题解 赢王
保证数据随机的那一档合法区间非常少
找性质：
于是可以求解一个区间的答案
于是只需要算 \(O(n)\) 个区间答案
需要将计算式写出，大力分类讨论拆 min
点击查看代码
#include <bits/stdc++.h>
using namespace std;
#define INF 0x3f3f3f3f
#define N 1000010
#define fir first
#define sec second
#define pb push_back
#define ll long long
//#define int long long
char buf[1<<21], *p1=buf, *p2=buf;
#define getchar() (p1==p2&&(p2=(p1=buf)+fread(buf, 1, 1<<21, stdin)), p1==p2?EOF:*p1++)
inline int read() {
int ans=0, f=1; char c=getchar();
while (!isdigit(c)) {if (c=='-') f=-f; c=getchar();}
while (isdigit(c)) {ans=(ans<<3)+(ans<<1)+(c^48); c=getchar();}
return ans*f;
}
int n, k;
int a[N];
const ll mod=998244353;
namespace force{
ll pre[N], ans;
map<vector<int>, bool> vis;
queue<pair<vector<int>, int>> q;
bool check(vector<int>& tem) {
for (auto it:tem) if (it%k) return 0;
return 1;
}
int bfs(vector<int> tem) {
vis.clear();
while (q.size()) q.pop();
q.push({tem, 0});
// cout<<"bfs: "; for (auto it:tem) cout<<it<<' '; cout<<endl;
while (q.size()) {
vector<int> u=q.front().fir; int dis=q.front().sec; q.pop();
if (check(u)) return dis;
for (int i=0; i+1<u.size(); ++i) {
--u[i]; ++u[i+1];
if (vis.find(u)==vis.end()) q.push({u, dis+1}), vis[u]=1;
++u[i]; --u[i+1];
++u[i]; --u[i+1];
if (vis.find(u)==vis.end()) q.push({u, dis+1}), vis[u]=1;
--u[i]; ++u[i+1];
}
}
assert(0);
return INF;
}
int calc(int l, int r) {
if ((pre[r]-pre[l-1])%k) return -1;
vector<int> tem;
for (int i=l; i<=r; ++i) tem.pb(a[i]);
return bfs(tem);
}
void solve() {
for (int i=1; i<=n; ++i) pre[i]=pre[i-1]+a[i];
for (int i=1; i<=n; ++i)
for (int j=i; j<=n; ++j)
ans+=calc(i, j);
cout<<(ans%mod+mod)%mod<<endl;
}
}
namespace task1{
ll pre[N], ans;
int f[110][220];
const int dlt=110;
#define f(a, b) f[a][b+dlt]
int dp(vector<int> tem) {
// cout<<"dp: "; for (auto it:tem) cout<<it<<' '; cout<<endl;
int n=tem.size()-1;
// cout<<"n: "<<n<<endl;
memset(f, 0x3f, sizeof(f));
f(0, 0)=0;
f(1, tem[1])=f(1, tem[1]-k)=0;
for (int i=1; i<=n; ++i) {
for (int j=-k; j<=k; ++j) {
// cout<<"ij: "<<i<<' '<<j<<' '<<f(i-1, j)<<endl;
// cout<<"ij: "<<i<<' '<<j<<' '<<f(i, j)<<endl;
if (j+tem[i]<=k) f(i, j+tem[i])=min(f(i, j+tem[i]), f(i-1, j)+abs(j));
if (j-k+tem[i]>=-k) f(i, j-k+tem[i])=min(f(i, j-k+tem[i]), f(i-1, j)+abs(j));
}
}
// f(n, k)=min(f(n, k), f(n-1, k-tem[n])+abs(k-tem[n]));
// f(n, 0)=min(f(n, 0), f(n-1, -tem[n])+abs(-tem[n]));
// cout<<"return: "<<min(f(n, 0), f(n, k))<<endl;
return min(f(n, 0), f(n, k));
}
int calc(int l, int r) {
if ((pre[r]-pre[l-1])%k) return -1;
vector<int> tem; tem.pb(0);
for (int i=l; i<=r; ++i) tem.pb(a[i]);
return dp(tem);
}
void solve() {
for (int i=1; i<=n; ++i) pre[i]=pre[i-1]+a[i];
for (int i=1; i<=n; ++i)
for (int j=i; j<=n; ++j)
ans+=calc(i, j);
cout<<(ans%mod+mod)%mod<<endl;
}
}
namespace task2{
ll ans, cnt;
vector<int> s[N];
pair<ll, ll> bit[N];
struct que{int l, r; ll k, c;};
vector<que> q[N];
int pre[N], rk[N], uni[N], usiz;
inline void add(int i, ll dat) {for (; i<=usiz; i+=i&-i) bit[i].fir+=dat, ++bit[i].sec;}
inline pair<ll, ll> query(int l, int r) {
pair<ll, ll> ans(0, 0); --l;
while (r>l) ans.fir+=bit[r].fir, ans.sec+=bit[r].sec, r-=r&-r;
while (l>r) ans.fir-=bit[l].fir, ans.sec-=bit[l].sec, l-=l&-l;
return ans;
}
void query(int r, ll t, ll c) {
// cout<<"query: "<<r<<' '<<t<<' '<<c<<endl;
ll dlt=0;
for (int i=1; i<=r; ++i)
if (pre[i]<=t) dlt=(dlt+min(t-pre[i], k-(t-pre[i])))%mod;
else dlt=(dlt+min(pre[i]-t, k-(pre[i]-t)))%mod;
// cout<<"dlt: "<<dlt<<endl;
ans=(ans+dlt*c)%mod;
}
void solve() {
cnt=1ll*n*(n+1)/2;
uni[++usiz]=0; s[1].pb(0);
for (int i=1; i<=n; ++i) uni[++usiz]=pre[i]=(pre[i-1]+a[i])%k;
// cout<<"pre: "; for (int i=1; i<=n; ++i) cout<<pre[i]<<' '; cout<<endl;
sort(uni+1, uni+usiz+1);
usiz=unique(uni+1, uni+usiz+1)-uni-1;
for (int i=1; i<=n; ++i) s[rk[i]=lower_bound(uni+1, uni+usiz+1, pre[i])-uni].pb(i);
for (int i=1; i<=usiz; ++i) {
// cout<<"i: "; cout<<s[i][0]<<' ';
ll t=s[i].size(); cnt-=t*(t-1)/2;
for (int j=1; j<s[i].size(); ++j) {
// cout<<s[i][j]<<' ';
// cout<<s[i][j-1]+1<<' '<<s[i][j]<<' '<<1ll*j*((ll)s[i].size()-j)%mod<<endl;
query(s[i][j-1], pre[s[i][j-1]], -1ll*j*((ll)s[i].size()-j)%mod);
query(s[i][j], pre[s[i][j-1]], 1ll*j*((ll)s[i].size()-j)%mod);
} //cout<<endl;
}
cout<<((ans-cnt)%mod+mod)%mod<<endl;
}
}
namespace task{
ll ans, cnt;
vector<int> s[N];
pair<ll, ll> bit[N];
struct que{ll l, r, k, c;};
vector<que> q[N];
int pre[N], rk[N], uni[N], usiz;
inline void add(int i, ll dat) {for (; i<=usiz; i+=i&-i) bit[i].fir+=dat, ++bit[i].sec;}
inline pair<ll, ll> query(int l, int r) {
pair<ll, ll> ans(0, 0); --l;
while (r>l) ans.fir+=bit[r].fir, ans.sec+=bit[r].sec, r-=r&-r;
while (l>r) ans.fir-=bit[l].fir, ans.sec-=bit[l].sec, l-=l&-l;
return ans;
}
// void query(int r, ll t, ll c) {
// ll dlt=0;
// for (int i=1; i<=r; ++i) {
// if (t<=pre[i] && pre[i]<=t+k/2) dlt+=pre[i]-t;
// if (t+k/2+1<=pre[i] && pre[i]<=k) dlt+=-pre[i]+t+k;
// if (t-k/2<=pre[i] && pre[i]<=t-1) dlt+=-pre[i]+t;
// if (0<=pre[i] && pre[i]<=t-k/2-1) dlt+=pre[i]+k-t;
// }
// ans=(ans+dlt*c)%mod;
// }
void query(int r, ll t, ll c) {
q[r].pb({t, t+k/2, c, c*-t%mod});
q[r].pb({t+k/2+1, k, -c, c*(t+k)%mod});
q[r].pb({t-k/2, t-1, -c, c*t%mod});
q[r].pb({0, t-k/2-1, c, c*(k-t)%mod});
}
void solve() {
cnt=1ll*n*(n+1)/2;
uni[++usiz]=0; s[1].pb(0);
for (int i=1; i<=n; ++i) uni[++usiz]=pre[i]=(pre[i-1]+a[i])%k;
// cout<<"pre: "; for (int i=1; i<=n; ++i) cout<<pre[i]<<' '; cout<<endl;
sort(uni+1, uni+usiz+1);
usiz=unique(uni+1, uni+usiz+1)-uni-1;
for (int i=1; i<=n; ++i) s[rk[i]=lower_bound(uni+1, uni+usiz+1, pre[i])-uni].pb(i);
for (int i=1; i<=usiz; ++i) {
// cout<<"i: "; cout<<s[i][0]<<' ';
ll t=s[i].size(); cnt-=t*(t-1)/2;
for (int j=1; j<s[i].size(); ++j) {
// cout<<s[i][j]<<' ';
// cout<<s[i][j-1]+1<<' '<<s[i][j]<<' '<<1ll*j*((ll)s[i].size()-j)%mod<<endl;
query(s[i][j-1], pre[s[i][j-1]], -1ll*j*((ll)s[i].size()-j)%mod);
query(s[i][j], pre[s[i][j-1]], 1ll*j*((ll)s[i].size()-j)%mod);
} //cout<<endl;
}
for (int i=1; i<=n; ++i) {
// cout<<"i: "<<i<<endl;
add(rk[i], pre[i]);
for (auto it:q[i]) {
// cout<<"("<<it.l<<','<<it.r<<") ";
if (it.l>it.r) continue;
it.l=lower_bound(uni+1, uni+usiz+1, it.l)-uni, it.r=upper_bound(uni+1, uni+usiz+1, it.r)-uni-1;
if (it.l>it.r) continue;
pair<ll, ll> t=query(it.l, it.r);
ans=(ans+t.fir*it.k+t.sec*it.c)%mod;
} //cout<<endl;
}
// cout<<ans<<endl;
cout<<((ans-cnt)%mod+mod)%mod<<endl;
}
}
signed main()
{
freopen("win.in", "r", stdin);
freopen("win.out", "w", stdout);
n=read(); k=read();
for (int i=1; i<=n; ++i) a[i]=read();
// if (n<=10) force::solve();
// else task1::solve();
task::solve();
return 0;
}
