We hoped you find our problems interesting. We apologize for the late editorial. Hopefully you were still able to enjoy our contest. ↵
↵
Anyway, here are the tutorials for each of the problems:↵
↵
###[problem:1397A]↵
↵
If the total number of occurrences of some character $c$ is not a multiple of $n$, then it is impossible to make all $n$ strings equal — because then it is impossible for all $n$ strings to have the same number of $c$.↵
↵
On the other hand, if the total number of occurrences of every character $c$ is a multiple of $n$, then it is always possible to make all $n$ strings equal. To achieve this, for every character $c$ we move exactly ((the total number of occurrences of $c$) $/$ $n$) characters $c$ to the end of each string, and by the end we will have all $n$ strings equal each other.↵
↵
We can easily check if the condition satisfies by counting the total number of occurrences of each character $c$ and check its divisibility by $n$. The final complexity is $O(S \cdot 26)$ or $O(S)$ where $S$ is the sum of lengths of all strings.↵
↵
<spoiler summary="C++ solution">```cpp↵
↵
#include <iostream>↵
#include <vector>↵
↵
using namespace std;↵
↵
int main()↵
{↵
int tests;↵
cin >> tests;↵
while (tests--) {↵
int n;↵
cin >> n;↵
↵
vector<int> cnt(26);↵
for (int i = 0; i < n; ++i) {↵
string s;↵
cin >> s;↵
for (char ch : s) {↵
++cnt[ch - 'a'];↵
}↵
}↵
↵
bool ans = true;↵
for (int i = 0; i < 26; ++i) {↵
if (cnt[i] % n != 0) {↵
ans = false;↵
break;↵
}↵
}↵
↵
cout << (ans ? "YES" : "NO") << endl;↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
numTests = int(input())↵
for testNo in range(numTests):↵
n = int(input())↵
cnt = [0 for i in range(26)]↵
for _ in range(n):↵
s = input()↵
for i in s:↵
cnt[ord(i) - 97] += 1↵
↵
ans = True↵
for i in range(26):↵
if cnt[i] % n != 0:↵
ans = False↵
break↵
↵
if ans:↵
print('YES')↵
else:↵
print('NO')↵
```↵
</spoiler>↵
↵
###[problem:1397B]↵
↵
First of all, the optimal way to reorder is to sort $a$ in non-decreasing order.↵
↵
<spoiler summary="Proof">↵
The cost to transform $a_i$ to $c^i$ is $\lvert a_i - c^i \rvert$, and $\lvert a_i - c^i \rvert + \lvert a_j - c^j \rvert \le \lvert a_j - c^i \rvert + \lvert a_i - c^j \rvert$ when $i < j$ and $a_i \le a_j$, thus it is optimal to have $a_i \le a_j$ for each $0 \le i < j < n$.↵
</spoiler>↵
↵
From now on, we assume $a$ is sorted in non-decreasing order.↵
↵
Denote $a_{max} = a_{n - 1}$ as the maximum value in $a$, $f(x) = \sum{\lvert a_i - x^i \rvert}$ as the minimum cost to transform $a$ into $\{x^0, x^1, \cdots, x^{n-1}\}$, and $c$ as the value where $f(c)$ is minimum.↵
↵
Note that $c^{n - 1} - a_{max} \le f(c) \le f(1)$, which implies $c^{n - 1} \le f(1) + a_{max}$.↵
↵
We enumerate $x$ from $1, 2, 3, \dots$ until $x^{n - 1}$ exceeds $f(1) + a_{max}$, calculate $f(x)$ in $O(n)$, and the final answer is the minimum among all calculated values. The final complexity is $O(n \cdot max(x))$.↵
↵
But why doesn't this get TLE? Because $f(1) = \sum{(a_i - 1)} < a_{max} \cdot n \le 10^9 \cdot n$, thus $x^{n - 1} \le f(1) + a_{max} \le 10^9 \cdot (n + 1)$. When $n = 3, 4, 5, 6$, $max(x)$ does not exceed $63245, 1709, 278, 93$ respectively; so we can see that $O(n \cdot max(x))$ comfortably fits in the time limit.↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <iostream>↵
#include <vector>↵
#include <algorithm>↵
#include <numeric>↵
↵
using namespace std;↵
↵
const int64_t INF = 1e17;↵
inline int64_t mul(int64_t a, int64_t b)↵
{↵
return (INF / a > b ? a * b : INF);↵
}↵
↵
inline int64_t add(int64_t a, int64_t b)↵
{↵
return (a + b >= INF ? INF : a + b);↵
}↵
↵
int main()↵
{↵
int n;↵
cin >> n;↵
↵
vector<int> a(n);↵
for (int &x : a) cin >> x;↵
sort(a.begin(), a.end());↵
↵
if (n <= 2) {↵
cout << a[0] - 1 << endl;↵
} else {↵
int64_t ans = accumulate(a.begin(), a.end(), 0ll) - n;↵
↵
for (int x = 1; ; ++x) {↵
int64_t curPow = 1, curCost = 0;↵
for (int i = 0; i < n; ++i) {↵
curCost = add(curCost, abs(a[i] - curPow));↵
curPow = mul(curPow, x);↵
}↵
↵
if (curPow == INF || curPow / x > ans + a[n - 1]) break;↵
ans = min(ans, curCost);↵
}↵
↵
cout << ans << endl;↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
n = int(input())↵
a = [int(x) for x in input().split()]↵
a.sort()↵
inf = 10**18↵
↵
if n <= 2:↵
print(a[0] - 1)↵
else:↵
ans = sum(a) - n↵
↵
for x in range(1, 10**9):↵
curPow = 1↵
curCost = 0↵
for i in range(n):↵
curCost += abs(a[i] - curPow)↵
curPow *= x↵
if curPow > inf:↵
break↵
↵
if curPow > inf:↵
break↵
if curPow / x > ans + a[n - 1]:↵
break↵
↵
ans = min(ans, curCost)↵
↵
print(ans)↵
```↵
</spoiler>↵
↵
↵
###[problem:1396A]↵
↵
In this problem, the answer is rather simple. Here is one possible solution to this task.↵
↵
<spoiler summary="Solution for n = 1">↵
$1 \space \space 1$ ↵
$0$ ↵
$1 \space \space 1$ ↵
$0$ ↵
$1 \space \space 1$ ↵
$-a_1$↵
</spoiler>↵
↵
<spoiler summary="Solution for n != 1">↵
$1 \space \space 1$ ↵
$-a_1$ ↵
$1 \space \space n$ ↵
$0, \space -n \cdot a_2, \space -n \cdot a_3, \space \dots , \space -n \cdot a_n$ ↵
$2 \space \space n$ ↵
$(n-1) \cdot a_2, \space (n-1) \cdot a_3, \space \dots , \space (n-1) \cdot a_n$↵
</spoiler>↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
using ll = long long;↵
↵
int main() {↵
ios_base::sync_with_stdio(false); cin.tie(nullptr);↵
int N;↵
cin >> N;↵
vector<ll> A(N);↵
for (int i = 0; i < N; ++i) cin >> A[i];↵
if (N == 1) {↵
for (int z = 0; z < 3; ++z) {↵
cout << "1 1\n";↵
cout << -A[0] << "\n";↵
A[0] = 0;↵
}↵
return 0;↵
}↵
cout << "1 " << N << "\n";↵
for (int i = 0; i + 1 < N; ++i) cout << -A[i] * N << " "; cout << "0\n";↵
cout << "1 " << N - 1 << "\n";↵
for (int i = 0; i + 1 < N; ++i) cout << A[i] * (N - 1) << " "; cout << "\n";↵
cout << N << " " << N << "\n";↵
cout << -A[N - 1] << "\n";↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
n = int(input())↵
a = list(map(int, input().split()))↵
↵
if n == 1:↵
print('1 1', -a[0], '1 1', '0', '1 1', '0', sep='\n')↵
exit(0)↵
↵
print(1, n)↵
for i in range(n):↵
print(-a[i] * n, end = ' ')↵
a[i] -= a[i] * n↵
print()↵
↵
print(1, n - 1)↵
for i in range(n - 1):↵
print(-a[i], end = ' ')↵
a[i] = 0↵
print()↵
↵
print(2, n)↵
for i in range(1, n):↵
print(-a[i], end = ' ')↵
a[i] = 0↵
print()↵
```↵
</spoiler>↵
↵
###[problem:1396B]↵
↵
Let us denote $S$ as the current total number of stones. ↵
↵
Consider the following cases:↵
↵
**Case A: There is a pile that has more than $\lfloor \frac{S}{2} \rfloor$ stones.**↵
↵
The first player (T) can always choose from this pile, thus he (T) is the winner.↵
↵
**Case B: Every pile has at most $\lfloor \frac{S}{2} \rfloor$ stones, and $S$ is even.**↵
↵
It can be proven that the second player (HL) always wins.↵
↵
<spoiler summary="Proof 1">↵
Let us prove by induction:↵
↵
When $S = 0$, the second player obviously wins. ↵
↵
When $S \geq 2$, consider the game state after the first player moves. If there is a pile that now has more than $\lfloor \frac{S}{2} \rfloor$ stones, then we arrive back at _case A_ where the next player to move wins. Otherwise, the second player ↵
can choose from any valid pile (note that the case condition implies that there are at least two non-empty piles before the ↵
first player's move). Now $S$ has been reduced by $2$, and every pile still has at most $\lfloor \frac{S}{2} \rfloor$ stones.↵
</spoiler>↵
↵
<spoiler summary="Proof 2">↵
The condition allows us to assign a perfect matching of stones, where one stone is matched with exactly one stone ↵
from a different pile. ↵
↵
A greedy way to create such a matching: Give each label $0, 1, \dots, S - 1$ to a different stone so that for every pair of ↵
stones with labels $l < r$ that are from the same pile, stones $l + 1, l + 2, \dots, r - 1$ are also from that pile; then ↵
match stones $i$ with $i + \frac{S}{2}$ for all $0 \le i < \frac{S}{2}$.↵
↵
For every stone that the first player removes, the second player can always remove its matching stone, until the first player ↵
can no longer make a move and loses.↵
</spoiler>↵
↵
**Case C: Every pile has at most $\lfloor \frac{S}{2} \rfloor$ stones, and $S$ is odd.**↵
↵
The first player (T) can choose from any pile, and we arrive back at _case B_ where the next player to move loses. ↵
↵
So the first player (T) wins if and only if there is a pile that has more than $\lfloor \frac{S}{2} \rfloor$ stones or $S$ ↵
is odd. This can be easily checked in $O(n)$.↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <iostream>↵
#include <vector>↵
#include <algorithm>↵
#include <numeric>↵
↵
using namespace std;↵
↵
int main()↵
{↵
int t;↵
cin >> t;↵
while (t--) {↵
int n;↵
cin >> n;↵
↵
vector<int> a(n);↵
for (int &x : a) cin >> x;↵
↵
int maxPile = *max_element(a.begin(), a.end());↵
int numStones = accumulate(a.begin(), a.end(), 0);↵
↵
if (maxPile * 2 > numStones || (numStones & 1)) cout << "T" << endl;↵
else cout << "HL" << endl;↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
t = int(input())↵
for _ in range(t):↵
n = int(input())↵
a = [int(x) for x in input().split()]↵
↵
maxPile = max(a)↵
numStones = sum(a)↵
↵
if maxPile * 2 > numStones or (numStones & 1):↵
print('T')↵
else:↵
print('HL')↵
```↵
</spoiler>↵
↵
###[problem:1396C]↵
↵
In this problem, it is useful to note that when the boss only has $1$ hp left, just use the pistol because it has the least reloading time. So there are 3 strategies we will use when playing at stage $i$ $(1 \le i \le n)$:↵
↵
* Take $a_i$ pistol shots to kill first $a_i$ monsters and shoot the boss with the AWP.↵
* Take $a_i + 1$ pistol shots and move back to this stage later to take another pistol shot to finish the boss.↵
* Use the laser gun and move back to this stage later to kill the boss with a pistol shot.↵
↵
**Observation:** We will always finish the game at stage $n$ or $n - 1$. Considering we are at stage $i$ $(i \le n - 1)$ and the boss at both stage $i$ stage $i - 1$ has $1$ hp left, we can spend $2 * d$ time to finish both these stages instead of going back later, which costs us exactly the same.↵
↵
Therefore, we will calculate $dp(i,0/1)$ as the minimum time to finish first $a_i - 1$ stages and 0/1 is the remaining hp of the boss at stage i. The transitions are easy to figure out by using 3 strategies as above. The only thing we should note is that we can actually finish the game at stage $n - 1$ by instantly kill the boss at stage $n$ with the AWP so we don't have to go back to this level later.↵
↵
Answer to the problem is $dp(n, 0)$. Time complexity: $O(n)$.↵
↵
<spoiler summary="C++ solution">```cpp↵
/*input↵
4 2 4 4 1↵
4 5 1 2↵
*/↵
#include <bits/stdc++.h>↵
using namespace std;↵
↵
int read() {↵
int x = 0, c = getchar();↵
for(; !(c > 47 && c < 58); c = getchar());↵
for(; (c > 47 && c < 58); c = getchar()) x = x * 10 + c - 48;↵
return x;↵
}↵
↵
void upd(long long &a, long long b) {↵
a = (a < b) ? a : b;↵
}↵
↵
const int N = 1e6 + 5;↵
↵
long long f[N][2];↵
int n, r1, r2, r3, d, a[N];↵
↵
int main(){ ↵
n = read(), r1 = read(), r2 = read(), r3 = read(), d = read();↵
for(int i = 1; i <= n; a[i ++] = read());↵
↵
for(int i = 2; i <= n; ++ i) f[i][0] = f[i][1] = 1e18;↵
↵
f[1][0] = 1ll * r1 * a[1] + r3;↵
f[1][1] = min(0ll + r2, 1ll * r1 * a[1] + r1);↵
for(int i = 1; i < n; ++ i) {↵
// 0 -> 0↵
// so we clear this one and the next one as well↵
upd(f[i + 1][0], f[i][0] + d + 1ll * r1 * a[i + 1] + r3);↵
↵
// 0 -> 1↵
// this one is cleared, but next one isnt↵
upd(f[i + 1][1], f[i][0] + d + min(0ll + r2, 1ll * r1 * a[i + 1] + r1));↵
↵
// 1 -> 0↵
upd(f[i + 1][0], f[i][1] + d + 1ll * r1 * a[i + 1] + r3 + 2 * d + r1);↵
upd(f[i + 1][0], f[i][1] + d + 1ll * r1 * a[i + 1] + r1 + d + r1 + d + r1);↵
upd(f[i + 1][0], f[i][1] + d + r2 + d + r1 + d + r1);↵
↵
// 1 -> 1↵
upd(f[i + 1][1], f[i][1] + d + r2 + d + r1 + d);↵
upd(f[i + 1][1], f[i][1] + d + 1ll * r1 * a[i + 1] + r1 + d + r1 + d);↵
↵
if(i == n - 1) {↵
upd(f[i + 1][0], f[i][1] + d + 1ll * r1 * a[i + 1] + r3 + d + r1);↵
}↵
}↵
cout << f[n][0] << endl;↵
}↵
```↵
</spoiler>↵
↵
[tutorial:1396D]↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
using ll = long long;↵
const int MOD = 1000000007;↵
↵
int L;↵
ll sum[8040];↵
int len[8040];↵
int last[8040];↵
int lazy[8040];↵
↵
void init(int v, int l, int r, const vector<int> &xs) {↵
len[v] = xs[r] - xs[l - 1];↵
if (l < r) {↵
int md = (l + r) >> 1;↵
init(v << 1, l, md, xs);↵
init(v << 1 | 1, md + 1, r, xs);↵
}↵
}↵
↵
void reset(int v, int l, int r, const vector<int> &go) {↵
lazy[v] = -1;↵
if (l == r) {↵
sum[v] = ll(len[v]) * (L - go[l]);↵
last[v] = go[l];↵
return;↵
}↵
int md = (l + r) >> 1;↵
reset(v << 1, l, md, go);↵
reset(v << 1 | 1, md + 1, r, go);↵
sum[v] = sum[v << 1] + sum[v << 1 | 1];↵
last[v] = last[v << 1 | 1];↵
}↵
↵
void push(int v, int l, int r) {↵
if (lazy[v] != -1) {↵
last[v] = lazy[v];↵
sum[v] = ll(len[v]) * (L - lazy[v]);↵
if (l < r) {↵
lazy[v << 1] = lazy[v];↵
lazy[v << 1 | 1] = lazy[v];↵
}↵
lazy[v] = -1;↵
}↵
}↵
↵
void modify(int v, int l, int r, int L, int R, int qv) {↵
push(v, l, r);↵
if (L > r || R < l) return;↵
if (L <= l && r <= R) {↵
lazy[v] = qv;↵
push(v, l, r);↵
return;↵
}↵
int md = (l + r) >> 1;↵
modify(v << 1, l, md, L, R, qv);↵
modify(v << 1 | 1, md + 1, r, L, R, qv);↵
sum[v] = sum[v << 1] + sum[v << 1 | 1];↵
last[v] = last[v << 1 | 1];↵
}↵
↵
int walk(int v, int l, int r, int qv) {↵
push(v, l, r);↵
if (last[v] <= qv) return -1;↵
if (l == r) return l;↵
int md = (l + r) >> 1;↵
int ans = walk(v << 1, l, md, qv);↵
if (ans == -1) ans = walk(v << 1 | 1, md + 1, r, qv);↵
return ans;↵
}↵
↵
int main() {↵
ios_base::sync_with_stdio(false); cin.tie(nullptr);↵
int N, K; cin >> N >> K >> L;↵
vector<int> X(N), Y(N), C(N);↵
vector<int> xs = {-1, L};↵
vector<int> ys = {-1, L};↵
for (int i = 0; i < N; ++i) {↵
cin >> X[i] >> Y[i] >> C[i];↵
--C[i];↵
xs.emplace_back(X[i]);↵
ys.emplace_back(Y[i]);↵
}↵
sort(xs.begin(), xs.end());↵
xs.resize(unique(xs.begin(), xs.end()) - xs.begin());↵
sort(ys.begin(), ys.end());↵
ys.resize(unique(ys.begin(), ys.end()) - ys.begin());↵
int NX = xs.size();↵
int NY = ys.size();↵
{↵
vector<int> order(N);↵
iota(order.begin(), order.end(), 0);↵
sort(order.begin(), order.end(), [&](int i, int j) {↵
return make_pair(Y[i], -X[i]) > make_pair(Y[j], -X[j]);↵
});↵
vector<int> newX(N), newY(N), newC(N);↵
for (int i = 0; i < N; ++i) {↵
newX[i] = X[order[i]];↵
newY[i] = Y[order[i]];↵
newC[i] = C[order[i]];↵
}↵
X.swap(newX), Y.swap(newY), C.swap(newC);↵
}↵
init(1, 1, NX - 2, xs);↵
int ans = 0;↵
for (int yr = 1; yr + 1 < NY; ++yr) {↵
vector<vector<int>> addAt(NX);↵
for (int i = 0; i < N; ++i) {↵
if (Y[i] <= ys[yr]) {↵
int xi = lower_bound(xs.begin(), xs.end(), X[i]) - xs.begin();↵
addAt[xi].emplace_back(C[i]);↵
}↵
}↵
int bad = K;↵
vector<int> cnts(K);↵
auto inc = [&](int z) {↵
if (++cnts[z] == 1) --bad;↵
};↵
auto dec = [&](int z) {↵
if (--cnts[z] == 0) ++bad;↵
};↵
vector<int> go(NX);↵
int ptr = 0;↵
for (int i = 1; i + 1 < NX; ++i) {↵
while (bad && ptr + 2 < NX) {↵
ptr++;↵
for (int z : addAt[ptr]) inc(z);↵
}↵
if (bad) go[i] = L;↵
else go[i] = xs[ptr];↵
for (int z : addAt[i]) dec(z);↵
}↵
reset(1, 1, NX - 2, go);↵
vector<int> prv(N);↵
vector<int> nxt(N);↵
vector<map<int, int>> mp(K);↵
for (int i = 0; i < N; ++i) {↵
if (Y[i] <= ys[yr]) {↵
auto it = mp[C[i]].lower_bound(X[i]);↵
if (it == mp[C[i]].end()) {↵
nxt[i] = -1;↵
} else {↵
nxt[i] = it->second;↵
}↵
it = mp[C[i]].upper_bound(X[i]);↵
if (it == mp[C[i]].begin()) {↵
prv[i] = -1;↵
} else {↵
prv[i] = prev(it)->second;↵
}↵
mp[C[i]][X[i]] = i;↵
}↵
}↵
auto remove = [&](int i) {↵
int xprv = (prv[i] == -1 ? -1 : X[prv[i]]);↵
int xcur = X[i];↵
int xnxt = (nxt[i] == -1 ? L : X[nxt[i]]);↵
int l = lower_bound(xs.begin(), xs.end(), xprv) - xs.begin() + 1;↵
int r = walk(1, 1, NX - 2, xnxt);↵
if (r == -1) r = NX - 1; --r;↵
r = min(r, int(lower_bound(xs.begin(), xs.end(), xcur) - xs.begin()));↵
if (l <= r) modify(1, 1, NX - 2, l, r, xnxt);↵
};↵
ptr = N - 1;↵
for (int yl = 1; yl <= yr; ++yl) {↵
ll add = sum[1] % MOD * (ys[yr + 1] - ys[yr]) % MOD * (ys[yl] - ys[yl - 1]) % MOD;↵
ans = (ans + add) % MOD;↵
while (ptr >= 0 && Y[ptr] == ys[yl]) remove(ptr--);↵
}↵
assert(sum[1] == 0);↵
}↵
cout << ans << "\n";↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
[tutorial:1396E]↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
using ll = long long;↵
↵
int main() {↵
ios_base::sync_with_stdio(false); cin.tie(nullptr);↵
int N; ll K;↵
cin >> N >> K;↵
vector<vector<int>> adj(N);↵
for (int i = 0; i < N - 1; ++i) {↵
int v, u;↵
cin >> v >> u;↵
adj[--v].emplace_back(--u);↵
adj[u].emplace_back(v);↵
}↵
vector<int> sz(N);↵
function<void(int, int)> dfs1 = [&](int v, int p) {↵
sz[v] = 1;↵
for (int u : adj[v]) if (u != p) {↵
dfs1(u, v);↵
sz[v] += sz[u];↵
}↵
};↵
dfs1(0, -1);↵
int root = 0;↵
for (int i = 1; i < N; ++i) {↵
if (sz[i] >= N / 2 && sz[i] < sz[root]) root = i;↵
}↵
vector<int> dist(N);↵
vector<int> top(N, -1);↵
vector<int> par(N, -1);↵
ll low = 0, high = 0;↵
function<void(int, int, int)> dfs2 = [&](int v, int p, int r) {↵
dist[v] = dist[p] + 1;↵
top[v] = r;↵
par[v] = p;↵
{↵
auto it = find(adj[v].begin(), adj[v].end(), p);↵
assert(it != adj[v].end());↵
adj[v].erase(it);↵
}↵
sz[v] = 1;↵
for (int u : adj[v]) {↵
dfs2(u, v, r);↵
sz[v] += sz[u];↵
}↵
low += (sz[v] & 1);↵
high += sz[v];↵
};↵
for (int v : adj[root]) {↵
dfs2(v, root, v);↵
}↵
if (low > K || high < K || (high - K) % 2) {↵
cout << "NO\n";↵
return 0;↵
}↵
set<pair<int, int>> sizes;↵
for (int v : adj[root]) {↵
sizes.emplace(sz[v], v);↵
}↵
vector<set<pair<int, int>>> lcas(N);↵
vector<int> deg(N);↵
for (int v = 0; v < N; ++v) deg[v] = adj[v].size();↵
for (int v = 0; v < N; ++v) if (v != root) {↵
if (deg[v] == 0) {↵
} else {↵
lcas[top[v]].emplace(dist[v], v);↵
}↵
}↵
vector<bool> matched(N);↵
function<void(int)> kill = [&](int v) {↵
assert(deg[v] == 0);↵
if (--deg[par[v]] == 0) {↵
v = par[v];↵
lcas[top[v]].erase(pair<int, int>(dist[v], v));↵
}↵
};↵
cout << "YES\n";↵
while (high > K) {↵
assert(sizes.size());↵
int v = (--sizes.end())->second;↵
sizes.erase(pair<int, int>(sz[v], v));↵
assert(lcas[v].size());↵
int mdist = (--lcas[v].end())->first;↵
if (high - 2 * mdist <= K) {↵
int x = lcas[v].lower_bound(pair<int, int>((high - K) / 2, -1))->second;↵
int y = -1;↵
for (int z : adj[x]) if (!matched[z]) {↵
y = z;↵
break;↵
}↵
high = K;↵
cout << x + 1 << " " << y + 1 << "\n";↵
matched[x] = true;↵
matched[y] = true;↵
break;↵
} else {↵
high -= 2 * mdist;↵
assert(lcas[v].size());↵
int u = (--lcas[v].end())->second;↵
vector<int> nxts;↵
while (nxts.size() < 2 && adj[u].size()) {↵
int w = adj[u].back();↵
adj[u].pop_back();↵
if (!matched[w]) {↵
nxts.emplace_back(w);↵
}↵
}↵
if (nxts.size() < 2) nxts.emplace_back(u);↵
assert(nxts.size() == 2);↵
cout << nxts[0] + 1 << " " << nxts[1] + 1 << "\n";↵
matched[nxts[0]] = true;↵
matched[nxts[1]] = true;↵
kill(nxts[0]);↵
kill(nxts[1]);↵
sz[v] -= 2;↵
if (sz[v]) sizes.emplace(sz[v], v);↵
}↵
}↵
vector<int> seq;↵
function<void(int)> dfs3 = [&](int v) {↵
if (!matched[v]) seq.emplace_back(v);↵
for (int u : adj[v]) dfs3(u);↵
};↵
dfs3(root);↵
int h = seq.size() / 2;↵
for (int i = 0; i < h; ++i) {↵
cout << seq[i] + 1 << " " << seq[i + h] + 1 << "\n";↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
Anyway, here are the tutorials for each of the problems:↵
↵
###[problem:1397A]↵
↵
If the total number of occurrences of some character $c$ is not a multiple of $n$, then it is impossible to make all $n$ strings equal — because then it is impossible for all $n$ strings to have the same number of $c$.↵
↵
On the other hand, if the total number of occurrences of every character $c$ is a multiple of $n$, then it is always possible to make all $n$ strings equal. To achieve this, for every character $c$ we move exactly ((the total number of occurrences of $c$) $/$ $n$) characters $c$ to the end of each string, and by the end we will have all $n$ strings equal each other.↵
↵
We can easily check if the condition satisfies by counting the total number of occurrences of each character $c$ and check its divisibility by $n$. The final complexity is $O(S \cdot 26)$ or $O(S)$ where $S$ is the sum of lengths of all strings.↵
↵
<spoiler summary="C++ solution">```cpp↵
↵
#include <iostream>↵
#include <vector>↵
↵
using namespace std;↵
↵
int main()↵
{↵
int tests;↵
cin >> tests;↵
while (tests--) {↵
int n;↵
cin >> n;↵
↵
vector<int> cnt(26);↵
for (int i = 0; i < n; ++i) {↵
string s;↵
cin >> s;↵
for (char ch : s) {↵
++cnt[ch - 'a'];↵
}↵
}↵
↵
bool ans = true;↵
for (int i = 0; i < 26; ++i) {↵
if (cnt[i] % n != 0) {↵
ans = false;↵
break;↵
}↵
}↵
↵
cout << (ans ? "YES" : "NO") << endl;↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
numTests = int(input())↵
for testNo in range(numTests):↵
n = int(input())↵
cnt = [0 for i in range(26)]↵
for _ in range(n):↵
s = input()↵
for i in s:↵
cnt[ord(i) - 97] += 1↵
↵
ans = True↵
for i in range(26):↵
if cnt[i] % n != 0:↵
ans = False↵
break↵
↵
if ans:↵
print('YES')↵
else:↵
print('NO')↵
```↵
</spoiler>↵
↵
###[problem:1397B]↵
↵
First of all, the optimal way to reorder is to sort $a$ in non-decreasing order.↵
↵
<spoiler summary="Proof">↵
The cost to transform $a_i$ to $c^i$ is $\lvert a_i - c^i \rvert$, and $\lvert a_i - c^i \rvert + \lvert a_j - c^j \rvert \le \lvert a_j - c^i \rvert + \lvert a_i - c^j \rvert$ when $i < j$ and $a_i \le a_j$, thus it is optimal to have $a_i \le a_j$ for each $0 \le i < j < n$.↵
</spoiler>↵
↵
From now on, we assume $a$ is sorted in non-decreasing order.↵
↵
Denote $a_{max} = a_{n - 1}$ as the maximum value in $a$, $f(x) = \sum{\lvert a_i - x^i \rvert}$ as the minimum cost to transform $a$ into $\{x^0, x^1, \cdots, x^{n-1}\}$, and $c$ as the value where $f(c)$ is minimum.↵
↵
Note that $c^{n - 1} - a_{max} \le f(c) \le f(1)$, which implies $c^{n - 1} \le f(1) + a_{max}$.↵
↵
We enumerate $x$ from $1, 2, 3, \dots$ until $x^{n - 1}$ exceeds $f(1) + a_{max}$, calculate $f(x)$ in $O(n)$, and the final answer is the minimum among all calculated values. The final complexity is $O(n \cdot max(x))$.↵
↵
But why doesn't this get TLE? Because $f(1) = \sum{(a_i - 1)} < a_{max} \cdot n \le 10^9 \cdot n$, thus $x^{n - 1} \le f(1) + a_{max} \le 10^9 \cdot (n + 1)$. When $n = 3, 4, 5, 6$, $max(x)$ does not exceed $63245, 1709, 278, 93$ respectively; so we can see that $O(n \cdot max(x))$ comfortably fits in the time limit.↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <iostream>↵
#include <vector>↵
#include <algorithm>↵
#include <numeric>↵
↵
using namespace std;↵
↵
const int64_t INF = 1e17;↵
inline int64_t mul(int64_t a, int64_t b)↵
{↵
return (INF / a > b ? a * b : INF);↵
}↵
↵
inline int64_t add(int64_t a, int64_t b)↵
{↵
return (a + b >= INF ? INF : a + b);↵
}↵
↵
int main()↵
{↵
int n;↵
cin >> n;↵
↵
vector<int> a(n);↵
for (int &x : a) cin >> x;↵
sort(a.begin(), a.end());↵
↵
if (n <= 2) {↵
cout << a[0] - 1 << endl;↵
} else {↵
int64_t ans = accumulate(a.begin(), a.end(), 0ll) - n;↵
↵
for (int x = 1; ; ++x) {↵
int64_t curPow = 1, curCost = 0;↵
for (int i = 0; i < n; ++i) {↵
curCost = add(curCost, abs(a[i] - curPow));↵
curPow = mul(curPow, x);↵
}↵
↵
if (curPow == INF || curPow / x > ans + a[n - 1]) break;↵
ans = min(ans, curCost);↵
}↵
↵
cout << ans << endl;↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
n = int(input())↵
a = [int(x) for x in input().split()]↵
a.sort()↵
inf = 10**18↵
↵
if n <= 2:↵
print(a[0] - 1)↵
else:↵
ans = sum(a) - n↵
↵
for x in range(1, 10**9):↵
curPow = 1↵
curCost = 0↵
for i in range(n):↵
curCost += abs(a[i] - curPow)↵
curPow *= x↵
if curPow > inf:↵
break↵
↵
if curPow > inf:↵
break↵
if curPow / x > ans + a[n - 1]:↵
break↵
↵
ans = min(ans, curCost)↵
↵
print(ans)↵
```↵
</spoiler>↵
↵
↵
###[problem:1396A]↵
↵
In this problem, the answer is rather simple. Here is one possible solution to this task.↵
↵
<spoiler summary="Solution for n = 1">↵
$1 \space \space 1$ ↵
$0$ ↵
$1 \space \space 1$ ↵
$0$ ↵
$1 \space \space 1$ ↵
$-a_1$↵
</spoiler>↵
↵
<spoiler summary="Solution for n != 1">↵
$1 \space \space 1$ ↵
$-a_1$ ↵
$1 \space \space n$ ↵
$0, \space -n \cdot a_2, \space -n \cdot a_3, \space \dots , \space -n \cdot a_n$ ↵
$2 \space \space n$ ↵
$(n-1) \cdot a_2, \space (n-1) \cdot a_3, \space \dots , \space (n-1) \cdot a_n$↵
</spoiler>↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
using ll = long long;↵
↵
int main() {↵
ios_base::sync_with_stdio(false); cin.tie(nullptr);↵
int N;↵
cin >> N;↵
vector<ll> A(N);↵
for (int i = 0; i < N; ++i) cin >> A[i];↵
if (N == 1) {↵
for (int z = 0; z < 3; ++z) {↵
cout << "1 1\n";↵
cout << -A[0] << "\n";↵
A[0] = 0;↵
}↵
return 0;↵
}↵
cout << "1 " << N << "\n";↵
for (int i = 0; i + 1 < N; ++i) cout << -A[i] * N << " "; cout << "0\n";↵
cout << "1 " << N - 1 << "\n";↵
for (int i = 0; i + 1 < N; ++i) cout << A[i] * (N - 1) << " "; cout << "\n";↵
cout << N << " " << N << "\n";↵
cout << -A[N - 1] << "\n";↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
n = int(input())↵
a = list(map(int, input().split()))↵
↵
if n == 1:↵
print('1 1', -a[0], '1 1', '0', '1 1', '0', sep='\n')↵
exit(0)↵
↵
print(1, n)↵
for i in range(n):↵
print(-a[i] * n, end = ' ')↵
a[i] -= a[i] * n↵
print()↵
↵
print(1, n - 1)↵
for i in range(n - 1):↵
print(-a[i], end = ' ')↵
a[i] = 0↵
print()↵
↵
print(2, n)↵
for i in range(1, n):↵
print(-a[i], end = ' ')↵
a[i] = 0↵
print()↵
```↵
</spoiler>↵
↵
###[problem:1396B]↵
↵
Let us denote $S$ as the current total number of stones. ↵
↵
Consider the following cases:↵
↵
**Case A: There is a pile that has more than $\lfloor \frac{S}{2} \rfloor$ stones.**↵
↵
The first player (T) can always choose from this pile, thus he (T) is the winner.↵
↵
**Case B: Every pile has at most $\lfloor \frac{S}{2} \rfloor$ stones, and $S$ is even.**↵
↵
It can be proven that the second player (HL) always wins.↵
↵
<spoiler summary="Proof 1">↵
Let us prove by induction:↵
↵
When $S = 0$, the second player obviously wins. ↵
↵
When $S \geq 2$, consider the game state after the first player moves. If there is a pile that now has more than $\lfloor \frac{S}{2} \rfloor$ stones, then we arrive back at _case A_ where the next player to move wins. Otherwise, the second player ↵
can choose from any valid pile (note that the case condition implies that there are at least two non-empty piles before the ↵
first player's move). Now $S$ has been reduced by $2$, and every pile still has at most $\lfloor \frac{S}{2} \rfloor$ stones.↵
</spoiler>↵
↵
<spoiler summary="Proof 2">↵
The condition allows us to assign a perfect matching of stones, where one stone is matched with exactly one stone ↵
from a different pile. ↵
↵
A greedy way to create such a matching: Give each label $0, 1, \dots, S - 1$ to a different stone so that for every pair of ↵
stones with labels $l < r$ that are from the same pile, stones $l + 1, l + 2, \dots, r - 1$ are also from that pile; then ↵
match stones $i$ with $i + \frac{S}{2}$ for all $0 \le i < \frac{S}{2}$.↵
↵
For every stone that the first player removes, the second player can always remove its matching stone, until the first player ↵
can no longer make a move and loses.↵
</spoiler>↵
↵
**Case C: Every pile has at most $\lfloor \frac{S}{2} \rfloor$ stones, and $S$ is odd.**↵
↵
The first player (T) can choose from any pile, and we arrive back at _case B_ where the next player to move loses. ↵
↵
So the first player (T) wins if and only if there is a pile that has more than $\lfloor \frac{S}{2} \rfloor$ stones or $S$ ↵
is odd. This can be easily checked in $O(n)$.↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <iostream>↵
#include <vector>↵
#include <algorithm>↵
#include <numeric>↵
↵
using namespace std;↵
↵
int main()↵
{↵
int t;↵
cin >> t;↵
while (t--) {↵
int n;↵
cin >> n;↵
↵
vector<int> a(n);↵
for (int &x : a) cin >> x;↵
↵
int maxPile = *max_element(a.begin(), a.end());↵
int numStones = accumulate(a.begin(), a.end(), 0);↵
↵
if (maxPile * 2 > numStones || (numStones & 1)) cout << "T" << endl;↵
else cout << "HL" << endl;↵
}↵
}↵
```↵
</spoiler>↵
↵
↵
<spoiler summary="Python solution">```python↵
t = int(input())↵
for _ in range(t):↵
n = int(input())↵
a = [int(x) for x in input().split()]↵
↵
maxPile = max(a)↵
numStones = sum(a)↵
↵
if maxPile * 2 > numStones or (numStones & 1):↵
print('T')↵
else:↵
print('HL')↵
```↵
</spoiler>↵
↵
###[problem:1396C]↵
↵
In this problem, it is useful to note that when the boss only has $1$ hp left, just use the pistol because it has the least reloading time. So there are 3 strategies we will use when playing at stage $i$ $(1 \le i \le n)$:↵
↵
* Take $a_i$ pistol shots to kill first $a_i$ monsters and shoot the boss with the AWP.↵
* Take $a_i + 1$ pistol shots and move back to this stage later to take another pistol shot to finish the boss.↵
* Use the laser gun and move back to this stage later to kill the boss with a pistol shot.↵
↵
**Observation:** We will always finish the game at stage $n$ or $n - 1$. Considering we are at stage $i$ $(i \le n - 1)$ and the boss at both stage $i$ stage $i - 1$ has $1$ hp left, we can spend $2 * d$ time to finish both these stages instead of going back later, which costs us exactly the same.↵
↵
Therefore, we will calculate $dp(i,0/1)$ as the minimum time to finish first $a_i - 1$ stages and 0/1 is the remaining hp of the boss at stage i. The transitions are easy to figure out by using 3 strategies as above. The only thing we should note is that we can actually finish the game at stage $n - 1$ by instantly kill the boss at stage $n$ with the AWP so we don't have to go back to this level later.↵
↵
Answer to the problem is $dp(n, 0)$. Time complexity: $O(n)$.↵
↵
<spoiler summary="C++ solution">```cpp↵
/*input↵
4 2 4 4 1↵
4 5 1 2↵
*/↵
#include <bits/stdc++.h>↵
using namespace std;↵
↵
int read() {↵
int x = 0, c = getchar();↵
for(; !(c > 47 && c < 58); c = getchar());↵
for(; (c > 47 && c < 58); c = getchar()) x = x * 10 + c - 48;↵
return x;↵
}↵
↵
void upd(long long &a, long long b) {↵
a = (a < b) ? a : b;↵
}↵
↵
const int N = 1e6 + 5;↵
↵
long long f[N][2];↵
int n, r1, r2, r3, d, a[N];↵
↵
int main(){ ↵
n = read(), r1 = read(), r2 = read(), r3 = read(), d = read();↵
for(int i = 1; i <= n; a[i ++] = read());↵
↵
for(int i = 2; i <= n; ++ i) f[i][0] = f[i][1] = 1e18;↵
↵
f[1][0] = 1ll * r1 * a[1] + r3;↵
f[1][1] = min(0ll + r2, 1ll * r1 * a[1] + r1);↵
for(int i = 1; i < n; ++ i) {↵
// 0 -> 0↵
// so we clear this one and the next one as well↵
upd(f[i + 1][0], f[i][0] + d + 1ll * r1 * a[i + 1] + r3);↵
↵
// 0 -> 1↵
// this one is cleared, but next one isnt↵
upd(f[i + 1][1], f[i][0] + d + min(0ll + r2, 1ll * r1 * a[i + 1] + r1));↵
↵
// 1 -> 0↵
upd(f[i + 1][0], f[i][1] + d + 1ll * r1 * a[i + 1] + r3 + 2 * d + r1);↵
upd(f[i + 1][0], f[i][1] + d + 1ll * r1 * a[i + 1] + r1 + d + r1 + d + r1);↵
upd(f[i + 1][0], f[i][1] + d + r2 + d + r1 + d + r1);↵
↵
// 1 -> 1↵
upd(f[i + 1][1], f[i][1] + d + r2 + d + r1 + d);↵
upd(f[i + 1][1], f[i][1] + d + 1ll * r1 * a[i + 1] + r1 + d + r1 + d);↵
↵
if(i == n - 1) {↵
upd(f[i + 1][0], f[i][1] + d + 1ll * r1 * a[i + 1] + r3 + d + r1);↵
}↵
}↵
cout << f[n][0] << endl;↵
}↵
```↵
</spoiler>↵
↵
[tutorial:1396D]↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
using ll = long long;↵
const int MOD = 1000000007;↵
↵
int L;↵
ll sum[8040];↵
int len[8040];↵
int last[8040];↵
int lazy[8040];↵
↵
void init(int v, int l, int r, const vector<int> &xs) {↵
len[v] = xs[r] - xs[l - 1];↵
if (l < r) {↵
int md = (l + r) >> 1;↵
init(v << 1, l, md, xs);↵
init(v << 1 | 1, md + 1, r, xs);↵
}↵
}↵
↵
void reset(int v, int l, int r, const vector<int> &go) {↵
lazy[v] = -1;↵
if (l == r) {↵
sum[v] = ll(len[v]) * (L - go[l]);↵
last[v] = go[l];↵
return;↵
}↵
int md = (l + r) >> 1;↵
reset(v << 1, l, md, go);↵
reset(v << 1 | 1, md + 1, r, go);↵
sum[v] = sum[v << 1] + sum[v << 1 | 1];↵
last[v] = last[v << 1 | 1];↵
}↵
↵
void push(int v, int l, int r) {↵
if (lazy[v] != -1) {↵
last[v] = lazy[v];↵
sum[v] = ll(len[v]) * (L - lazy[v]);↵
if (l < r) {↵
lazy[v << 1] = lazy[v];↵
lazy[v << 1 | 1] = lazy[v];↵
}↵
lazy[v] = -1;↵
}↵
}↵
↵
void modify(int v, int l, int r, int L, int R, int qv) {↵
push(v, l, r);↵
if (L > r || R < l) return;↵
if (L <= l && r <= R) {↵
lazy[v] = qv;↵
push(v, l, r);↵
return;↵
}↵
int md = (l + r) >> 1;↵
modify(v << 1, l, md, L, R, qv);↵
modify(v << 1 | 1, md + 1, r, L, R, qv);↵
sum[v] = sum[v << 1] + sum[v << 1 | 1];↵
last[v] = last[v << 1 | 1];↵
}↵
↵
int walk(int v, int l, int r, int qv) {↵
push(v, l, r);↵
if (last[v] <= qv) return -1;↵
if (l == r) return l;↵
int md = (l + r) >> 1;↵
int ans = walk(v << 1, l, md, qv);↵
if (ans == -1) ans = walk(v << 1 | 1, md + 1, r, qv);↵
return ans;↵
}↵
↵
int main() {↵
ios_base::sync_with_stdio(false); cin.tie(nullptr);↵
int N, K; cin >> N >> K >> L;↵
vector<int> X(N), Y(N), C(N);↵
vector<int> xs = {-1, L};↵
vector<int> ys = {-1, L};↵
for (int i = 0; i < N; ++i) {↵
cin >> X[i] >> Y[i] >> C[i];↵
--C[i];↵
xs.emplace_back(X[i]);↵
ys.emplace_back(Y[i]);↵
}↵
sort(xs.begin(), xs.end());↵
xs.resize(unique(xs.begin(), xs.end()) - xs.begin());↵
sort(ys.begin(), ys.end());↵
ys.resize(unique(ys.begin(), ys.end()) - ys.begin());↵
int NX = xs.size();↵
int NY = ys.size();↵
{↵
vector<int> order(N);↵
iota(order.begin(), order.end(), 0);↵
sort(order.begin(), order.end(), [&](int i, int j) {↵
return make_pair(Y[i], -X[i]) > make_pair(Y[j], -X[j]);↵
});↵
vector<int> newX(N), newY(N), newC(N);↵
for (int i = 0; i < N; ++i) {↵
newX[i] = X[order[i]];↵
newY[i] = Y[order[i]];↵
newC[i] = C[order[i]];↵
}↵
X.swap(newX), Y.swap(newY), C.swap(newC);↵
}↵
init(1, 1, NX - 2, xs);↵
int ans = 0;↵
for (int yr = 1; yr + 1 < NY; ++yr) {↵
vector<vector<int>> addAt(NX);↵
for (int i = 0; i < N; ++i) {↵
if (Y[i] <= ys[yr]) {↵
int xi = lower_bound(xs.begin(), xs.end(), X[i]) - xs.begin();↵
addAt[xi].emplace_back(C[i]);↵
}↵
}↵
int bad = K;↵
vector<int> cnts(K);↵
auto inc = [&](int z) {↵
if (++cnts[z] == 1) --bad;↵
};↵
auto dec = [&](int z) {↵
if (--cnts[z] == 0) ++bad;↵
};↵
vector<int> go(NX);↵
int ptr = 0;↵
for (int i = 1; i + 1 < NX; ++i) {↵
while (bad && ptr + 2 < NX) {↵
ptr++;↵
for (int z : addAt[ptr]) inc(z);↵
}↵
if (bad) go[i] = L;↵
else go[i] = xs[ptr];↵
for (int z : addAt[i]) dec(z);↵
}↵
reset(1, 1, NX - 2, go);↵
vector<int> prv(N);↵
vector<int> nxt(N);↵
vector<map<int, int>> mp(K);↵
for (int i = 0; i < N; ++i) {↵
if (Y[i] <= ys[yr]) {↵
auto it = mp[C[i]].lower_bound(X[i]);↵
if (it == mp[C[i]].end()) {↵
nxt[i] = -1;↵
} else {↵
nxt[i] = it->second;↵
}↵
it = mp[C[i]].upper_bound(X[i]);↵
if (it == mp[C[i]].begin()) {↵
prv[i] = -1;↵
} else {↵
prv[i] = prev(it)->second;↵
}↵
mp[C[i]][X[i]] = i;↵
}↵
}↵
auto remove = [&](int i) {↵
int xprv = (prv[i] == -1 ? -1 : X[prv[i]]);↵
int xcur = X[i];↵
int xnxt = (nxt[i] == -1 ? L : X[nxt[i]]);↵
int l = lower_bound(xs.begin(), xs.end(), xprv) - xs.begin() + 1;↵
int r = walk(1, 1, NX - 2, xnxt);↵
if (r == -1) r = NX - 1; --r;↵
r = min(r, int(lower_bound(xs.begin(), xs.end(), xcur) - xs.begin()));↵
if (l <= r) modify(1, 1, NX - 2, l, r, xnxt);↵
};↵
ptr = N - 1;↵
for (int yl = 1; yl <= yr; ++yl) {↵
ll add = sum[1] % MOD * (ys[yr + 1] - ys[yr]) % MOD * (ys[yl] - ys[yl - 1]) % MOD;↵
ans = (ans + add) % MOD;↵
while (ptr >= 0 && Y[ptr] == ys[yl]) remove(ptr--);↵
}↵
assert(sum[1] == 0);↵
}↵
cout << ans << "\n";↵
return 0;↵
}↵
```↵
</spoiler>↵
↵
[tutorial:1396E]↵
↵
<spoiler summary="C++ solution">```cpp↵
#include <bits/stdc++.h>↵
↵
using namespace std;↵
↵
using ll = long long;↵
↵
int main() {↵
ios_base::sync_with_stdio(false); cin.tie(nullptr);↵
int N; ll K;↵
cin >> N >> K;↵
vector<vector<int>> adj(N);↵
for (int i = 0; i < N - 1; ++i) {↵
int v, u;↵
cin >> v >> u;↵
adj[--v].emplace_back(--u);↵
adj[u].emplace_back(v);↵
}↵
vector<int> sz(N);↵
function<void(int, int)> dfs1 = [&](int v, int p) {↵
sz[v] = 1;↵
for (int u : adj[v]) if (u != p) {↵
dfs1(u, v);↵
sz[v] += sz[u];↵
}↵
};↵
dfs1(0, -1);↵
int root = 0;↵
for (int i = 1; i < N; ++i) {↵
if (sz[i] >= N / 2 && sz[i] < sz[root]) root = i;↵
}↵
vector<int> dist(N);↵
vector<int> top(N, -1);↵
vector<int> par(N, -1);↵
ll low = 0, high = 0;↵
function<void(int, int, int)> dfs2 = [&](int v, int p, int r) {↵
dist[v] = dist[p] + 1;↵
top[v] = r;↵
par[v] = p;↵
{↵
auto it = find(adj[v].begin(), adj[v].end(), p);↵
assert(it != adj[v].end());↵
adj[v].erase(it);↵
}↵
sz[v] = 1;↵
for (int u : adj[v]) {↵
dfs2(u, v, r);↵
sz[v] += sz[u];↵
}↵
low += (sz[v] & 1);↵
high += sz[v];↵
};↵
for (int v : adj[root]) {↵
dfs2(v, root, v);↵
}↵
if (low > K || high < K || (high - K) % 2) {↵
cout << "NO\n";↵
return 0;↵
}↵
set<pair<int, int>> sizes;↵
for (int v : adj[root]) {↵
sizes.emplace(sz[v], v);↵
}↵
vector<set<pair<int, int>>> lcas(N);↵
vector<int> deg(N);↵
for (int v = 0; v < N; ++v) deg[v] = adj[v].size();↵
for (int v = 0; v < N; ++v) if (v != root) {↵
if (deg[v] == 0) {↵
} else {↵
lcas[top[v]].emplace(dist[v], v);↵
}↵
}↵
vector<bool> matched(N);↵
function<void(int)> kill = [&](int v) {↵
assert(deg[v] == 0);↵
if (--deg[par[v]] == 0) {↵
v = par[v];↵
lcas[top[v]].erase(pair<int, int>(dist[v], v));↵
}↵
};↵
cout << "YES\n";↵
while (high > K) {↵
assert(sizes.size());↵
int v = (--sizes.end())->second;↵
sizes.erase(pair<int, int>(sz[v], v));↵
assert(lcas[v].size());↵
int mdist = (--lcas[v].end())->first;↵
if (high - 2 * mdist <= K) {↵
int x = lcas[v].lower_bound(pair<int, int>((high - K) / 2, -1))->second;↵
int y = -1;↵
for (int z : adj[x]) if (!matched[z]) {↵
y = z;↵
break;↵
}↵
high = K;↵
cout << x + 1 << " " << y + 1 << "\n";↵
matched[x] = true;↵
matched[y] = true;↵
break;↵
} else {↵
high -= 2 * mdist;↵
assert(lcas[v].size());↵
int u = (--lcas[v].end())->second;↵
vector<int> nxts;↵
while (nxts.size() < 2 && adj[u].size()) {↵
int w = adj[u].back();↵
adj[u].pop_back();↵
if (!matched[w]) {↵
nxts.emplace_back(w);↵
}↵
}↵
if (nxts.size() < 2) nxts.emplace_back(u);↵
assert(nxts.size() == 2);↵
cout << nxts[0] + 1 << " " << nxts[1] + 1 << "\n";↵
matched[nxts[0]] = true;↵
matched[nxts[1]] = true;↵
kill(nxts[0]);↵
kill(nxts[1]);↵
sz[v] -= 2;↵
if (sz[v]) sizes.emplace(sz[v], v);↵
}↵
}↵
vector<int> seq;↵
function<void(int)> dfs3 = [&](int v) {↵
if (!matched[v]) seq.emplace_back(v);↵
for (int u : adj[v]) dfs3(u);↵
};↵
dfs3(root);↵
int h = seq.size() / 2;↵
for (int i = 0; i < h; ++i) {↵
cout << seq[i] + 1 << " " << seq[i + h] + 1 << "\n";↵
}↵
}↵
```↵
</spoiler>↵
↵
