Codeforces Round #822 (Div.2) Editorial
Codeforces Round #822 (Div.2) Editorial
Thank you for your participation!
1734A — Select Three Sticks
We first sort the array in non-decreasing order.
Denote the indices of the elements that we choose from to be , , and , where , and the final value (after performing the operations) of the concerned elements to be .
The minimum required number of operations is then . It is well-known that such expression attains its minimum value when is the median of , , and . Since the array has already been sorted, it is best to assign to be .
Our expression then becomes . We would like to minimize the value of , which implies should be as small as possible since is sorted. It is clear that taking would minimize the value of the expression. Similarly, we can show that we can take to minimize the value of the expression.
Therefore, the only possible values of the triplets are of the form for positive integers , and we can iterate through all such triplets and find the best one.
The time complexity is per case due to sorting.
#include <bits/stdc++.h>
using namespace std;
void solve(){
int n;
cin>>n;
int a[n+1];
for (int i=1; i<=n; i++){
cin>>a[i];
}
sort(a+1,a+n+1);
int ans=2e9;
for (int i=3; i<=n; i++){
ans=min(ans,a[i]-a[i-2]);
}
cout<<ans<<'\n';
}
int main(){
int t;
cin>>t;
for (int i=1; i<=t; i++){
solve();
}
}
1734B — Bright, Nice, Brilliant
Note that the brightnesses of the rooms on the -th floor is at most . This is because in room , only rooms, namely, , , , can reach to through some number of staircases.
It is also possible to find a configuration of torches in the pyramid such that the brightnesses of the rooms on the -th floor is exactly , i.e. it attains the upper bound.
The configuration is as follows: Room contains a torch if and only if it is the leftmost room () or the rightmost room () on the -th floor.
This is valid because for all rooms , it can be reached from , , , , and , , , . In other words, room has brightness , so the pyramid is nice.
#include <bits/stdc++.h>
using namespace std;
int main() {
int t;
cin >> t;
while (t--) {
int n;
cin >> n;
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= i; j++) {
cout << (j == 1 || j == i) << ' ';
}
cout << '\n';
}
}
}
1734C — Removing Smallest Multiples
One operation should be used to remove every element not belonging to .
Let be an element not belonging to . Suppose a -cost operation removes value , then must be divisible by . Furthermore, the multiples must have been already removed from , where we write .
Since removed elements stay removed, the above is only possible if all of does not belong to .
For each , let be the smallest integer satisfying the above condition. As we can always remove using a -cost operation, and in particular exists.
The total cost must be at least . We claim that this cost can be achieved.
To do so, we should remove the required elements in ascending order. When removing , we assume all with have already been removed. At this state, an -cost operation would be able to remove .
It remains to find the values . To do so efficiently, we can perform the above process in a bottom-up manner similar to the Sieve of Eratosthenes. Please refer to the code below for implementation details. The overall complexity is .
#include <bits/stdc++.h>
using namespace std;
void solve() {
int n;
cin >> n;
bool a[n + 1];
string str;
cin >> str;
for (int i = 1; i <= n; i++) {
a[i] = (str[i - 1] == '1');
}
long long ans = 0;
int cost[n + 1];
for (int i = n; i >= 1; i--) {
for (int j = i; j <= n; j += i) {
if (a[j]) break;
cost[j] = i;
}
}
for (int i = 1; i <= n; i++) {
if (!a[i]) ans += cost[i];
}
cout << ans << '\n';
}
int main() {
int t;
cin >> t;
while (t--) solve();
}
1734D — Slime Escape
Let's call a group of slime good if their total health is at least , or if defeating this group allows you to reach the exits.
We partition the slimes into good groups in a two-pointer like manner. To form the groups to the right, start from position , then find the smallest position such that slimes from through form a good group. We do the same starting from again. Repeat this process until slimes to the right are partitioned into groups, which can be done by maintaining the sum of health. We partition the left slimes into groups in a similar way.
We can observe that in an optimal strategy, we may assume the player absorbs group-by-group.
Assuming there is a valid strategy to reach the exit. Let be the first group to the left, and be the first group to the right. Without loss of generality, assume all slimes of are absorbed before all slimes of are absorbed first. Suppose there are slimes in and slimes in .
Suppose we did not absorb all of at a time. For example, if there are slimes in and or more slimes in , the beginning of our strategy may look like this:
Consider instead the strategy where we move all `L' to the front. We claim that the strategy stays valid. That is, the following strategy
is at least as good.
We check that at every moment of strategy , there is some point in time which we would have less than or equal amount of health.
For the state that we take moves to the left in , we compare it with any moment in that we have taken moves to the left. To reach , we take moves to the right. We can check that under the assumption, . Since is the smallest good group to the right, taking these extra right moves must have reduced our health. So, we have more health at than in .
Now consider the state in where we take all left moves and more right moves. We should compare it with any moment in that we have taken right moves and some left moves. We can check that under the assumption . Since is the smallest good group, taking only moves to the left gives us less health compared to taking all moves to the left. Therefore, we have more health at than in .
Therefore, if is valid, must be valid. So, we only need to consider strategies of the second form.
By applying this claim recursively, we notice that we only need to consider strategies that absorb whole groups at a time.
This works for all strategies, we just referred to the starting strategy as , and the better strategy as to simplify notations.
For any good group, since the total health is positive, there is no drawback to absorbing a good group. In other words, whenever it is possible to absorb a good group, we will absorb it.
For each group , we calculate the ``requirement'' of the group — the lowest health we can begin with, such that we can absorb the group while maintaining non-negative health at all times. The requirement of a group of slime with health can be expressed as
Finally, we can simply simulate the process. We repeatedly attempt to absorb good groups to the left or to the right. We keep track of the current health, initially equal to . Whenever we consider whether to absorb a group or not, we absorb it if and only if the current health is at least as high as its requirement. Otherwise, we ignore it for now and attempt to do so for the group on the other side. If it is possible to reach a state where either all left groups or all right groups are absorbed, then we can win the game. If at some point, it is not possible to absorb the left group nor the right group, then we lose.
The overall complexity is .
It is also possible to use a range / segment tree form the groups instead of using two-pointers, in which case its complexity would be .
#include <bits/stdc++.h>
using namespace std;
void solve() {
long long n, k, P;
cin >> n >> k;
k--;
long long L[n];
for (int i = 0; i < n; i++) cin >> L[i];
P = L[k];
L[k] = 0;
long long ps[n + 1];
ps[0] = 0;
for (int i = 1; i <= n; i++) ps[i] = ps[i - 1] + L[i - 1];
int l = k, r = k + 1;
vector<pair<long long, long long> > LG, RG;
for (int i = k - 1; i >= 0; i--) {
if (ps[i] <= ps[l] || i == 0) {
lpng long worst = 0, cur = 0;
for (int j = l - 1; j >= i; j--) {
cur += L[j];
worst = min(worst, cur);
}
LG.push_back({cur, -worst});
l = i;
}
}
for (int i = k + 2; i <= n; i++) {
if (ps[i] >= ps[r] || i == n) {
long long worst = 0, cur = 0;
for (int j = r; j <= i - 1; j++) {
cur += L[j];
worst = min(worst, cur);
}
RG.push_back({cur, -worst});
r = i;
}
}
reverse(LG.begin(), LG.end());
reverse(RG.begin(), RG.end());
long long curp = P;
while (true) {
bool acted = false;
if (!LG.empty() && curp >= LG.back().second) {
curp += LG.back().first;
LG.pop_back();
acted = true;
}
if (!RG.empty() && curp >= RG.back().second) {
curp += RG.back().first;
RG.pop_back();
acted = true;
}
if (LG.empty() || RG.empty()) {
cout << "YES\n";
return;
}
if (!acted) {
cout << "NO\n";
return;
}
}
}
int main() {
int t;
cin >> t;
while (t--) solve();
}
1734E — Rectangular Congruences
We say a matrix to be good if it satisfies the congruence condition (the second condition).
When we have a good matrix, we can add any value to a whole row while maintaining the congruence relation. The same is true for adding the same value to a whole column.
Suppose we have any good matrix , then by adding to the -th row for each of , we obtain a good matrix that has the desired values on the diagonal.
In fact, there are a lot of possible constructions. We present a few of them here:
. This needs special handling when .
.
The coolest part is that all quadratic polynomials in the form are valid for all integers and
As a bonus, we prove that the general quadratic polynomial gives a good construction.
Since we can add values to a whole row or column, and are also constant on the rows and columns, adding them by has no effect. So we may just assume .
So it suffices to show that satisfies the condition. We can see directly that . As , , , and is a prime, this expression must be nonzero .
Here are some extra observations that may enable one to find a good matrix more quickly.
For each , the values over all must be a permutation of .
A good matrix stays good if we permute the rows or permute the columns. Therefore, we can show that there exists some good matrix with , and , . Assuming this, it should not be difficult to discover yields one of the good matrices.
Let be the two dimensional difference array of , that is, . Then, the condition becomes sum of any rectangles of must be nonzero . It is easy to see is valid. This corresponds to the solution that .
#include <bits/stdc++.h>
using namespace std;
int want[355];
int board[355][355];
int main() {
int n;
cin >> n;
for(int i = 0; i < n ; i ++){
cin >> want[i];
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
board[i][j] = (i * j) % n;
}
}
for(int i = 0; i < n; i ++){
int extra = (want[i] + n - board[i][i] )% n ;
for (int j = 0; j < n; j++) {
board[i][j] += extra;
board[i][j] %= n;
}
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
cout << board[i][j] << ' ';
}
cout << endl;
}
}
1734F — Zeros and Ones
Observe that the -th character is `1' if and only if has an odd number of set bits in its binary representation. Both solutions make use of this fact.
The constraints allows solutions of up to . Yet, both of the model solution runs in .
1. Digit DP solution
The question can be reformulated as follows: How many integers between and inclusive have the property that the total number of set bits of and is an odd number?
This can be solved with digit DP. We process the bit position from down to . We maintain three states:
, a boolean value;
, an integer between and inclusive; and
, a boolean value.
We can thus conclude the following: the number of trailing zeros is all we need to decide the answer.
After processing each bit , we should have the following: the number of integers between and inclusive which have the following property:
the total number of set bits of and is equal to ;
the number of trailing `1's of is equal to ;
the boolean value (where is the Iverson bracket).
Now onto the transitions. Suppose we are adding the -th digit, and let be the new digit of , and be the -th digit of .
If , then after digit will be transited to after digit ;
if , then after digit will be transited to after digit ;
if , then after digit will be transited to after digit
The final answer is the total number of values for which and .
The above solution runs in per query. There is a simple way to optimize this to : note that we only need to keep parity of .
There are many other digit DP approaches that give similar time complexity. The constraints should allow most of them pass.
val Boolean.chi:Int get() = if(this) 1 else 0 //characteristic function
fun Int.has(i:Int):Boolean = (this and (1 shl i) != 0)
fun Long.has(i:Int):Boolean = (this and (1L shl i) != 0L)
fun main(){
repeat(readLine()!!.toInt()){
val (n,m) = (readLine()!!.split(" ").map { it.toLong() })
var DP = Array(2){Array(2){LongArray(64)}}
DP[0][0][0] = 1
for(bit in 60 downTo 0 ){
var new = Array(2){Array(2){LongArray(64)}}
for(under in 0 until 2){
for(ans in 0 until 2){
for(nd in 0 until 2){
if(under == 0 && nd == 1 && !m.has(bit)) continue
val nunder = ((under == 1) || (nd == 0 && m.has(bit))).chi
val np = nd + n.has(bit).chi
for(trail in 0 until 62){
var nans = ans
if(nd == 1){
nans = 1 - nans
}
var newtrail:Int
if(np == 0){
newtrail = 0
}else if(np == 1){
nans = 1 - nans
newtrail = trail + 1
}else if(np == 2){
nans = (nans + trail + 1) %2
newtrail = 0
}else{
error("oh no")
}
new[nunder][nans][newtrail] += DP[under][ans][trail]
}
}
}
}
DP = new
}
var ret = 0L
for(trail in 0 until 64){
ret += DP[1][1][trail]
}
println(ret)
}
}
2. Recursive Solution, by darkkcyan
Define the function the parity of bit one in the number .
We have thus reworded the statement into evaluating the follow expression:
The formula can be further transformed as:
since holds true for all non-negative integers and .
Imagine we construct a grid and assign the value at row and column to be . Then, is sum of a diagonal of length which starts at either or . Without loss of generality, we use in this editorial.
The grid can be constructed similarly to the way we construct the string . We start with a -by- matrix .
Then, the matrix of size can be constructed as follows:
where is the matrix but with flipped bits.
Here is another way of constructing the grid: let be an infinite chess board with alternating colors, similar to a regular chess board, but with each of the cells being size . For example, , and in an -by- grid is as follows:
.*.*.*.*
*.*.*.*.
.*.*.*.*
*.*.*.*.
.*.*.*.*
*.*.*.*.
.*.*.*.*
*.*.*.*.
..**..**
..**..**
**..**..
**..**..
..**..**
..**..**
**..**..
**..**..
....****
....****
....****
....****
****....
****....
****....
****....
We claim that our grid is the of all chess board . The proof is easy: is constructed by -ing the -th bit of the the row and column number.
We are therefore motivated to proceed in the following way: if we drop the least significant bit (by making it to be ), we are still solving a very similar problem to the original problem, because dropping the first bit is similar to removing . And when we shift to , it is a recursion of the same problem!
Going back to the problem, where we are computing sum of a diagonal of length . If is odd, we can make it odd by adding the last element to the result and decreasing by one. Now, is even, and we can utilize the recurrence as follows:
remove .
scale the board down by 2 (including and ). By doing so, becomes .
solve the new problem.
scale the board up again and add back.
from the result of the scaled down problem, some how calculate the result of the original problem
The result of the scaled down problem is the number of -by- cells with value . From the number of -by- cells with value , we compute the number of cells with value as well. It is not hard to observe that it crosses the -by- cells at all places. The only thing that matters is the parity of .
If is even, then the diagonal crosses the diagonal of the -by- cells. In the scaled-down version, the diagonal is still a single diagonal starting at ; otherwise,
if is odd, it crosses the corner of the -by- cells. In the scaled-down version, the diagonal is actually neighboring diagonals starting at and .
Also, the -by- cells with values and respectively will also have the form:
.* *.
*. .*
From here we have everything we need to compute the result of the original problem.
Overall, the number of states we have to visit is .
import random
cache = {}
def popcount(n):
res = 0
while n:
res += 1
n &= n - 1
return res
def solve(n, k):
if k == 0:
return 0
if k == 1:
return popcount(n) & 1
if k % 2 == 1:
t = solve(n, k - 1)
x = popcount((k - 1) ^ (n + k - 1)) & 1
# print(t, x)
return t + x
if (n, k) in cache:
return cache[(n, k)]
if n % 2 == 0:
one_cell = 2
zero_cell = 0
cnt1 = solve(n // 2, k // 2)
cnt0 = k // 2 - cnt1
else:
one_cell = 0
zero_cell = 1
cnt1 = solve(n // 2, k // 2) + solve(n // 2 + 1, k // 2)
cnt0 = k - cnt1
res = one_cell * cnt1 + zero_cell * cnt0
# print(f"n = {n}, k = {k}, cnt1 = {cnt1}, cnt0 = {cnt0}, one_cell = {one_cell}, zero_cell = {zero_cell}")
cache[(n, k)] = res
return res
t = int(input())
for _ in range(t):
cache.clear()
n, k = map(int, input().split())
print(solve(n, k))
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