#### [1043A — Elections](http://codeforces.me/contest/1043/problem/A)↵
↵
<spoiler summary="Tutorial">↵
We can observe that result cannot exceed $201$ — Awruk gets at least $101$ votes from one person and Elodreip cannot get more than $100$ votes from one person. So we can iterate over every possible integer from $1$ to $201$ and check if Awruk wins with $k$ set to this integer. We have to remember that $k$ — $a_i$ is always at least $0$, so we have to check this condition too. Complexity $O(n * M)$, where $M$ denotes maximum possible value of $a_i$. Try to solve it in $O(n)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
int n;↵
int mx = 0, sum = 0;↵
↵
int main(){↵
scanf("%d", &n);↵
for(int i = 1; i <= n; ++i){↵
int a; scanf("%d", &a);↵
mx = max(mx, a);↵
sum += a;↵
}↵
↵
sum *= 2;↵
sum += n;↵
sum /= n;↵
↵
printf("%d", max(sum, mx));↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043B — Lost Array](http://codeforces.me/contest/1043/problem/B)↵
↵
<spoiler summary="Tutorial">↵
First, let's observe that we can replace array $a_i$ with array $b_i$ = $a_i$ $-$ $a_{i-1}$, because all we care about are differences between neighboring elements.↵
Now, we can see that our lost array can have length $d$ if and only if for every $j$ such that $j$ $+$ $d$ $\leqslant$ $n$, $b_j$ $=$ $b_{j+d}$.↵
So we can iterate over every possible $d$ from $1$ to $n$ and check if it is correct in $O(n)$. Complexity of whole algorithm is $O(n^2)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 1007;↵
↵
int n;↵
int in[N];↵
↵
bool ok(int d){↵
for(int i = 0; i + d < n; ++i)↵
if(in[i + 1] - in[i] != in[i + d + 1] - in[i + d])↵
return false;↵
return true;↵
}↵
↵
int main(){↵
scanf("%d", &n);↵
for(int i = 1; i <= n; ++i)↵
scanf("%d", &in[i]);↵
↵
vector <int> res;↵
for(int i = 1; i <= n; ++i)↵
if(ok(i))↵
res.push_back(i);↵
↵
printf("%d\n", res.size());↵
for(int v: res)↵
printf("%d ", v);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043C — Smallest Word](http://codeforces.me/contest/1043/problem/C)↵
↵
<spoiler summary="Tutorial">↵
Basically in problem we are given a word in which for every $i$ we can reverse prefix of first $i$ elements and we want to get the smallest lexicographically word. We will show that we can always achieve word in form $a^{j}b^{n-j}$.↵
↵
Let's say that we solved our problem for prefix of length $i$ and for this prefix we have word $a^{j}b^{i-j}$ (at the beginning it's just empty word). If our next letter is $b$ then we do nothing, because we will get word $a^{j}b^{i-j+1}$ which is still the smallest lexicographically word. Otherwise we want to reverse prefix of length $i$, add letter $a$ and reverse prefix of length $i$ $+$ $1$, so we get a word $a^{j+1}b^{i-j}$, which is still fine for us.↵
↵
There is still a problem — what if we have already reversed prefix $i$ and we just said that we will reverse it second time. But instead of reversing it second time, we can deny it's first reverse.↵
↵
Final complexity is $O(n)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 1007;↵
↵
string s;↵
bool write[N];↵
↵
int main(){↵
cin >> s;↵
for(int i = 1; i < s.size(); ++i)↵
if(s[i] == 'a'){↵
write[i - 1] ^= 1;↵
write[i] = 1;↵
}↵
↵
for(int i = 0; i < s.size(); ++i)↵
printf("%d%c", write[i], i + 1 == (int)s.size() ? '\n' : ' ');↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043D — Mysterious Crime](http://codeforces.me/contest/1043/problem/D)↵
↵
<spoiler summary="Tutorial">↵
Deleting prefix and suffix is nothing more than taking a subarray. If subarray is common for all permutations then it has to appear in first permutation. We renumber all permutations such that first permutation is $1$, $2$, ..., $n$ $-$ $1$, $n$.↵
↵
Now for every $i$ in every permutation we count how long is subarray starting at $i$ which looks like $i$, $i$ $+$ $1$, ..., $i$ $+$ $k$. It can be easily done in $O(n)$ for one permutation with two pointers technique.↵
↵
Now for every element $i$ we compute $reach[i]$ equal the longest subarray starting in $i$ which looks like $i$, $i$ $+$ $1$, ..., $i$ $+$ $k$ and it apears in all subarrays. It is just minimum over previously calculated values for all permutations.↵
↵
Now we can see that our result is $\sum\limits_{i = 1}^{n}{reach[i] - i}$. Final complexity $O(nm)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
typedef long long int LL;↵
↵
const int N = 1e5 + 7;↵
↵
int n, m;↵
int mn[N];↵
int ren[N];↵
int perm[15][N];↵
↵
int main(){↵
scanf("%d %d", &n, &m);↵
for(int i = 1; i <= m; ++i)↵
for(int j = 1; j <= n; ++j)↵
scanf("%d", &perm[i][j]);↵
↵
for(int i = 1; i <= n; ++i)↵
ren[perm[1][i]] = i;↵
↵
for(int i = 1; i <= m; ++i)↵
for(int j = 1; j <= n; ++j)↵
perm[i][j] = ren[perm[i][j]];↵
↵
for(int i = 1; i <= n; ++i)↵
mn[i] = n;↵
↵
for(int i = 1; i <= m; ++i){↵
int cur = 1;↵
for(int j = 1; j <= n; ++j){↵
if(cur < j)↵
++cur;↵
↵
while(cur < n && perm[i][cur + 1] == perm[i][cur] + 1)↵
++cur;↵
mn[perm[i][j]] = min(mn[perm[i][j]], perm[i][cur]);↵
}↵
}↵
↵
LL res = 0;↵
int now = 1;↵
while(now <= n){↵
int cur = mn[now] - now + 1;↵
res += 1LL * (cur + 1) * cur / 2LL;↵
now = mn[now] + 1;↵
}↵
↵
printf("%lld\n", res);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043E — Train Hard, Win Easy](http://codeforces.me/contest/1043/problem/E)↵
↵
<spoiler summary="Tutorial">↵
Let's compute result if there are no edges, we can add them later. If there are no edges then result for pair ($i$, $j$) is min($x_i$ $+$ $y_j$, $x_j$ $+$ $y_i$). First let's fix $i$ for which we want to compute result. Then calculate result with all pairs $j$ such that $x_i$ $+$ $y_j$ $\leqslant$ $x_j$ $+$ $y_i$. After some transformations we get that $x_i$ $-$ $y_i$ $\leqslant$ $x_j$ $-$ $y_j$. Similarly we have that $y_i$ $+$ $x_j$ $<$ $x_i$ $+$ $y_j$ if $x_i$ $-$ $y_i$ $>$ $y_j$ $-$ $x_j$.↵
↵
So let's sort over differences of $x_i$ $-$ $y_i$ and compute prefix sums of $x_i$ and suffix sums of $y_i$. Now we can compute for every $i$ result in $O(1)$. Then we can iterate over every edge ($u$, $v$) and subtract min($x_u$ $+$ $y_v$, $x_v$ $+$ $y_u$) from result of $u$ and $v$.↵
↵
Complexity $O(nlogn)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
typedef long long int LL;↵
↵
#define st first↵
#define nd second↵
#define PII pair <int, int>↵
↵
const int N = 3e5 + 7;↵
↵
int n, m;↵
PII diff[N];↵
int place[N];↵
vector <int> G[N];↵
↵
LL ans[N];↵
int x[N], y[N];↵
LL pref[N], suf[N];↵
↵
int main(){↵
scanf("%d %d", &n, &m);↵
for(int i = 1; i <= n; ++i){↵
scanf("%d %d", &x[i], &y[i]);↵
diff[i] = {y[i] - x[i], i};↵
}↵
↵
for(int i = 1; i <= m; ++i){↵
int u, v;↵
scanf("%d %d", &u, &v);↵
↵
G[u].push_back(v);↵
G[v].push_back(u);↵
}↵
↵
sort(diff + 1, diff + n + 1);↵
for(int i = 1; i <= n; ++i)↵
place[diff[i].nd] = i;↵
↵
for(int i = 1; i <= n; ++i)↵
pref[i] = pref[i - 1] + y[diff[i].nd];↵
↵
for(int i = n; i >= 1; --i)↵
suf[i] = suf[i + 1] + x[diff[i].nd];↵
↵
for(int i = 1; i <= n; ++i){↵
int u = diff[i].nd;↵
LL res = pref[i - 1] + suf[i + 1] + 1LL * (i - 1) * x[u] + 1LL * (n - i) * y[u];↵
↵
for(int v: G[u])↵
res -= min(x[u] + y[v], x[v] + y[u]);↵
ans[u] = res;↵
}↵
↵
for(int i = 1; i <= n; ++i)↵
printf("%lld ", ans[i]);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:Mateuszrze,2018-10-28]↵
↵
#### [1043F — Make It One](http://codeforces.me/contest/1043/problem/F)↵
↵
<spoiler summary="Tutorial">↵
First let's observe that if there exists valid subset then it's size is at most $7$ (because product of $7$ smallest primes is bigger then $3 * 10^5$).↵
Let's define $dp[i][j]$ — number of ways to pick $i$ different elements such that their gcd is equal to $j$. We can use inclusion--exclusion principle to calculate it. Then $dp[i][j]$ = $\binom{cnt_j}{i}$ — $\sum\limits_{k = 2 * j}^{\infty}{dp[i][k]}$, where $cnt_j$ denotes number of $a_i$ such that $j$ $|$ $a_i$. Because for $k$ $>$ $3 * 10^5$, $dp[i][k]$ $=$ $0$ we have to check only $k$ $\leqslant$ $3 * 10^5$.↵
↵
Our answer is the smallest $i$ such that $dp[i][1]$ is non-zero. Since $dp[i][j]$ can be quite big we should compute it modulo some big prime.↵
↵
Final complexity is $O(logM * (n + M))$, where M is equal to maximum of $a_i$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution 1">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 3e5 + 7;↵
const int MX = 1e9 + 7;↵
↵
int n;↵
int cnt[N];↵
int sil[N];↵
int odw[N];↵
int dp[20][N];↵
↵
int fast(int a, int b){↵
int ret = 1;↵
while(b){↵
if(b & 1)↵
ret = (1LL * ret * a)%MX;↵
↵
b >>= 1;↵
a = (1LL * a * a)%MX;↵
}↵
↵
return ret;↵
}↵
↵
int newton(int a, int b){↵
if(b < 0 || a < b) return 0;↵
return (((1LL * sil[a] * odw[b])%MX) * odw[a - b])%MX;↵
}↵
↵
void sub(int &a, int b){↵
a -= b;↵
if(a < 0)↵
a += MX;↵
}↵
↵
int main(){↵
sil[0] = 1;↵
for(int i = 1; i < N; ++i)↵
sil[i] = (1LL * sil[i - 1] * i)%MX;↵
↵
odw[N - 1] = fast(sil[N - 1], MX - 2);↵
for(int i = N - 1; i >= 1; --i)↵
odw[i - 1] = (1LL * odw[i] * i)%MX;↵
↵
scanf("%d", &n);↵
for(int i = 1; i <= n; ++i){↵
int a;↵
scanf("%d", &a);↵
cnt[a]++;↵
}↵
↵
for(int i = 1; i < N; ++i)↵
for(int j = i + i; j < N; j += i)↵
cnt[i] += cnt[j];↵
↵
for(int i = 1; i < 20; ++i){↵
for(int j = N - 1; j >= 1; --j){↵
dp[i][j] = newton(cnt[j], i);↵
for(int k = j + j; k < N; k += j)↵
sub(dp[i][j], dp[i][k]);↵
}↵
↵
if(dp[i][1] > 0){↵
printf("%d\n", i);↵
return 0;↵
}↵
}↵
↵
puts("-1");↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
<spoiler summary="Solution 2">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 3e5 + 7;↵
↵
int n;↵
int cnt[N];↵
int roz[N];↵
int dist[N];↵
queue <int> Q;↵
vector <int> dv[N];↵
↵
int base(int a){↵
int ret = 1;↵
while(a > 1){↵
if(ret%roz[a] != 0)↵
ret *= roz[a];↵
a /= roz[a];↵
}↵
↵
return ret;↵
}↵
↵
void getEdges(int u, int d){↵
vector <int> cur;↵
vector <int> val;↵
↵
while(u > 1){↵
cur.push_back(roz[u]);↵
u /= roz[u];↵
}↵
↵
for(int v: cur)↵
u *= v;↵
↵
int T = 1 << (int)cur.size();↵
val.resize(T);↵
↵
for(int i = 0; i < T; ++i){↵
val[i] = u;↵
for(int j = 0; j < (int)cur.size(); ++j)↵
if(i & (1 << j))↵
val[i] /= cur[j];↵
}↵
↵
for(int i = 0; i < T; ++i){↵
int s = 0;↵
for(int j = i; true; j = (j - 1) & i){↵
if(__builtin_popcount(i ^ j) & 1)↵
s -= cnt[val[j]];↵
else↵
s += cnt[val[j]];↵
↵
if(j == 0)↵
break;↵
}↵
↵
assert(s >= 0);↵
if(s && dist[val[i]] == -1){↵
dist[val[i]] = d;↵
Q.push(val[i]);↵
}↵
}↵
}↵
↵
int main(){↵
scanf("%d", &n);↵
for(int i = 2; i < N; ++i){↵
if(roz[i] != 0)↵
continue;↵
↵
for(int j = i; j < N; j += i)↵
roz[j] = i;↵
}↵
↵
for(int i = 1; i < N; ++i)↵
for(int j = i; j < N; j += i)↵
dv[j].push_back(i);↵
↵
for(int i = 1; i < N; ++i)↵
dist[i] = -1;↵
↵
for(int i = 1; i <= n; ++i){↵
int a;↵
scanf("%d", &a);↵
a = base(a);↵
↵
if(dist[a] != -1)↵
continue;↵
↵
dist[a] = 1;↵
Q.push(a);↵
↵
for(int v: dv[a])↵
cnt[v]++;↵
}↵
↵
while(!Q.empty()){↵
int u = Q.front();↵
Q.pop();↵
getEdges(u, dist[u] + 1);↵
}↵
↵
printf("%d\n", dist[1]);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043G — Speckled Band](http://codeforces.me/contest/1043/problem/G)↵
↵
<spoiler summary="Tutorial">↵
Let's solve the problem for some string $s$ for any time.↵
↵
Let's say, that partition of string $s$ into $k$ strings $s_1 s_2 \ldots s_{i_1}, s_{i_1 + 1} \ldots s_{i_2}, \ldots, {s_{i_{k-1} + 1}} \ldots s_{i_k}$ is good if at least one pair of this strings are equal. We want to find a minimal possible number of different strings in all good partitions.↵
↵
It's easy to see, that the answer is $-1$ if and only if all symbols in $s$ are different. And if we have two equal symbols $s_i = s_j$ ($i < j$) we can cut a string into strings $s_1 \ldots s_{i-1}, s_i, s_{i+1} \ldots s_{j-1}, s_j, s_{j+1} \ldots s_n$ and it is a good partition. In this partition there is at most $4$ different strings. ↵
↵
So the answer can be $-1$, $1$, $2$, $3$, $4$.↵
↵
The answer is $-1$ if all symbols in $s$ are different (case $0$).↵
↵
The answer is $1$ if the string $s = aaa \ldots a$, for some string $a$ (case $1$).↵
↵
The answer is $2$ if the string $s$ is $aab$, $aba$ or $baa$ for some strings $a$ and $b$ (case $2$). ↵
↵
The answer is $3$ if the string $s$ is $baac$, $bcaa$ or $aabc$ for some strings $a$, $b$, $c$. In two last cases it's easy to see, that $|a|=1$ (case $3$).↵
↵
To solve our problem let's build suffix array with lcp for string $s$. And let's find $lt_i$~--- minimal possible number $r$, such that $s_i s_{i+1} \ldots s_{r}$ is a tandem (the string, that can be presented as $aa$ for some string $a$) and $rt_i$~--- maximal possible number $l$ such that $s_l s_{l+1} \ldots s_{i}$ is a tandem. This numbers can be found using Main and Lorentz algorithm for finding tandem repetitions in the string.↵
↵
Now we can solve query for segment $[l, r]$:↵
\begin{itemize}↵
\item Case $0$: if $r - l \geq 26$, there exists equal symbols, otherwise we can check it by $O(r - l)$;↵
\item Case $1$: to check that $s[l \ldots r] = aa \ldots a$ we can see that $|a|$ is a divisor of $(r - l + 1)$ and $(r - l + 1) / |a|$ is a prime number (if we take a longest possible string $a$). So we should check only $O(log(n))$ lenghts of string $a$;↵
\item Case $2$: $s = aab$ $\Longleftrightarrow$ $lt_l \leq r$, $s = baa$ $\Longleftrightarrow$ $rt_r \geq l$. In the last case we should check, that $s[l \ldots r]$ has a border. It's the most interesting part of the problem, let's solve it in the end;↵
\item Case $3$: $s = abac$ $\Longleftrightarrow$ $s_l$ exists on $s_{l+1} \ldots s_r$ (can be done using prefix sums), $s = baca$ $\Longleftrightarrow$ $s_r$ exists on $s_l \ldots s_{r-1}$ (can be done using prefix sums). To check $s = baac$ we can check, that $lt_i \leq r$ for some $l \leq i \leq r$, that can be done using minimum on segment in the array $lt$.↵
\end{itemize}↵
↵
Now we should the hardest part of this problem~--- we have some segments $[l, r]$. For all of them, we should check that the border of $s[l \ldots r]$ exists. Here I know two methods, that uses only suffix array. Easiest of them:↵
↵
We have segment $[l, r]$. Let's check for all lengths $b \leq \sqrt{n}$, that $s[l \ldots (l + b - 1)] = s[(r - b + 1) \ldots r]$. If we don't find border, if it exists, it's length $> \sqrt{n}$. Let's define $i$~--- maximal index $i$ such that $lcp(l, i) \geq r - i + 1$, and string $s[i \ldots r]$ is a border of $s[l \ldots r]$. So $lcp(l, i) > \sqrt{n}$. But it's easy to see, that the distance between $l$ and $i$ in suffix array $\leq \sqrt{n}$ , so we need to check only $O(\sqrt{n})$ variants of $i$.↵
↵
Another method can check that border exists for all segments $[l, r]$ using offline algorithm by $O(q \cdot log(n)^2)$ time.↵
↵
So the total complexity will be $O((n + q) \cdot \sqrt{n})$ or $O((n + q) \cdot log(n)^2)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution 1">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int BIG = 1e9 + 239;↵
const int M = 2 * 1e5 + 239;↵
const int L = 19;↵
const int A = 30;↵
const int T = (1 << 19);↵
const int two = 2;↵
↵
int flm[two][M];↵
↵
inline void z_function(string &s, int c)↵
{↵
int n = s.length();↵
flm[c][0] = 0;↵
int l = 0;↵
int r = 0;↵
for (int i = 1; i < n; i++)↵
{↵
flm[c][i] = min(flm[c][i - l], r - i);↵
if (flm[c][i] < 0) flm[c][i] = 0;↵
while (i + flm[c][i] < n && s[flm[c][i]] == s[i + flm[c][i]]) flm[c][i]++;↵
if (i + flm[c][i] > r)↵
{↵
l = i;↵
r = i + flm[c][i];↵
}↵
} ↵
}↵
↵
int a[M], lcp[M], pos[M];↵
↵
inline void suffix_array(string s)↵
{↵
s += (char)(31);↵
int n = s.length();↵
vector<pair<char, int> > v;↵
for (int i = 0; i < n; i++)↵
v.push_back(make_pair(s[i], i));↵
sort(v.begin(), v.end());↵
vector<pair<int, int> > num;↵
int last = 0;↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, v[i].second));↵
if (v[i].first != v[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, v.back().second));↵
vector<int> u(n);↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
int d = 1;↵
vector<pair<pair<int, int>, int> > t;↵
vector<vector<pair<int, int> > > h;↵
while (d < n)↵
{↵
t.clear();↵
h.clear();↵
h.resize(n);↵
for (int i = 0; i < n; i++)↵
{↵
int l = num[i].second - d;↵
if (l < 0) l += n;↵
h[u[l]].push_back(make_pair(num[i].first, l));↵
}↵
for (int i = 0; i < n; i++)↵
for (pair<int, int> r : h[i])↵
t.push_back(make_pair(make_pair(i, r.first), r.second));↵
last = 0;↵
num.clear();↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, t[i].second));↵
if (t[i].first != t[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, t.back().second));↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
d <<= 1;↵
} ↵
for (int i = 1; i < n; i++) a[i - 1] = num[i].second;↵
}↵
↵
string s; ↵
↵
inline void kasai()↵
{↵
int n = s.size();↵
suffix_array(s); ↵
for (int i = 0; i < n; i++)↵
pos[a[i]] = i;↵
int k = 0;↵
for (int i = 0; i < n; i++)↵
{↵
if (pos[i] == n - 1) continue;↵
while (s[i + k] == s[a[pos[i] + 1] + k] && a[pos[i] + 1] + k < n && i + k < n) k++;↵
lcp[pos[i]] = k;↵
k = max(0, k - 1);↵
}↵
}↵
↵
int n, lw[M], rw[M]; ↵
string prr, rvl; ↵
vector<int> open_l[M], close_l[M]; ↵
vector<int> open_r[M], close_r[M];↵
multiset<int> nw;↵
↵
inline void func(int l, int r)↵
{↵
if (r - l == 1) return;↵
int mid = (l + r) >> 1;↵
func(l, mid);↵
func(mid, r);↵
rvl = "";↵
for (int i = mid - 1; i >= l; i--) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid; i < r; i++) prr += s[i];↵
prr += '#';↵
for (int i = l; i < mid; i++) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = l; c < mid; c++)↵
{↵
int k1 = 0;↵
if (c > l) k1 = flm[0][mid - c];↵
int k2 = flm[1][r - mid + 1 + c - l];↵
int len = mid - c;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{↵
open_l[c - rg].push_back((2 * len));↵
close_l[c - lg].push_back((2 * len));↵
open_r[c - rg + 2 * len - 1].push_back((2 * len));↵
close_r[c - lg + 2 * len - 1].push_back((2 * len));↵
} ↵
}↵
rvl = "";↵
for (int i = mid; i < r; i++) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid - 1; i >= l; i--) prr += s[i];↵
prr += '#';↵
for (int i = r - 1; i >= mid; i--) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = mid; c < r; c++)↵
{↵
int k1 = 0;↵
if (c != r - 1) k1 = flm[0][c + 1 - mid];↵
int k2 = flm[1][r - c + mid - l];↵
int len = c - mid + 1;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{ ↵
open_l[c + lg - 2 * len + 1].push_back((2 * len));↵
close_l[c + rg - 2 * len + 1].push_back((2 * len));↵
open_r[c + lg].push_back((2 * len));↵
close_r[c + rg].push_back((2 * len));↵
} ↵
}↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_l[i]) nw.insert(len);↵
if (!nw.empty()) lw[i] = min(lw[i], *nw.begin());↵
for (int len : close_l[i]) nw.erase(nw.lower_bound(len));↵
open_l[i].clear();↵
close_l[i].clear();↵
} ↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_r[i]) nw.insert(len);↵
if (!nw.empty()) rw[i] = min(rw[i], *nw.begin());↵
for (int len : close_r[i]) nw.erase(nw.lower_bound(len));↵
open_r[i].clear();↵
close_r[i].clear();↵
}↵
}↵
↵
int mn[L][M], st2[M], lc[L][M], kol[A][M];↵
↵
inline int getmin(int l, int r)↵
{↵
int u = st2[r - l + 1];↵
return min(mn[u][l], mn[u][r - (1 << u) + 1]);↵
}↵
↵
inline int gett(int l, int r)↵
{↵
if (l == r) return (n - l);↵
l = pos[l];↵
r = pos[r];↵
if (l > r) swap(l, r);↵
r--;↵
int u = st2[r - l + 1];↵
return min(lc[u][l], lc[u][r - (1 << u) + 1]);↵
}↵
↵
int q, la[M], ra[M], has[M], mnp[M];↵
↵
inline void init()↵
{↵
for (int i = 0; i < n; i++) mn[0][i] = lw[i];↵
for (int i = 0; i < n - 1; i++) lc[0][i] = lcp[i];↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n)↵
{↵
mn[i][j] = mn[i - 1][j];↵
continue;↵
}↵
mn[i][j] = min(mn[i - 1][j], mn[i - 1][r]);↵
}↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n - 1; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n - 1)↵
{↵
lc[i][j] = lc[i - 1][j];↵
continue;↵
} ↵
lc[i][j] = min(lc[i - 1][j], lc[i - 1][r]);↵
} ↵
st2[1] = 0;↵
for (int i = 2; i <= n; i++)↵
{↵
st2[i] = st2[i - 1];↵
if ((1 << (st2[i] + 1)) <= i) st2[i]++;↵
} ↵
}↵
↵
int in[L][M], gl[T];↵
↵
inline void build(int i, int l, int r)↵
{↵
if (i == 0) gl[i] = 0;↵
else gl[i] = gl[(i - 1) / 2] + 1;↵
if (r - l == 1)↵
{↵
in[gl[i]][l] = a[l];↵
return;↵
}↵
int mid = (l + r) >> 1;↵
build(2 * i + 1, l, mid);↵
build(2 * i + 2, mid, r);↵
merge(in[gl[i] + 1] + l, in[gl[i] + 1] + mid, in[gl[i] + 1] + mid, in[gl[i] + 1] + r, in[gl[i]] + l);↵
}↵
↵
inline bool is_on(int i, int l, int r, int ql, int qr, int xl, int xr)↵
{↵
if (r <= ql || qr <= l) return false;↵
if (ql <= l && r <= qr)↵
{↵
int it = upper_bound(in[gl[i]] + l, in[gl[i]] + r, xl) - in[gl[i]] - l;↵
if (it == r - l) return false;↵
return (in[gl[i]][it + l] <= xr); ↵
}↵
int mid = (l + r) >> 1;↵
if (is_on(2 * i + 1, l, mid, ql, qr, xl, xr)) return true;↵
return is_on(2 * i + 2, mid, r, ql, qr, xl, xr);↵
}↵
↵
inline bool check_all(int l, int r)↵
{↵
int len = (r - l + 1);↵
int pl = pos[l];↵
int lf = pl;↵
int rf = n;↵
while (rf - lf > 1)↵
{↵
int h = (lf + rf) / 2;↵
if (gett(a[h], l) >= len) lf = h;↵
else rf = h;↵
}↵
int rg = lf + 1;↵
lf = -1;↵
rf = pl;↵
while (rf - lf > 1)↵
{↵
int h = (lf + rf) / 2;↵
if (gett(a[h], l) >= len) rf = h;↵
else lf = h;↵
}↵
int lg = rf;↵
return is_on(0, 0, n, lg, rg, l, r);↵
}↵
↵
int par[M];↵
set<int> cmp[M];↵
set<pair<int, int> > sc[M];↵
vector<int> ok[M];↵
vector<pair<int, int> > mg[M];↵
↵
inline void border_check()↵
{↵
for (int i = 0; i < q; i++) has[i] = false;↵
for (int i = 0; i < q; i++) ok[ra[i] - la[i]].push_back(i);↵
for (int i = 0; i < n - 1; i++) mg[lcp[i]].push_back(make_pair(a[i], a[i + 1]));↵
for (int i = 0; i < n; i++)↵
{↵
par[i] = i;↵
cmp[i].insert(i);↵
}↵
for (int c = n; c >= 1; c--)↵
{↵
for (int i : ok[c - 1])↵
sc[par[la[i]]].insert(make_pair(ra[i], i)); ↵
for (pair<int, int> t : mg[c])↵
{↵
int l = par[t.first];↵
int r = par[t.second];↵
if (cmp[l].size() > cmp[r].size()) swap(l, r);↵
for (int x : cmp[l])↵
while (true)↵
{↵
auto it = sc[r].lower_bound(make_pair(x, 0));↵
if (it == sc[r].end() || it->first > x + c - 1) break;↵
has[it->second] = true;↵
sc[r].erase(it);↵
}↵
for (pair<int, int> u : sc[l])↵
{↵
int id = u.second;↵
auto uk = cmp[r].lower_bound(ra[id] - c + 1);↵
if (uk != cmp[r].end() && (*uk) <= ra[id]) ↵
has[id] = true;↵
else ↵
sc[r].insert(u);↵
}↵
for (int x : cmp[l])↵
{↵
par[x] = r;↵
cmp[r].insert(x);↵
}↵
} ↵
} ↵
for (int i = 0; i < q; i++)↵
if (!has[i] && check_all(la[i], ra[i]))↵
{↵
has[i] = true;↵
continue;↵
}↵
}↵
↵
inline bool checkno(int l, int r)↵
{↵
if (r - l + 1 > 26) return false;↵
vector<int> kol(26, 0);↵
for (int x = l; x <= r; x++)↵
{↵
if (kol[s[x] - 'a'] > 0) return false;↵
kol[s[x] - 'a']++;↵
} ↵
return true;↵
}↵
↵
inline bool try_kol(int l, int r, int p)↵
{↵
int len = (r - l + 1) / p;↵
return (gett(l, l + len) >= (r - l + 1 - len));↵
}↵
↵
inline bool ison(int l, int r, char x)↵
{↵
return (kol[x - 'a'][r + 1] > kol[x - 'a'][l]); ↵
}↵
↵
inline int query(int l, int r, int id)↵
{ ↵
if (checkno(l, r)) return -1;↵
int len = (r - l + 1);↵
while (len > 1)↵
{↵
int p = mnp[len];↵
if (try_kol(l, r, p)) return 1;↵
while ((len % p) == 0) len /= p;↵
}↵
if (lw[l] <= r) return 2;↵
if (rw[r] >= l) return 2;↵
if (has[id]) return 2;↵
if (ison(l + 1, r, s[l])) return 3;↵
if (ison(l, r - 1, s[r])) return 3;↵
if (getmin(l, r) <= r) return 3; ↵
return 4;↵
}↵
↵
int main()↵
{↵
cin.sync_with_stdio(0); ↵
cin >> n >> s;↵
memset(mnp, -1, sizeof(mnp));↵
for (int i = 2; i <= n; i++)↵
if (mnp[i] == -1)↵
for (int j = i; j <= n; j += i)↵
if (mnp[j] == -1)↵
mnp[j] = i;↵
memset(kol[0], 0, sizeof(kol[0]));↵
for (int i = 0; i < n; i++)↵
{↵
for (int x = 0; x < 26; x++) kol[x][i + 1] = kol[x][i];↵
kol[s[i] - 'a'][i + 1]++;↵
}↵
for (int i = 0; i < n; i++)↵
{↵
lw[i] = n + 1;↵
rw[i] = n + 1;↵
}↵
func(0, n); ↵
for (int i = 0; i < n; i++)↵
{↵
if (lw[i] == n + 1) lw[i] = n;↵
else lw[i] += (i - 1); ↵
if (rw[i] == n + 1) rw[i] = -1;↵
else rw[i] = (i - rw[i] + 1);↵
}↵
kasai(); ↵
init(); ↵
build(0, 0, n); ↵
cin >> q;↵
for (int i = 0; i < q; i++)↵
{↵
cin >> la[i] >> ra[i];↵
la[i]--, ra[i]--; ↵
}↵
border_check();↵
for (int i = 0; i < q; i++) cout << query(la[i], ra[i], i) << "\n";↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
↵
<spoiler summary="Solution 2">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
#define TIME (clock() * 1.0 / CLOCKS_PER_SEC)↵
↵
const int BIG = 1e9 + 239;↵
const int M = 2 * 1e5 + 239;↵
const int L = 19;↵
const int A = 30;↵
const int T = (1 << 19);↵
const int two = 2;↵
↵
int flm[two][M];↵
↵
inline void z_function(string &s, int c)↵
{↵
int n = s.length();↵
flm[c][0] = 0;↵
int l = 0;↵
int r = 0;↵
for (int i = 1; i < n; i++)↵
{↵
flm[c][i] = min(flm[c][i - l], r - i);↵
if (flm[c][i] < 0) flm[c][i] = 0;↵
while (i + flm[c][i] < n && s[flm[c][i]] == s[i + flm[c][i]]) flm[c][i]++;↵
if (i + flm[c][i] > r)↵
{↵
l = i;↵
r = i + flm[c][i];↵
}↵
} ↵
}↵
↵
int a[M], lcp[M], pos[M];↵
↵
inline void suffix_array(string s)↵
{↵
s += (char)(31);↵
int n = s.length();↵
vector<pair<char, int> > v;↵
for (int i = 0; i < n; i++)↵
v.push_back(make_pair(s[i], i));↵
sort(v.begin(), v.end());↵
vector<pair<int, int> > num;↵
int last = 0;↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, v[i].second));↵
if (v[i].first != v[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, v.back().second));↵
vector<int> u(n);↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
int d = 1;↵
vector<pair<pair<int, int>, int> > t;↵
vector<vector<pair<int, int> > > h;↵
while (d < n)↵
{↵
t.clear();↵
h.clear();↵
h.resize(n);↵
for (int i = 0; i < n; i++)↵
{↵
int l = num[i].second - d;↵
if (l < 0) l += n;↵
h[u[l]].push_back(make_pair(num[i].first, l));↵
}↵
for (int i = 0; i < n; i++)↵
for (pair<int, int> r : h[i])↵
t.push_back(make_pair(make_pair(i, r.first), r.second));↵
last = 0;↵
num.clear();↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, t[i].second));↵
if (t[i].first != t[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, t.back().second));↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
d <<= 1;↵
} ↵
for (int i = 1; i < n; i++) a[i - 1] = num[i].second;↵
}↵
↵
string s; ↵
↵
inline void kasai()↵
{↵
int n = s.size();↵
suffix_array(s); ↵
for (int i = 0; i < n; i++)↵
pos[a[i]] = i;↵
int k = 0;↵
for (int i = 0; i < n; i++)↵
{↵
if (pos[i] == n - 1) continue;↵
while (s[i + k] == s[a[pos[i] + 1] + k] && a[pos[i] + 1] + k < n && i + k < n) k++;↵
lcp[pos[i]] = k;↵
k = max(0, k - 1);↵
}↵
}↵
↵
int n, lw[M], rw[M]; ↵
string prr, rvl; ↵
vector<int> open_l[M], close_l[M]; ↵
vector<int> open_r[M], close_r[M];↵
multiset<int> nw;↵
↵
inline void func(int l, int r)↵
{↵
if (r - l == 1) return;↵
int mid = (l + r) >> 1;↵
func(l, mid);↵
func(mid, r);↵
rvl = "";↵
for (int i = mid - 1; i >= l; i--) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid; i < r; i++) prr += s[i];↵
prr += '#';↵
for (int i = l; i < mid; i++) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = l; c < mid; c++)↵
{↵
int k1 = 0;↵
if (c > l) k1 = flm[0][mid - c];↵
int k2 = flm[1][r - mid + 1 + c - l];↵
int len = mid - c;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{↵
open_l[c - rg].push_back((2 * len));↵
close_l[c - lg].push_back((2 * len));↵
open_r[c - rg + 2 * len - 1].push_back((2 * len));↵
close_r[c - lg + 2 * len - 1].push_back((2 * len));↵
} ↵
}↵
rvl = "";↵
for (int i = mid; i < r; i++) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid - 1; i >= l; i--) prr += s[i];↵
prr += '#';↵
for (int i = r - 1; i >= mid; i--) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = mid; c < r; c++)↵
{↵
int k1 = 0;↵
if (c != r - 1) k1 = flm[0][c + 1 - mid];↵
int k2 = flm[1][r - c + mid - l];↵
int len = c - mid + 1;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{ ↵
open_l[c + lg - 2 * len + 1].push_back((2 * len));↵
close_l[c + rg - 2 * len + 1].push_back((2 * len));↵
open_r[c + lg].push_back((2 * len));↵
close_r[c + rg].push_back((2 * len));↵
} ↵
}↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_l[i]) nw.insert(len);↵
if (!nw.empty()) lw[i] = min(lw[i], *nw.begin());↵
for (int len : close_l[i]) nw.erase(nw.lower_bound(len));↵
open_l[i].clear();↵
close_l[i].clear();↵
} ↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_r[i]) nw.insert(len);↵
if (!nw.empty()) rw[i] = min(rw[i], *nw.begin());↵
for (int len : close_r[i]) nw.erase(nw.lower_bound(len));↵
open_r[i].clear();↵
close_r[i].clear();↵
}↵
}↵
↵
int mn[L][M], st2[M], lc[L][M], kol[A][M];↵
↵
inline int getmin(int l, int r)↵
{↵
int u = st2[r - l + 1];↵
return min(mn[u][l], mn[u][r - (1 << u) + 1]);↵
}↵
↵
inline int gett(int l, int r)↵
{↵
if (l == r) return (n - l);↵
l = pos[l];↵
r = pos[r];↵
if (l > r) swap(l, r);↵
r--;↵
int u = st2[r - l + 1];↵
return min(lc[u][l], lc[u][r - (1 << u) + 1]);↵
}↵
↵
int q, mnp[M];↵
↵
inline void init()↵
{↵
for (int i = 0; i < n; i++) mn[0][i] = lw[i];↵
for (int i = 0; i < n - 1; i++) lc[0][i] = lcp[i];↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n)↵
{↵
mn[i][j] = mn[i - 1][j];↵
continue;↵
}↵
mn[i][j] = min(mn[i - 1][j], mn[i - 1][r]);↵
}↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n - 1; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n - 1)↵
{↵
lc[i][j] = lc[i - 1][j];↵
continue;↵
} ↵
lc[i][j] = min(lc[i - 1][j], lc[i - 1][r]);↵
} ↵
st2[1] = 0;↵
for (int i = 2; i <= n; i++)↵
{↵
st2[i] = st2[i - 1];↵
if ((1 << (st2[i] + 1)) <= i) st2[i]++;↵
} ↵
}↵
↵
int u, len;↵
↵
inline bool check(int l, int r)↵
{↵
for (int t = 1; t <= min(len, ((r - l + 1) / 2)); t++)↵
if (gett(l, r - t + 1) >= t)↵
return true;↵
for (int i = max(0, pos[l] - len); i <= min(n - 1, pos[l] + len); i++)↵
if (l < a[i] && a[i] <= r && gett(l, a[i]) >= r - a[i] + 1) ↵
return true;↵
return false;↵
}↵
↵
inline bool checkno(int l, int r)↵
{↵
if (r - l + 1 > 26) return false;↵
vector<int> kol(26, 0);↵
for (int x = l; x <= r; x++)↵
{↵
if (kol[s[x] - 'a'] > 0) return false;↵
kol[s[x] - 'a']++;↵
} ↵
return true;↵
}↵
↵
inline bool try_kol(int l, int r, int p)↵
{↵
int len = (r - l + 1) / p;↵
return (gett(l, l + len) >= (r - l + 1 - len));↵
}↵
↵
inline bool ison(int l, int r, char x)↵
{↵
return (kol[x - 'a'][r + 1] > kol[x - 'a'][l]); ↵
}↵
↵
inline int query(int l, int r)↵
{ ↵
if (checkno(l, r)) return -1;↵
int len = (r - l + 1);↵
while (len > 1)↵
{↵
int p = mnp[len];↵
if (try_kol(l, r, p)) return 1;↵
while ((len % p) == 0) len /= p;↵
}↵
if (lw[l] <= r) return 2;↵
if (rw[r] >= l) return 2;↵
if (check(l, r)) return 2;↵
if (ison(l + 1, r, s[l])) return 3;↵
if (ison(l, r - 1, s[r])) return 3;↵
if (getmin(l, r) <= r) return 3; ↵
return 4;↵
}↵
↵
int main()↵
{↵
cin.sync_with_stdio(0); ↵
cin >> n >> s;↵
memset(mnp, -1, sizeof(mnp));↵
for (int i = 2; i <= n; i++)↵
if (mnp[i] == -1)↵
for (int j = i; j <= n; j += i)↵
if (mnp[j] == -1)↵
mnp[j] = i;↵
memset(kol[0], 0, sizeof(kol[0]));↵
for (int i = 0; i < n; i++)↵
{↵
for (int x = 0; x < 26; x++) kol[x][i + 1] = kol[x][i];↵
kol[s[i] - 'a'][i + 1]++;↵
}↵
for (int i = 0; i < n; i++)↵
{↵
lw[i] = n + 1;↵
rw[i] = n + 1;↵
}↵
func(0, n); ↵
for (int i = 0; i < n; i++)↵
{↵
if (lw[i] == n + 1) lw[i] = n;↵
else lw[i] += (i - 1); ↵
if (rw[i] == n + 1) rw[i] = -1;↵
else rw[i] = (i - rw[i] + 1);↵
}↵
kasai(); ↵
init(); ↵
cin >> q;↵
for (int i = 0; i < n; i++)↵
if (i * i >= n)↵
{↵
len = i + 3;↵
break;↵
}↵
for (int i = 0; i < q; i++)↵
{↵
int l, r;↵
cin >> l >> r;↵
l--, r--;↵
cout << query(l, r) << "\n"; ↵
}↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:isaf27,2018-10-28]
↵
<spoiler summary="Tutorial">↵
We can observe that result cannot exceed $201$ — Awruk gets at least $101$ votes from one person and Elodreip cannot get more than $100$ votes from one person. So we can iterate over every possible integer from $1$ to $201$ and check if Awruk wins with $k$ set to this integer. We have to remember that $k$ — $a_i$ is always at least $0$, so we have to check this condition too. Complexity $O(n * M)$, where $M$ denotes maximum possible value of $a_i$. Try to solve it in $O(n)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
int n;↵
int mx = 0, sum = 0;↵
↵
int main(){↵
scanf("%d", &n);↵
for(int i = 1; i <= n; ++i){↵
int a; scanf("%d", &a);↵
mx = max(mx, a);↵
sum += a;↵
}↵
↵
sum *= 2;↵
sum += n;↵
sum /= n;↵
↵
printf("%d", max(sum, mx));↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043B — Lost Array](http://codeforces.me/contest/1043/problem/B)↵
↵
<spoiler summary="Tutorial">↵
First, let's observe that we can replace array $a_i$ with array $b_i$ = $a_i$ $-$ $a_{i-1}$, because all we care about are differences between neighboring elements.↵
Now, we can see that our lost array can have length $d$ if and only if for every $j$ such that $j$ $+$ $d$ $\leqslant$ $n$, $b_j$ $=$ $b_{j+d}$.↵
So we can iterate over every possible $d$ from $1$ to $n$ and check if it is correct in $O(n)$. Complexity of whole algorithm is $O(n^2)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 1007;↵
↵
int n;↵
int in[N];↵
↵
bool ok(int d){↵
for(int i = 0; i + d < n; ++i)↵
if(in[i + 1] - in[i] != in[i + d + 1] - in[i + d])↵
return false;↵
return true;↵
}↵
↵
int main(){↵
scanf("%d", &n);↵
for(int i = 1; i <= n; ++i)↵
scanf("%d", &in[i]);↵
↵
vector <int> res;↵
for(int i = 1; i <= n; ++i)↵
if(ok(i))↵
res.push_back(i);↵
↵
printf("%d\n", res.size());↵
for(int v: res)↵
printf("%d ", v);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043C — Smallest Word](http://codeforces.me/contest/1043/problem/C)↵
↵
<spoiler summary="Tutorial">↵
Basically in problem we are given a word in which for every $i$ we can reverse prefix of first $i$ elements and we want to get the smallest lexicographically word. We will show that we can always achieve word in form $a^{j}b^{n-j}$.↵
↵
Let's say that we solved our problem for prefix of length $i$ and for this prefix we have word $a^{j}b^{i-j}$ (at the beginning it's just empty word). If our next letter is $b$ then we do nothing, because we will get word $a^{j}b^{i-j+1}$ which is still the smallest lexicographically word. Otherwise we want to reverse prefix of length $i$, add letter $a$ and reverse prefix of length $i$ $+$ $1$, so we get a word $a^{j+1}b^{i-j}$, which is still fine for us.↵
↵
There is still a problem — what if we have already reversed prefix $i$ and we just said that we will reverse it second time. But instead of reversing it second time, we can deny it's first reverse.↵
↵
Final complexity is $O(n)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 1007;↵
↵
string s;↵
bool write[N];↵
↵
int main(){↵
cin >> s;↵
for(int i = 1; i < s.size(); ++i)↵
if(s[i] == 'a'){↵
write[i - 1] ^= 1;↵
write[i] = 1;↵
}↵
↵
for(int i = 0; i < s.size(); ++i)↵
printf("%d%c", write[i], i + 1 == (int)s.size() ? '\n' : ' ');↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043D — Mysterious Crime](http://codeforces.me/contest/1043/problem/D)↵
↵
<spoiler summary="Tutorial">↵
Deleting prefix and suffix is nothing more than taking a subarray. If subarray is common for all permutations then it has to appear in first permutation. We renumber all permutations such that first permutation is $1$, $2$, ..., $n$ $-$ $1$, $n$.↵
↵
Now for every $i$ in every permutation we count how long is subarray starting at $i$ which looks like $i$, $i$ $+$ $1$, ..., $i$ $+$ $k$. It can be easily done in $O(n)$ for one permutation with two pointers technique.↵
↵
Now for every element $i$ we compute $reach[i]$ equal the longest subarray starting in $i$ which looks like $i$, $i$ $+$ $1$, ..., $i$ $+$ $k$ and it apears in all subarrays. It is just minimum over previously calculated values for all permutations.↵
↵
Now we can see that our result is $\sum\limits_{i = 1}^{n}{reach[i] - i}$. Final complexity $O(nm)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
typedef long long int LL;↵
↵
const int N = 1e5 + 7;↵
↵
int n, m;↵
int mn[N];↵
int ren[N];↵
int perm[15][N];↵
↵
int main(){↵
scanf("%d %d", &n, &m);↵
for(int i = 1; i <= m; ++i)↵
for(int j = 1; j <= n; ++j)↵
scanf("%d", &perm[i][j]);↵
↵
for(int i = 1; i <= n; ++i)↵
ren[perm[1][i]] = i;↵
↵
for(int i = 1; i <= m; ++i)↵
for(int j = 1; j <= n; ++j)↵
perm[i][j] = ren[perm[i][j]];↵
↵
for(int i = 1; i <= n; ++i)↵
mn[i] = n;↵
↵
for(int i = 1; i <= m; ++i){↵
int cur = 1;↵
for(int j = 1; j <= n; ++j){↵
if(cur < j)↵
++cur;↵
↵
while(cur < n && perm[i][cur + 1] == perm[i][cur] + 1)↵
++cur;↵
mn[perm[i][j]] = min(mn[perm[i][j]], perm[i][cur]);↵
}↵
}↵
↵
LL res = 0;↵
int now = 1;↵
while(now <= n){↵
int cur = mn[now] - now + 1;↵
res += 1LL * (cur + 1) * cur / 2LL;↵
now = mn[now] + 1;↵
}↵
↵
printf("%lld\n", res);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043E — Train Hard, Win Easy](http://codeforces.me/contest/1043/problem/E)↵
↵
<spoiler summary="Tutorial">↵
Let's compute result if there are no edges, we can add them later. If there are no edges then result for pair ($i$, $j$) is min($x_i$ $+$ $y_j$, $x_j$ $+$ $y_i$). First let's fix $i$ for which we want to compute result. Then calculate result with all pairs $j$ such that $x_i$ $+$ $y_j$ $\leqslant$ $x_j$ $+$ $y_i$. After some transformations we get that $x_i$ $-$ $y_i$ $\leqslant$ $x_j$ $-$ $y_j$. Similarly we have that $y_i$ $+$ $x_j$ $<$ $x_i$ $+$ $y_j$ if $x_i$ $-$ $y_i$ $>$ $y_j$ $-$ $x_j$.↵
↵
So let's sort over differences of $x_i$ $-$ $y_i$ and compute prefix sums of $x_i$ and suffix sums of $y_i$. Now we can compute for every $i$ result in $O(1)$. Then we can iterate over every edge ($u$, $v$) and subtract min($x_u$ $+$ $y_v$, $x_v$ $+$ $y_u$) from result of $u$ and $v$.↵
↵
Complexity $O(nlogn)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
typedef long long int LL;↵
↵
#define st first↵
#define nd second↵
#define PII pair <int, int>↵
↵
const int N = 3e5 + 7;↵
↵
int n, m;↵
PII diff[N];↵
int place[N];↵
vector <int> G[N];↵
↵
LL ans[N];↵
int x[N], y[N];↵
LL pref[N], suf[N];↵
↵
int main(){↵
scanf("%d %d", &n, &m);↵
for(int i = 1; i <= n; ++i){↵
scanf("%d %d", &x[i], &y[i]);↵
diff[i] = {y[i] - x[i], i};↵
}↵
↵
for(int i = 1; i <= m; ++i){↵
int u, v;↵
scanf("%d %d", &u, &v);↵
↵
G[u].push_back(v);↵
G[v].push_back(u);↵
}↵
↵
sort(diff + 1, diff + n + 1);↵
for(int i = 1; i <= n; ++i)↵
place[diff[i].nd] = i;↵
↵
for(int i = 1; i <= n; ++i)↵
pref[i] = pref[i - 1] + y[diff[i].nd];↵
↵
for(int i = n; i >= 1; --i)↵
suf[i] = suf[i + 1] + x[diff[i].nd];↵
↵
for(int i = 1; i <= n; ++i){↵
int u = diff[i].nd;↵
LL res = pref[i - 1] + suf[i + 1] + 1LL * (i - 1) * x[u] + 1LL * (n - i) * y[u];↵
↵
for(int v: G[u])↵
res -= min(x[u] + y[v], x[v] + y[u]);↵
ans[u] = res;↵
}↵
↵
for(int i = 1; i <= n; ++i)↵
printf("%lld ", ans[i]);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:Mateuszrze,2018-10-28]↵
↵
#### [1043F — Make It One](http://codeforces.me/contest/1043/problem/F)↵
↵
<spoiler summary="Tutorial">↵
First let's observe that if there exists valid subset then it's size is at most $7$ (because product of $7$ smallest primes is bigger then $3 * 10^5$).↵
Let's define $dp[i][j]$ — number of ways to pick $i$ different elements such that their gcd is equal to $j$. We can use inclusion--exclusion principle to calculate it. Then $dp[i][j]$ = $\binom{cnt_j}{i}$ — $\sum\limits_{k = 2 * j}^{\infty}{dp[i][k]}$, where $cnt_j$ denotes number of $a_i$ such that $j$ $|$ $a_i$. Because for $k$ $>$ $3 * 10^5$, $dp[i][k]$ $=$ $0$ we have to check only $k$ $\leqslant$ $3 * 10^5$.↵
↵
Our answer is the smallest $i$ such that $dp[i][1]$ is non-zero. Since $dp[i][j]$ can be quite big we should compute it modulo some big prime.↵
↵
Final complexity is $O(logM * (n + M))$, where M is equal to maximum of $a_i$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution 1">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 3e5 + 7;↵
const int MX = 1e9 + 7;↵
↵
int n;↵
int cnt[N];↵
int sil[N];↵
int odw[N];↵
int dp[20][N];↵
↵
int fast(int a, int b){↵
int ret = 1;↵
while(b){↵
if(b & 1)↵
ret = (1LL * ret * a)%MX;↵
↵
b >>= 1;↵
a = (1LL * a * a)%MX;↵
}↵
↵
return ret;↵
}↵
↵
int newton(int a, int b){↵
if(b < 0 || a < b) return 0;↵
return (((1LL * sil[a] * odw[b])%MX) * odw[a - b])%MX;↵
}↵
↵
void sub(int &a, int b){↵
a -= b;↵
if(a < 0)↵
a += MX;↵
}↵
↵
int main(){↵
sil[0] = 1;↵
for(int i = 1; i < N; ++i)↵
sil[i] = (1LL * sil[i - 1] * i)%MX;↵
↵
odw[N - 1] = fast(sil[N - 1], MX - 2);↵
for(int i = N - 1; i >= 1; --i)↵
odw[i - 1] = (1LL * odw[i] * i)%MX;↵
↵
scanf("%d", &n);↵
for(int i = 1; i <= n; ++i){↵
int a;↵
scanf("%d", &a);↵
cnt[a]++;↵
}↵
↵
for(int i = 1; i < N; ++i)↵
for(int j = i + i; j < N; j += i)↵
cnt[i] += cnt[j];↵
↵
for(int i = 1; i < 20; ++i){↵
for(int j = N - 1; j >= 1; --j){↵
dp[i][j] = newton(cnt[j], i);↵
for(int k = j + j; k < N; k += j)↵
sub(dp[i][j], dp[i][k]);↵
}↵
↵
if(dp[i][1] > 0){↵
printf("%d\n", i);↵
return 0;↵
}↵
}↵
↵
puts("-1");↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
<spoiler summary="Solution 2">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int N = 3e5 + 7;↵
↵
int n;↵
int cnt[N];↵
int roz[N];↵
int dist[N];↵
queue <int> Q;↵
vector <int> dv[N];↵
↵
int base(int a){↵
int ret = 1;↵
while(a > 1){↵
if(ret%roz[a] != 0)↵
ret *= roz[a];↵
a /= roz[a];↵
}↵
↵
return ret;↵
}↵
↵
void getEdges(int u, int d){↵
vector <int> cur;↵
vector <int> val;↵
↵
while(u > 1){↵
cur.push_back(roz[u]);↵
u /= roz[u];↵
}↵
↵
for(int v: cur)↵
u *= v;↵
↵
int T = 1 << (int)cur.size();↵
val.resize(T);↵
↵
for(int i = 0; i < T; ++i){↵
val[i] = u;↵
for(int j = 0; j < (int)cur.size(); ++j)↵
if(i & (1 << j))↵
val[i] /= cur[j];↵
}↵
↵
for(int i = 0; i < T; ++i){↵
int s = 0;↵
for(int j = i; true; j = (j - 1) & i){↵
if(__builtin_popcount(i ^ j) & 1)↵
s -= cnt[val[j]];↵
else↵
s += cnt[val[j]];↵
↵
if(j == 0)↵
break;↵
}↵
↵
assert(s >= 0);↵
if(s && dist[val[i]] == -1){↵
dist[val[i]] = d;↵
Q.push(val[i]);↵
}↵
}↵
}↵
↵
int main(){↵
scanf("%d", &n);↵
for(int i = 2; i < N; ++i){↵
if(roz[i] != 0)↵
continue;↵
↵
for(int j = i; j < N; j += i)↵
roz[j] = i;↵
}↵
↵
for(int i = 1; i < N; ++i)↵
for(int j = i; j < N; j += i)↵
dv[j].push_back(i);↵
↵
for(int i = 1; i < N; ++i)↵
dist[i] = -1;↵
↵
for(int i = 1; i <= n; ++i){↵
int a;↵
scanf("%d", &a);↵
a = base(a);↵
↵
if(dist[a] != -1)↵
continue;↵
↵
dist[a] = 1;↵
Q.push(a);↵
↵
for(int v: dv[a])↵
cnt[v]++;↵
}↵
↵
while(!Q.empty()){↵
int u = Q.front();↵
Q.pop();↵
getEdges(u, dist[u] + 1);↵
}↵
↵
printf("%d\n", dist[1]);↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:FCB1234,2018-10-28]↵
↵
#### [1043G — Speckled Band](http://codeforces.me/contest/1043/problem/G)↵
↵
<spoiler summary="Tutorial">↵
Let's solve the problem for some string $s$ for any time.↵
↵
Let's say, that partition of string $s$ into $k$ strings $s_1 s_2 \ldots s_{i_1}, s_{i_1 + 1} \ldots s_{i_2}, \ldots, {s_{i_{k-1} + 1}} \ldots s_{i_k}$ is good if at least one pair of this strings are equal. We want to find a minimal possible number of different strings in all good partitions.↵
↵
It's easy to see, that the answer is $-1$ if and only if all symbols in $s$ are different. And if we have two equal symbols $s_i = s_j$ ($i < j$) we can cut a string into strings $s_1 \ldots s_{i-1}, s_i, s_{i+1} \ldots s_{j-1}, s_j, s_{j+1} \ldots s_n$ and it is a good partition. In this partition there is at most $4$ different strings. ↵
↵
So the answer can be $-1$, $1$, $2$, $3$, $4$.↵
↵
The answer is $-1$ if all symbols in $s$ are different (case $0$).↵
↵
The answer is $1$ if the string $s = aaa \ldots a$, for some string $a$ (case $1$).↵
↵
The answer is $2$ if the string $s$ is $aab$, $aba$ or $baa$ for some strings $a$ and $b$ (case $2$). ↵
↵
The answer is $3$ if the string $s$ is $baac$, $bcaa$ or $aabc$ for some strings $a$, $b$, $c$. In two last cases it's easy to see, that $|a|=1$ (case $3$).↵
↵
To solve our problem let's build suffix array with lcp for string $s$. And let's find $lt_i$~--- minimal possible number $r$, such that $s_i s_{i+1} \ldots s_{r}$ is a tandem (the string, that can be presented as $aa$ for some string $a$) and $rt_i$~--- maximal possible number $l$ such that $s_l s_{l+1} \ldots s_{i}$ is a tandem. This numbers can be found using Main and Lorentz algorithm for finding tandem repetitions in the string.↵
↵
Now we can solve query for segment $[l, r]$:↵
\begin{itemize}↵
\item Case $0$: if $r - l \geq 26$, there exists equal symbols, otherwise we can check it by $O(r - l)$;↵
\item Case $1$: to check that $s[l \ldots r] = aa \ldots a$ we can see that $|a|$ is a divisor of $(r - l + 1)$ and $(r - l + 1) / |a|$ is a prime number (if we take a longest possible string $a$). So we should check only $O(log(n))$ lenghts of string $a$;↵
\item Case $2$: $s = aab$ $\Longleftrightarrow$ $lt_l \leq r$, $s = baa$ $\Longleftrightarrow$ $rt_r \geq l$. In the last case we should check, that $s[l \ldots r]$ has a border. It's the most interesting part of the problem, let's solve it in the end;↵
\item Case $3$: $s = abac$ $\Longleftrightarrow$ $s_l$ exists on $s_{l+1} \ldots s_r$ (can be done using prefix sums), $s = baca$ $\Longleftrightarrow$ $s_r$ exists on $s_l \ldots s_{r-1}$ (can be done using prefix sums). To check $s = baac$ we can check, that $lt_i \leq r$ for some $l \leq i \leq r$, that can be done using minimum on segment in the array $lt$.↵
\end{itemize}↵
↵
Now we should the hardest part of this problem~--- we have some segments $[l, r]$. For all of them, we should check that the border of $s[l \ldots r]$ exists. Here I know two methods, that uses only suffix array. Easiest of them:↵
↵
We have segment $[l, r]$. Let's check for all lengths $b \leq \sqrt{n}$, that $s[l \ldots (l + b - 1)] = s[(r - b + 1) \ldots r]$. If we don't find border, if it exists, it's length $> \sqrt{n}$. Let's define $i$~--- maximal index $i$ such that $lcp(l, i) \geq r - i + 1$, and string $s[i \ldots r]$ is a border of $s[l \ldots r]$. So $lcp(l, i) > \sqrt{n}$. But it's easy to see, that the distance between $l$ and $i$ in suffix array $\leq \sqrt{n}$ , so we need to check only $O(\sqrt{n})$ variants of $i$.↵
↵
Another method can check that border exists for all segments $[l, r]$ using offline algorithm by $O(q \cdot log(n)^2)$ time.↵
↵
So the total complexity will be $O((n + q) \cdot \sqrt{n})$ or $O((n + q) \cdot log(n)^2)$.↵
</spoiler>↵
↵
↵
<spoiler summary="Solution 1">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
const int BIG = 1e9 + 239;↵
const int M = 2 * 1e5 + 239;↵
const int L = 19;↵
const int A = 30;↵
const int T = (1 << 19);↵
const int two = 2;↵
↵
int flm[two][M];↵
↵
inline void z_function(string &s, int c)↵
{↵
int n = s.length();↵
flm[c][0] = 0;↵
int l = 0;↵
int r = 0;↵
for (int i = 1; i < n; i++)↵
{↵
flm[c][i] = min(flm[c][i - l], r - i);↵
if (flm[c][i] < 0) flm[c][i] = 0;↵
while (i + flm[c][i] < n && s[flm[c][i]] == s[i + flm[c][i]]) flm[c][i]++;↵
if (i + flm[c][i] > r)↵
{↵
l = i;↵
r = i + flm[c][i];↵
}↵
} ↵
}↵
↵
int a[M], lcp[M], pos[M];↵
↵
inline void suffix_array(string s)↵
{↵
s += (char)(31);↵
int n = s.length();↵
vector<pair<char, int> > v;↵
for (int i = 0; i < n; i++)↵
v.push_back(make_pair(s[i], i));↵
sort(v.begin(), v.end());↵
vector<pair<int, int> > num;↵
int last = 0;↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, v[i].second));↵
if (v[i].first != v[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, v.back().second));↵
vector<int> u(n);↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
int d = 1;↵
vector<pair<pair<int, int>, int> > t;↵
vector<vector<pair<int, int> > > h;↵
while (d < n)↵
{↵
t.clear();↵
h.clear();↵
h.resize(n);↵
for (int i = 0; i < n; i++)↵
{↵
int l = num[i].second - d;↵
if (l < 0) l += n;↵
h[u[l]].push_back(make_pair(num[i].first, l));↵
}↵
for (int i = 0; i < n; i++)↵
for (pair<int, int> r : h[i])↵
t.push_back(make_pair(make_pair(i, r.first), r.second));↵
last = 0;↵
num.clear();↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, t[i].second));↵
if (t[i].first != t[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, t.back().second));↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
d <<= 1;↵
} ↵
for (int i = 1; i < n; i++) a[i - 1] = num[i].second;↵
}↵
↵
string s; ↵
↵
inline void kasai()↵
{↵
int n = s.size();↵
suffix_array(s); ↵
for (int i = 0; i < n; i++)↵
pos[a[i]] = i;↵
int k = 0;↵
for (int i = 0; i < n; i++)↵
{↵
if (pos[i] == n - 1) continue;↵
while (s[i + k] == s[a[pos[i] + 1] + k] && a[pos[i] + 1] + k < n && i + k < n) k++;↵
lcp[pos[i]] = k;↵
k = max(0, k - 1);↵
}↵
}↵
↵
int n, lw[M], rw[M]; ↵
string prr, rvl; ↵
vector<int> open_l[M], close_l[M]; ↵
vector<int> open_r[M], close_r[M];↵
multiset<int> nw;↵
↵
inline void func(int l, int r)↵
{↵
if (r - l == 1) return;↵
int mid = (l + r) >> 1;↵
func(l, mid);↵
func(mid, r);↵
rvl = "";↵
for (int i = mid - 1; i >= l; i--) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid; i < r; i++) prr += s[i];↵
prr += '#';↵
for (int i = l; i < mid; i++) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = l; c < mid; c++)↵
{↵
int k1 = 0;↵
if (c > l) k1 = flm[0][mid - c];↵
int k2 = flm[1][r - mid + 1 + c - l];↵
int len = mid - c;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{↵
open_l[c - rg].push_back((2 * len));↵
close_l[c - lg].push_back((2 * len));↵
open_r[c - rg + 2 * len - 1].push_back((2 * len));↵
close_r[c - lg + 2 * len - 1].push_back((2 * len));↵
} ↵
}↵
rvl = "";↵
for (int i = mid; i < r; i++) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid - 1; i >= l; i--) prr += s[i];↵
prr += '#';↵
for (int i = r - 1; i >= mid; i--) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = mid; c < r; c++)↵
{↵
int k1 = 0;↵
if (c != r - 1) k1 = flm[0][c + 1 - mid];↵
int k2 = flm[1][r - c + mid - l];↵
int len = c - mid + 1;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{ ↵
open_l[c + lg - 2 * len + 1].push_back((2 * len));↵
close_l[c + rg - 2 * len + 1].push_back((2 * len));↵
open_r[c + lg].push_back((2 * len));↵
close_r[c + rg].push_back((2 * len));↵
} ↵
}↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_l[i]) nw.insert(len);↵
if (!nw.empty()) lw[i] = min(lw[i], *nw.begin());↵
for (int len : close_l[i]) nw.erase(nw.lower_bound(len));↵
open_l[i].clear();↵
close_l[i].clear();↵
} ↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_r[i]) nw.insert(len);↵
if (!nw.empty()) rw[i] = min(rw[i], *nw.begin());↵
for (int len : close_r[i]) nw.erase(nw.lower_bound(len));↵
open_r[i].clear();↵
close_r[i].clear();↵
}↵
}↵
↵
int mn[L][M], st2[M], lc[L][M], kol[A][M];↵
↵
inline int getmin(int l, int r)↵
{↵
int u = st2[r - l + 1];↵
return min(mn[u][l], mn[u][r - (1 << u) + 1]);↵
}↵
↵
inline int gett(int l, int r)↵
{↵
if (l == r) return (n - l);↵
l = pos[l];↵
r = pos[r];↵
if (l > r) swap(l, r);↵
r--;↵
int u = st2[r - l + 1];↵
return min(lc[u][l], lc[u][r - (1 << u) + 1]);↵
}↵
↵
int q, la[M], ra[M], has[M], mnp[M];↵
↵
inline void init()↵
{↵
for (int i = 0; i < n; i++) mn[0][i] = lw[i];↵
for (int i = 0; i < n - 1; i++) lc[0][i] = lcp[i];↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n)↵
{↵
mn[i][j] = mn[i - 1][j];↵
continue;↵
}↵
mn[i][j] = min(mn[i - 1][j], mn[i - 1][r]);↵
}↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n - 1; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n - 1)↵
{↵
lc[i][j] = lc[i - 1][j];↵
continue;↵
} ↵
lc[i][j] = min(lc[i - 1][j], lc[i - 1][r]);↵
} ↵
st2[1] = 0;↵
for (int i = 2; i <= n; i++)↵
{↵
st2[i] = st2[i - 1];↵
if ((1 << (st2[i] + 1)) <= i) st2[i]++;↵
} ↵
}↵
↵
int in[L][M], gl[T];↵
↵
inline void build(int i, int l, int r)↵
{↵
if (i == 0) gl[i] = 0;↵
else gl[i] = gl[(i - 1) / 2] + 1;↵
if (r - l == 1)↵
{↵
in[gl[i]][l] = a[l];↵
return;↵
}↵
int mid = (l + r) >> 1;↵
build(2 * i + 1, l, mid);↵
build(2 * i + 2, mid, r);↵
merge(in[gl[i] + 1] + l, in[gl[i] + 1] + mid, in[gl[i] + 1] + mid, in[gl[i] + 1] + r, in[gl[i]] + l);↵
}↵
↵
inline bool is_on(int i, int l, int r, int ql, int qr, int xl, int xr)↵
{↵
if (r <= ql || qr <= l) return false;↵
if (ql <= l && r <= qr)↵
{↵
int it = upper_bound(in[gl[i]] + l, in[gl[i]] + r, xl) - in[gl[i]] - l;↵
if (it == r - l) return false;↵
return (in[gl[i]][it + l] <= xr); ↵
}↵
int mid = (l + r) >> 1;↵
if (is_on(2 * i + 1, l, mid, ql, qr, xl, xr)) return true;↵
return is_on(2 * i + 2, mid, r, ql, qr, xl, xr);↵
}↵
↵
inline bool check_all(int l, int r)↵
{↵
int len = (r - l + 1);↵
int pl = pos[l];↵
int lf = pl;↵
int rf = n;↵
while (rf - lf > 1)↵
{↵
int h = (lf + rf) / 2;↵
if (gett(a[h], l) >= len) lf = h;↵
else rf = h;↵
}↵
int rg = lf + 1;↵
lf = -1;↵
rf = pl;↵
while (rf - lf > 1)↵
{↵
int h = (lf + rf) / 2;↵
if (gett(a[h], l) >= len) rf = h;↵
else lf = h;↵
}↵
int lg = rf;↵
return is_on(0, 0, n, lg, rg, l, r);↵
}↵
↵
int par[M];↵
set<int> cmp[M];↵
set<pair<int, int> > sc[M];↵
vector<int> ok[M];↵
vector<pair<int, int> > mg[M];↵
↵
inline void border_check()↵
{↵
for (int i = 0; i < q; i++) has[i] = false;↵
for (int i = 0; i < q; i++) ok[ra[i] - la[i]].push_back(i);↵
for (int i = 0; i < n - 1; i++) mg[lcp[i]].push_back(make_pair(a[i], a[i + 1]));↵
for (int i = 0; i < n; i++)↵
{↵
par[i] = i;↵
cmp[i].insert(i);↵
}↵
for (int c = n; c >= 1; c--)↵
{↵
for (int i : ok[c - 1])↵
sc[par[la[i]]].insert(make_pair(ra[i], i)); ↵
for (pair<int, int> t : mg[c])↵
{↵
int l = par[t.first];↵
int r = par[t.second];↵
if (cmp[l].size() > cmp[r].size()) swap(l, r);↵
for (int x : cmp[l])↵
while (true)↵
{↵
auto it = sc[r].lower_bound(make_pair(x, 0));↵
if (it == sc[r].end() || it->first > x + c - 1) break;↵
has[it->second] = true;↵
sc[r].erase(it);↵
}↵
for (pair<int, int> u : sc[l])↵
{↵
int id = u.second;↵
auto uk = cmp[r].lower_bound(ra[id] - c + 1);↵
if (uk != cmp[r].end() && (*uk) <= ra[id]) ↵
has[id] = true;↵
else ↵
sc[r].insert(u);↵
}↵
for (int x : cmp[l])↵
{↵
par[x] = r;↵
cmp[r].insert(x);↵
}↵
} ↵
} ↵
for (int i = 0; i < q; i++)↵
if (!has[i] && check_all(la[i], ra[i]))↵
{↵
has[i] = true;↵
continue;↵
}↵
}↵
↵
inline bool checkno(int l, int r)↵
{↵
if (r - l + 1 > 26) return false;↵
vector<int> kol(26, 0);↵
for (int x = l; x <= r; x++)↵
{↵
if (kol[s[x] - 'a'] > 0) return false;↵
kol[s[x] - 'a']++;↵
} ↵
return true;↵
}↵
↵
inline bool try_kol(int l, int r, int p)↵
{↵
int len = (r - l + 1) / p;↵
return (gett(l, l + len) >= (r - l + 1 - len));↵
}↵
↵
inline bool ison(int l, int r, char x)↵
{↵
return (kol[x - 'a'][r + 1] > kol[x - 'a'][l]); ↵
}↵
↵
inline int query(int l, int r, int id)↵
{ ↵
if (checkno(l, r)) return -1;↵
int len = (r - l + 1);↵
while (len > 1)↵
{↵
int p = mnp[len];↵
if (try_kol(l, r, p)) return 1;↵
while ((len % p) == 0) len /= p;↵
}↵
if (lw[l] <= r) return 2;↵
if (rw[r] >= l) return 2;↵
if (has[id]) return 2;↵
if (ison(l + 1, r, s[l])) return 3;↵
if (ison(l, r - 1, s[r])) return 3;↵
if (getmin(l, r) <= r) return 3; ↵
return 4;↵
}↵
↵
int main()↵
{↵
cin.sync_with_stdio(0); ↵
cin >> n >> s;↵
memset(mnp, -1, sizeof(mnp));↵
for (int i = 2; i <= n; i++)↵
if (mnp[i] == -1)↵
for (int j = i; j <= n; j += i)↵
if (mnp[j] == -1)↵
mnp[j] = i;↵
memset(kol[0], 0, sizeof(kol[0]));↵
for (int i = 0; i < n; i++)↵
{↵
for (int x = 0; x < 26; x++) kol[x][i + 1] = kol[x][i];↵
kol[s[i] - 'a'][i + 1]++;↵
}↵
for (int i = 0; i < n; i++)↵
{↵
lw[i] = n + 1;↵
rw[i] = n + 1;↵
}↵
func(0, n); ↵
for (int i = 0; i < n; i++)↵
{↵
if (lw[i] == n + 1) lw[i] = n;↵
else lw[i] += (i - 1); ↵
if (rw[i] == n + 1) rw[i] = -1;↵
else rw[i] = (i - rw[i] + 1);↵
}↵
kasai(); ↵
init(); ↵
build(0, 0, n); ↵
cin >> q;↵
for (int i = 0; i < q; i++)↵
{↵
cin >> la[i] >> ra[i];↵
la[i]--, ra[i]--; ↵
}↵
border_check();↵
for (int i = 0; i < q; i++) cout << query(la[i], ra[i], i) << "\n";↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
↵
<spoiler summary="Solution 2">↵
~~~~~↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
#define TIME (clock() * 1.0 / CLOCKS_PER_SEC)↵
↵
const int BIG = 1e9 + 239;↵
const int M = 2 * 1e5 + 239;↵
const int L = 19;↵
const int A = 30;↵
const int T = (1 << 19);↵
const int two = 2;↵
↵
int flm[two][M];↵
↵
inline void z_function(string &s, int c)↵
{↵
int n = s.length();↵
flm[c][0] = 0;↵
int l = 0;↵
int r = 0;↵
for (int i = 1; i < n; i++)↵
{↵
flm[c][i] = min(flm[c][i - l], r - i);↵
if (flm[c][i] < 0) flm[c][i] = 0;↵
while (i + flm[c][i] < n && s[flm[c][i]] == s[i + flm[c][i]]) flm[c][i]++;↵
if (i + flm[c][i] > r)↵
{↵
l = i;↵
r = i + flm[c][i];↵
}↵
} ↵
}↵
↵
int a[M], lcp[M], pos[M];↵
↵
inline void suffix_array(string s)↵
{↵
s += (char)(31);↵
int n = s.length();↵
vector<pair<char, int> > v;↵
for (int i = 0; i < n; i++)↵
v.push_back(make_pair(s[i], i));↵
sort(v.begin(), v.end());↵
vector<pair<int, int> > num;↵
int last = 0;↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, v[i].second));↵
if (v[i].first != v[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, v.back().second));↵
vector<int> u(n);↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
int d = 1;↵
vector<pair<pair<int, int>, int> > t;↵
vector<vector<pair<int, int> > > h;↵
while (d < n)↵
{↵
t.clear();↵
h.clear();↵
h.resize(n);↵
for (int i = 0; i < n; i++)↵
{↵
int l = num[i].second - d;↵
if (l < 0) l += n;↵
h[u[l]].push_back(make_pair(num[i].first, l));↵
}↵
for (int i = 0; i < n; i++)↵
for (pair<int, int> r : h[i])↵
t.push_back(make_pair(make_pair(i, r.first), r.second));↵
last = 0;↵
num.clear();↵
for (int i = 0; i < n - 1; i++)↵
{↵
num.push_back(make_pair(last, t[i].second));↵
if (t[i].first != t[i + 1].first) last++;↵
}↵
num.push_back(make_pair(last, t.back().second));↵
for (int i = 0; i < n; i++) u[num[i].second] = num[i].first;↵
d <<= 1;↵
} ↵
for (int i = 1; i < n; i++) a[i - 1] = num[i].second;↵
}↵
↵
string s; ↵
↵
inline void kasai()↵
{↵
int n = s.size();↵
suffix_array(s); ↵
for (int i = 0; i < n; i++)↵
pos[a[i]] = i;↵
int k = 0;↵
for (int i = 0; i < n; i++)↵
{↵
if (pos[i] == n - 1) continue;↵
while (s[i + k] == s[a[pos[i] + 1] + k] && a[pos[i] + 1] + k < n && i + k < n) k++;↵
lcp[pos[i]] = k;↵
k = max(0, k - 1);↵
}↵
}↵
↵
int n, lw[M], rw[M]; ↵
string prr, rvl; ↵
vector<int> open_l[M], close_l[M]; ↵
vector<int> open_r[M], close_r[M];↵
multiset<int> nw;↵
↵
inline void func(int l, int r)↵
{↵
if (r - l == 1) return;↵
int mid = (l + r) >> 1;↵
func(l, mid);↵
func(mid, r);↵
rvl = "";↵
for (int i = mid - 1; i >= l; i--) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid; i < r; i++) prr += s[i];↵
prr += '#';↵
for (int i = l; i < mid; i++) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = l; c < mid; c++)↵
{↵
int k1 = 0;↵
if (c > l) k1 = flm[0][mid - c];↵
int k2 = flm[1][r - mid + 1 + c - l];↵
int len = mid - c;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{↵
open_l[c - rg].push_back((2 * len));↵
close_l[c - lg].push_back((2 * len));↵
open_r[c - rg + 2 * len - 1].push_back((2 * len));↵
close_r[c - lg + 2 * len - 1].push_back((2 * len));↵
} ↵
}↵
rvl = "";↵
for (int i = mid; i < r; i++) rvl += s[i];↵
z_function(rvl, 0);↵
prr = "";↵
for (int i = mid - 1; i >= l; i--) prr += s[i];↵
prr += '#';↵
for (int i = r - 1; i >= mid; i--) prr += s[i];↵
z_function(prr, 1); ↵
for (int c = mid; c < r; c++)↵
{↵
int k1 = 0;↵
if (c != r - 1) k1 = flm[0][c + 1 - mid];↵
int k2 = flm[1][r - c + mid - l];↵
int len = c - mid + 1;↵
int lg = max(len - k2, 0);↵
int rg = min(len - 1, k1);↵
if (rg >= lg)↵
{ ↵
open_l[c + lg - 2 * len + 1].push_back((2 * len));↵
close_l[c + rg - 2 * len + 1].push_back((2 * len));↵
open_r[c + lg].push_back((2 * len));↵
close_r[c + rg].push_back((2 * len));↵
} ↵
}↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_l[i]) nw.insert(len);↵
if (!nw.empty()) lw[i] = min(lw[i], *nw.begin());↵
for (int len : close_l[i]) nw.erase(nw.lower_bound(len));↵
open_l[i].clear();↵
close_l[i].clear();↵
} ↵
for (int i = l; i < r; i++)↵
{↵
for (int len : open_r[i]) nw.insert(len);↵
if (!nw.empty()) rw[i] = min(rw[i], *nw.begin());↵
for (int len : close_r[i]) nw.erase(nw.lower_bound(len));↵
open_r[i].clear();↵
close_r[i].clear();↵
}↵
}↵
↵
int mn[L][M], st2[M], lc[L][M], kol[A][M];↵
↵
inline int getmin(int l, int r)↵
{↵
int u = st2[r - l + 1];↵
return min(mn[u][l], mn[u][r - (1 << u) + 1]);↵
}↵
↵
inline int gett(int l, int r)↵
{↵
if (l == r) return (n - l);↵
l = pos[l];↵
r = pos[r];↵
if (l > r) swap(l, r);↵
r--;↵
int u = st2[r - l + 1];↵
return min(lc[u][l], lc[u][r - (1 << u) + 1]);↵
}↵
↵
int q, mnp[M];↵
↵
inline void init()↵
{↵
for (int i = 0; i < n; i++) mn[0][i] = lw[i];↵
for (int i = 0; i < n - 1; i++) lc[0][i] = lcp[i];↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n)↵
{↵
mn[i][j] = mn[i - 1][j];↵
continue;↵
}↵
mn[i][j] = min(mn[i - 1][j], mn[i - 1][r]);↵
}↵
for (int i = 1; i < L; i++)↵
for (int j = 0; j < n - 1; j++)↵
{↵
int r = (j + (1 << (i - 1)));↵
if (r >= n - 1)↵
{↵
lc[i][j] = lc[i - 1][j];↵
continue;↵
} ↵
lc[i][j] = min(lc[i - 1][j], lc[i - 1][r]);↵
} ↵
st2[1] = 0;↵
for (int i = 2; i <= n; i++)↵
{↵
st2[i] = st2[i - 1];↵
if ((1 << (st2[i] + 1)) <= i) st2[i]++;↵
} ↵
}↵
↵
int u, len;↵
↵
inline bool check(int l, int r)↵
{↵
for (int t = 1; t <= min(len, ((r - l + 1) / 2)); t++)↵
if (gett(l, r - t + 1) >= t)↵
return true;↵
for (int i = max(0, pos[l] - len); i <= min(n - 1, pos[l] + len); i++)↵
if (l < a[i] && a[i] <= r && gett(l, a[i]) >= r - a[i] + 1) ↵
return true;↵
return false;↵
}↵
↵
inline bool checkno(int l, int r)↵
{↵
if (r - l + 1 > 26) return false;↵
vector<int> kol(26, 0);↵
for (int x = l; x <= r; x++)↵
{↵
if (kol[s[x] - 'a'] > 0) return false;↵
kol[s[x] - 'a']++;↵
} ↵
return true;↵
}↵
↵
inline bool try_kol(int l, int r, int p)↵
{↵
int len = (r - l + 1) / p;↵
return (gett(l, l + len) >= (r - l + 1 - len));↵
}↵
↵
inline bool ison(int l, int r, char x)↵
{↵
return (kol[x - 'a'][r + 1] > kol[x - 'a'][l]); ↵
}↵
↵
inline int query(int l, int r)↵
{ ↵
if (checkno(l, r)) return -1;↵
int len = (r - l + 1);↵
while (len > 1)↵
{↵
int p = mnp[len];↵
if (try_kol(l, r, p)) return 1;↵
while ((len % p) == 0) len /= p;↵
}↵
if (lw[l] <= r) return 2;↵
if (rw[r] >= l) return 2;↵
if (check(l, r)) return 2;↵
if (ison(l + 1, r, s[l])) return 3;↵
if (ison(l, r - 1, s[r])) return 3;↵
if (getmin(l, r) <= r) return 3; ↵
return 4;↵
}↵
↵
int main()↵
{↵
cin.sync_with_stdio(0); ↵
cin >> n >> s;↵
memset(mnp, -1, sizeof(mnp));↵
for (int i = 2; i <= n; i++)↵
if (mnp[i] == -1)↵
for (int j = i; j <= n; j += i)↵
if (mnp[j] == -1)↵
mnp[j] = i;↵
memset(kol[0], 0, sizeof(kol[0]));↵
for (int i = 0; i < n; i++)↵
{↵
for (int x = 0; x < 26; x++) kol[x][i + 1] = kol[x][i];↵
kol[s[i] - 'a'][i + 1]++;↵
}↵
for (int i = 0; i < n; i++)↵
{↵
lw[i] = n + 1;↵
rw[i] = n + 1;↵
}↵
func(0, n); ↵
for (int i = 0; i < n; i++)↵
{↵
if (lw[i] == n + 1) lw[i] = n;↵
else lw[i] += (i - 1); ↵
if (rw[i] == n + 1) rw[i] = -1;↵
else rw[i] = (i - rw[i] + 1);↵
}↵
kasai(); ↵
init(); ↵
cin >> q;↵
for (int i = 0; i < n; i++)↵
if (i * i >= n)↵
{↵
len = i + 3;↵
break;↵
}↵
for (int i = 0; i < q; i++)↵
{↵
int l, r;↵
cin >> l >> r;↵
l--, r--;↵
cout << query(l, r) << "\n"; ↵
}↵
return 0;↵
}↵
~~~~~↵
</spoiler>↵
↵
Author: [user:isaf27,2018-10-28]
