;
cal(i,j);
}
return;
}
for(i=0;i<ca;i++)c[i]=b[cb];
for(i=0;i<cb;i++)d[i]=a[ca];
work0(ca,cb,a,c);
work0(cb,ca,b,d);
work1(ca,cb,a,d);
work1(cb,ca,b,c);
for(i=0;i<ca;i++)f[xr][yl+i+1]=c[i];
for(i=0;i<cb;i++)f[xl+i+1][yr]=d[i];
f[xr][yr]=c[ca-1]+d[cb-1];
}
int main(){
UnsafeMod::set_mod(P);
for(g[K]=((UnsafeMod)G).pow((P-1)/N),ng[K]=g[K].pow(P-2),i=K-1;~i;i--)g[i]=g[i+1]*g[i+1],ng[i]=ng[i+1]*ng[i+1];
for(i=0;i<=K;i++){
gw[1<<i]=ngw[1<<i]=1;
for(j=1;j<1<<i;j++){
gw[(1<<i)+j]=gw[(1<<i)+j-1]*g[i];
ngw[(1<<i)+j]=ngw[(1<<i)+j-1]*ng[i];
}
}
for(inv[0]=inv[1]=1,i=2;i<=N;i++)inv[i]=1LL*(P-inv[P%i])*(P/i)%P;
for(rev[0]=rev[1]=1,i=2;i<=N;i++)rev[i]=rev[i-1]*inv[i];
for(fac[0]=i=1;i<=N;i++)fac[i]=fac[i-1]*i;
scanf("%d",&Case);
while(Case--){
scanf("%d",&n);
for(i=1;i<=n;i++)scanf("%d",&lim[i]);
n++;
lim[n]=lim[n-1];
for(i=1;i<=n;i++)f[i].clear();
build(1,n,1);
for(i=1;i<=n;i++)solve(i);
printf("%u\n",f[n][lim[n]].get());
}
}
J. Time Limit
#include<cstdio>
#include<algorithm>
using namespace std;
int Case,n,i,x,ans;
int main(){
scanf("%d",&Case);
while(Case--){
scanf("%d",&n);
for(i=1;i<=n;i++){
scanf("%d",&x);
if(i==1)ans=x*3;
else ans=max(ans,x+1);
}
while(ans%2)ans++;
printf("%d\n",ans);
}
}
