Educational Codeforces Round 69 Editorial

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Tutorial

Solution (adedalic)

fun main(args: Array<String>) {
    val T = readLine()!!.toInt()
    for (tc in 1..T) {
        val n = readLine()!!.toInt()
        val a = readLine()!!.split(' ').map { it.toInt() }.sortedDescending()
        println(minOf(a[1] - 1, n - 2))
    }
}

1197B - Pillars

Idea: BledDest

Tutorial

Solution (BledDest)

#include<bits/stdc++.h>

using namespace std;

int main()
{
	int n;
	scanf("%d", &n);
	vector<int> a(n);
	for(int i = 0; i < n; i++)
		scanf("%d", &a[i]);
	
	int pos = max_element(a.begin(), a.end()) - a.begin();
	bool res = true;
	for(int i = 0; i < pos; i++)
		res &= (a[i] < a[i + 1]);
	for(int i = pos; i < n - 1; i++)
		res &= (a[i] > a[i + 1]);

	if(res)
		puts("YES");
	else
		puts("NO");
}

1197C - Array Splitting

Idea: Roms

Tutorial

Solution (Roms)

#include<bits/stdc++.h>

using namespace std;

const int N = int(3e5) + 99;

int n, k;
int a[N];

int main(){
	cin >> n >> k;
	for(int i = 0; i < n; ++i)
		cin >> a[i];
	
	vector <int> v;
	for(int i = 1; i < n; ++i) v.push_back(a[i - 1] - a[i]);
	
	sort(v.begin(), v.end());
	
	int res = a[n - 1] - a[0];
	for(int i = 0; i < k - 1; ++i) res += v[i];
	
	cout << res << endl;
 	return 0;
}

1197D - Yet Another Subarray Problem

Idea: BledDest

Tutorial

Solution (Roms)

#include <bits/stdc++.h>

using namespace std;

const int N = int(3e5) + 99;

int n, m, k;
int a[N];
long long bst[N];
long long psum[N];

long long sum(int l, int r){
    l = max(l, 0);
	return psum[r] - (l == 0? 0 : psum[l - 1]);
}

int main() {
	cin >> n >> m >> k;
	for(int i = 0; i < n; ++i){
		cin >> a[i];
		psum[i] = a[i] + (i == 0? 0 : psum[i - 1]);
	}

	long long res = 0;
	for(int len = 1; len <= m && len <= n; ++len)
			res = max(res, sum(0, len - 1) - k);
	for(int i = 0; i < n; ++i){
		if(i + 1 >= m){
			long long nbst = sum(i - m + 1, i) - k;
			if(i - m >= 0) nbst += bst[i - m];
			bst[i] = max(bst[i], + nbst);
		}

		for(int len = 0; len < m && i + len < n; ++len)
			res = max(res, bst[i] + sum(i + 1, i + len) - k * (len > 0));
	}	

	cout << res << endl;
	return 0;
}

1197E - Culture Code

Idea: adedalic

Tutorial

Solution (adedalic)

#include<bits/stdc++.h>

using namespace std;

#define fore(i, l, r) for(int i = int(l); i < int(r); i++)
#define sz(a) int((a).size())

#define x first
#define y second

typedef long long li;
typedef long double ld;
typedef pair<int, int> pt;

const int INF = int(1e9) + 555;
const li INF64 = li(1e18);
const ld EPS = 1e-9;

const int MOD = int(1e9) + 7;

int norm(int a) {
	if(a >= MOD) a -= MOD;
	if(a < 0) a += MOD;
	return a;
}

pt combine(const pt &a, const pt &b) {
	if(a.x < b.x)
		return a;
	if(a.x > b.x)
		return b;
	return {a.x, norm(a.y + b.y)};
}

int n;
vector<pt> p;

inline bool read() {
	if(!(cin >> n))
		return false;
	p.resize(n);
	fore(i, 0, n)
		cin >> p[i].x >> p[i].y;
	
	return true;
}

vector<pt> T;

void setVal(int pos, const pt &val) {
	T[pos += n] = val;
	for(pos >>= 1; pos > 0; pos >>= 1)
		T[pos] = combine(T[2 * pos], T[2 * pos + 1]);
}

pt getMin(int l, int r) {
	pt ans = {INF, 0};
	for(l += n, r += n; l < r; l >>= 1, r >>= 1) {
		if(l & 1)
			ans = combine(T[l++], ans);
		if(r & 1)
			ans = combine(T[--r], ans);
	}
	return ans;
}

inline void solve() {
	auto comp = [](const pt &a, const pt &b) {
		if(a.y != b.y)
			return a.y < b.y;
		return a.x < b.x;
	};
	
	sort(p.begin(), p.end(), comp);
	
	T.assign(2 * n, {INF, 0});
	for (int i = n - 1; i >= 0; i--) {
		int pos = int(lower_bound(p.begin(), p.end(), pt(0, p[i].x), comp) - p.begin());
		
		if(pos >= n) {
			setVal(i, {p[i].y, 1});
			continue;
		}
		pt bst = getMin(pos, n);
		setVal(i, {bst.x - (p[i].x - p[i].y), bst.y});
	}
	
	cout << getMin(0, n).y << endl;
}

int main() {
#ifdef _DEBUG
	freopen("input.txt", "r", stdin);
	int tt = clock();
#endif
	ios_base::sync_with_stdio(false);
	cin.tie(0), cout.tie(0);
	cout << fixed << setprecision(15);
	
	if(read()) {
		solve();
		
#ifdef _DEBUG
		cerr << "TIME = " << clock() - tt << endl;
		tt = clock();
#endif
	}
	return 0;
}

1197F - Coloring Game

Idea: BledDest

Tutorial

Solution (BledDest)

#include<bits/stdc++.h>

using namespace std;

const int MOD = 998244353;

int add(int x, int y)
{
	return (x + y) % MOD;
}

int mul(int x, int y)
{
	return (x * 1ll * y) % MOD;
}

typedef vector<int> vec;
typedef vector<vec> mat;

vec mul(const mat& a, const vec& b)
{
	int n = a.size();
	int m = b.size();
	vector<int> c(m);
	for(int i = 0; i < n; i++)
		for(int j = 0; j < m; j++)
			c[i] = add(c[i], mul(b[j], a[i][j]));
	return c;
}

mat add(const mat& a, const mat& b)
{
	int n = a.size();
	int m = a[0].size();
	mat c(n, vec(m, 0));
	for(int i = 0; i < n; i++)
		for(int j = 0; j < m; j++)
			c[i][j] = add(a[i][j], b[i][j]);
	return c;
}

mat mul(const mat& a, const mat& b)
{
	int x = a.size();
	int y = b.size();
	int z = b[0].size();
	mat c(x, vec(z, 0));
	for(int i = 0; i < x; i++)
		for(int j = 0; j < y; j++)
			for(int k = 0; k < z; k++)
				c[i][k] = add(c[i][k], mul(a[i][j], b[j][k]));
	return c;
}

mat binpow(mat a, int d)
{
	int n = a.size();
	mat c = mat(n, vec(n, 0));
	for(int i = 0; i < n; i++) c[i][i] = 1;
	while(d > 0)
	{
		if(d % 2 == 1) c = mul(c, a);
		a = mul(a, a);
		d /= 2;
	}
	return c;
}

int f[3][3];

int extend(int color, vector<int> last_numbers)
{
	vector<int> used(4, 0);
	for(int i = 0; i < 3; i++)
		if(f[color][i])
			used[last_numbers[i]] = 1;
	for(int i = 0; i <= 3; i++)
		if(used[i] == 0)
			return i;
	return 3;
}

vector<int> extend_state(int color, vector<int> last_numbers)
{
	int z = extend(color, last_numbers);
	last_numbers.insert(last_numbers.begin(), z);
	last_numbers.pop_back();
	return last_numbers;
}

vector<int> int2state(int x)
{
	vector<int> res;
	for(int i = 0; i < 3; i++)
	{
		res.push_back(x % 4);
		x /= 4;
	}
	return res;
}

int state2int(const vector<int>& x)
{
	int res = 0;
	int deg = 1;
	for(auto y : x)
	{
		res += deg * y;
		deg *= 4;
	}
	return res;
}

mat form_matrix(int color)
{
	mat res(64, vec(64, 0));
	for(int i = 0; i < 64; i++)
	{
		int j = state2int(extend_state(color, int2state(i)));
		res[j][i] = add(res[j][i], 1);
	}
	return res;
}

mat color_matrices[3];
mat full_matrix;

vector<pair<int, int> > colored[1043];
int len[1043];
int dp[1043][4];

mat full_pows[31];

void precalc_pows()
{
	full_pows[0] = full_matrix;
	for(int i = 0; i <= 30; i++)
		full_pows[i + 1] = mul(full_pows[i], full_pows[i]);
}

vec powmul(int d, vec b)
{
	for(int i = 0; i <= 30; i++)
	{
		if(d % 2 == 1) 
			b = mul(full_pows[i], b);
		d /= 2;
	}
	return b;
}

int main()
{
	int n;
	cin >> n;
	for(int i = 0; i < n; i++)
		cin >> len[i];
	int m;
	cin >> m;
	for(int i = 0; i < m; i++)
	{
		int x, y, c;
		cin >> x >> y >> c;
		--x;
		--y;
		--c;
		colored[x].push_back(make_pair(y, c));
	}
	for(int i = 0; i < n; i++)
		sort(colored[i].begin(), colored[i].end());
	for(int i = 0; i < 3; i++)
		for(int j = 0; j < 3; j++)
			cin >> f[i][j];
	
	for(int i = 0; i < 3; i++)
		color_matrices[i] = form_matrix(i);
	full_matrix = color_matrices[0];
	for(int i = 1; i < 3; i++)
		full_matrix = add(full_matrix, color_matrices[i]);
	precalc_pows();
		
	dp[0][0] = 1;
	for(int i = 0; i < n; i++)
	{
		vec cur(64);
		cur[state2int({3, 3, 3})] = 1;
		int last = 0;
		for(auto x : colored[i])
		{
			cur = powmul(x.first - last, cur);
			cur = mul(color_matrices[x.second], cur);
			last = x.first + 1;
		}
		cur = powmul(len[i] - last, cur);
		for(int j = 0; j < 4; j++)
			for(int k = 0; k < 64; k++)
			{
				vector<int> s = int2state(k);
				dp[i + 1][j ^ s[0]] = add(dp[i + 1][j ^ s[0]], mul(dp[i][j], cur[k]));
			}
	}
	cout << dp[n][0] << endl;
}

ll brfc(vector<ld> a,ll m,ll k){ ll ans=0; // for(ll i=1;i<a.size();++i){ // cout<<a[i]<<" "; // } // cout<<endl; for(ll i=1;i<a.size();++i){ ll sum=0; for(ll j=i;j<a.size();++j){ sum+=a[j]; ll val=sum-k*((j-i+m)/m); ans=max(ans,val); } } return ans; }

Comments (95)

Show archived | Write comment?

okwedook

5 years ago, # |

← Rev. 2 →

+40

I challenge you to write D in $$$O(n)$$$

Solution

First of all if $$$m \geq n$$$, let's make $$$m = n$$$.

Let's us look at some segment $$$[l, r]$$$. The actual cost of it is $$$\sum_{l}^{r} a_i - \left \lfloor{\frac{r - l + 1}{m}}\right \rfloor \cdot k - (k$$$ or $$$0)$$$ depending on the segment. Let's subtract $$$k$$$ from each $$$a_{m \cdot i}$$$. Now we can be sure that the cost of $$$[l, r]$$$ is $$$\sum_{l}^{r} a_i - (k$$$ or $$$0)$$$ depending on the segment. How to differ those cases? Let's look at a half-interval $$$(l - 1, r]$$$. $$$\sum_{l}^{r} a_i$$$ is equal to $$$prefsum_r - prefsum_{l - 1}$$$. How do we define whether we should subtract $$$k$$$ or not? Let's look at $$$r \pmod m$$$ and $$$l - 1 \pmod m$$$. If $$$l - 1 \pmod m \geq r \pmod m$$$ than we alredy subtracted the right amount of $$$k$$$'s. Otherwise we should subtract another $$$k$$$.

Let's write it in $$$O(n \cdot m)$$$. Let's maintain $$$m$$$ values $$$mn_i =$$$ minimal prefsum of some index $$$j$$$ such that $$$i \equiv j \pmod m$$$. Now let's run from $$$0$$$ to $$$n - 1$$$ with index $$$i$$$. On each iteration let's consider all segments such that right bound is $$$i$$$. Now run through $$$mn$$$ and and update our answer if we could. After that let's make $$$mn_{i \pmod{m}} = min(mn_{i \pmod{m}}, prefsum_i)$$$.

Now how to make it $$$O(n)$$$. let's notice that answer for $$$i$$$'th index is $$$prefsum_i$$$ — $$$min(min(mn[0..i$$$ $$$\%$$$ $$$m - 1]) + k,$$$ $$$ min(mn[i$$$ $$$\%$$$ $$$m..m - 1]))$$$. So basically we need suffix minimum, prefix minimum and update one value (the first value in current suffix). Let's process each block of length m separately. For each block let's update suffix minimum in $$$m - 1$$$ updates. Let's define prefix minimum as $$$p = \infty$$$. Now while running through $$$i$$$ to $$$i + m$$$ let's update $$$mn_{i \pmod{m}}$$$ with $$$prefsum_i$$$ and after updating answer with $$$prefsum_i - min(p + k, suffmn_{i \% m})$$$ update $$$p$$$ as $$$min(p, mn_{i \% m})$$$. On each block it takes $$$O(m)$$$ iterations, and there are at most $$$\left \lceil{\frac{n}{m}}\right \rceil$$$ blocks. So it's $$$\left \lceil{\frac{n}{m}}\right \rceil \cdot O(m) = O(n)$$$, as $$$m \leq n$$$.

You can also notice that this solution doesn't need to store all the array, so the only memory it takes is $$$m$$$ values of $$$mn$$$ and other $$$O(1)$$$ values. So it's $$$O(n)$$$ in time and $$$O(min(n, m))$$$ in memory. This solution is the fastest of all and also uses 0KB of memory.

UPD: you can also optimize the solution I've written below using queue minimum. But it's not as nice and causes bigger constant.

→ Reply

Phanix

What's the intuition behind D?

5 years ago, # ^ |

+15

I can explain a different (easier) solution. Let's define $$$dp_i$$$ as answer, such that $$$i$$$ is the last used element. Now $$$dp_i = max(0, a_i - k, max(dp[j] - k + \sum_{j + 1}^{i} a_t)$$$, where $$$i - j \leq m$$$. The intuition is as follows: we need to divide all the segment, into subsegments, each length $$$\leq m$$$.

kanishk779

I used the technique but the it fails on test case 19 here is the link to it.

It's a little hard to understand your code. This is the implementation of what I wrote.

Spheniscine

Hm that's kinda weird, cause when I tried to implement it here (https://codeforces.net/contest/1197/submission/57633664) I ran into problems with it applying the "k" penalty more times than it should

In the end I implemented a solution that uses a second "tolerance" value ( = (-len) mod m ) to indicate how many more values the subarray could take before the k penalty should be applied again: https://codeforces.net/contest/1197/submission/57643293 but now I'm puzzled as to why your code works without it! :P

EDIT: Oh, I just realized that the first code sample had an issue where I wasn't getting the correct results for subarrays [0..i], where 1 < i < m . This bit me later in the butt in another test. The code now works without needing a tolerance term: https://codeforces.net/contest/1197/submission/57646083

jigsaw7

Can you explain a bit more the meaning of each term in the recursive formula in terms of the decision it represents?

Basically we want to find the best segment, such that after splitting it into $$$cnt$$$ subsegments each length $$$\leq m$$$, $$$\sum_{l}^{r} a_i - cnt \cdot k$$$ is maximum. So when we want to count $$$dp_i$$$, we take all lengths $$$1$$$ to $$$m$$$ and try to make $$$[i - len + 1, i]$$$ the last of subsegments. That's how we update dp. This solution relies on the fact that $$$k \geq 0$$$.

_pastor_

length ≤m, ∑rlai−cnt⋅k is maximum.

Can you plz explain this

Edit-ok i got that and you are very smart:)

I'm glad you did it by yourself =)

༼ つ ಥ_ಥ ༽つ thanx

spookywooky

4 years ago, # ^ |

Can you explain $$$best_i = \max\limits_{0 \le len, i - len \cdot m \ge 0} (sum(i-len \cdot m + 1, i) - len * k)$$$

What is this the "best" of? How comes this formular?

i will explain my solution which is similar to that solution (my sol https://codeforces.net/contest/1197/submission/80940806)

1.dp[i] will represent the subarray with max cost ending at i

2.now we will iterate from the last index 'i' up to when the length becomes m because the greatest integer of length less than m will be 1 so after that we don't have to because it will be same as dp[i-j] where j is the starting index

3.now we will just maintain 'sum' of value let say we are at jth index from the last index i so sum =a[j]+a[j+1]...a[i] now we dp[i]=max(dp[i],sum-k) 'sum-k' because the length i-j+1 is less than or equal m also we have to do dp[i]=max(dp[i],dp[j-1]+sum-k)

What is the key idea, or key observation to come up with that solution?

StinsonOvO

so for this solution, how does it take into account segments with length > m?

Every segment consists of subsegments of length $$$\leq m$$$. For each small subsegment you have to subtract $$$k$$$. So each segment is just continuous merge of small subsegments. That's why we can apply DP. Each segment is either a small segment or a merge of segment and a small segment.

nishu2002

14 months ago, # ^ |

why always subtract? let us say i=7, j=5,m=4 then ceil(i/m)=ceil(j/m).isn't it?

Because the task is equivalent to all segments, which have the length of at most m. Our DP does just that, when we try to update it with last $$$m$$$ segments.

prabhat7298

This easier solution is O(m*n) right?

Yes

pmitkov

You can check my solution, Only $$$j = i - m$$$ needs to be considered, Otherwise just take max subarray ending at $$$i$$$ with length $$$\leq m$$$.

Yes, your solution is actually the same, I simply don't differ those cases. It still works in $$$O(n\cdot m)$$$.

It can actually be improved to $$$O(n)$$$ easily by also storing the 2 best subarrays of length $$$\leq m$$$ for $$$i - 1$$$ so you don't have to do a for loop.

husong

consider this case: say $$$j = i - 1, dp_j = a_j - k$$$, when you try to update the value for $$$dp_i$$$ using $$$dp_j - k + a_i$$$, $$$k$$$ will be deducted twice when you calculate $$$dp_i$$$. Is that the case or am I misunderstanding something?

Yes, in this case it subtracts twice, but we will find the best answer anyway.

Is it ok for you to explain why this is the case?

I assume that [i, i] is the last small segment. It means that before i there were small segments only of length m. This leads to another subtraction as [i, i] is a different segment.

I mean why we can still find the optimal answer despite that k is subtracted twice

This is the situation, where length is x * m + 1, we have to subtract k x + 1 times.

I kindda get your idea after re-reading your comments. In your method, although $$$dp[i]$$$ is miscalculated when we try to update it using $$$dp[i-1]$$$, the optimal answer will surface out when we try to update it using $$$dp[i-2]$$$, is that the case?

Yes, we choose the maximal value of all, so we will get the best answer.

I think I finally get it. Thanks for answering my questions patiently!

Jon.Snow

Thanks for clear explanation!

_Bishop_

When you use the transition $$$dp[j] - k + \sum_{j+1}^{i} a_t$$$ , then you extend the optimal segment ending at $$$j$$$. How do we decide the $$$-k$$$ part here?

Consider it this way: adding a segment of length $$$\leq m$$$ costs you $$$k$$$ money.

But may be when we add the segment, then the number of segments doesn't increase....Suppose $$$m = 5$$$ and $$$dp[j]$$$ comprises of segments of size $$$3,5,5,5$$$. If we use one more element to update $$$dp[i]$$$ then the segments will be $$$4,5,5,5$$$ . Here $$$-k$$$ won't be there.

We will overlook both $$$4, 5, 5, 5$$$ and $$$3, 5, 5, 5$$$. I do not understand your usage of word "segment". We'll just try and add every last possible segment $$$[j+1, i]$$$ and update $$$dp[i]$$$, which is the best answer for all segments ending at position $$$i$$$.

In the above example with $$$m = 5$$$, I meant that since $$$dp[j]$$$ means the answer for position $$$j$$$ ,let's say optimal value is for the interval $$$[j - 17, j]$$$ which means value of $$$dp[j]$$$ is $$$\sum_{j-17}^{j} a_t - 4k$$$ . When we update something like $$$dp[j+1]$$$ using $$$dp[j]$$$ then we will take $$$dp[j] - k + a[j+1]$$$ , but it should only be $$$dp[j] - a[j+1]$$$ because now the interval becomes of size 19 and we have already paid $$$4k$$$ in $$$dp[j]$$$...

I think you just don't get the dp approach. Yes, we will overlook ALL possible decompositions and choose the best one. Please look at this thread: https://codeforces.net/blog/entry/68615?#comment-530099

make_85

I don't understand the solution for C, I guess this solution works because the initial array is sorted? Is there known algorithm that leads to this solution using differences between adjacent elements?

agrawal117

https://codeforces.net/blog/entry/68553?#comment-529278 may be this comment will help u !!

practice_

Let a[1], a[2].. a[n] be the elements in the array. Let 1 <= i < j < k < l....< n be the k-1 indices we choose for dividing the array into k segments. So, cost of division = (a[i] — a[1]) + (a[j] — a[i+1]) + .... + (a[k] — a[j+1]) + (a[n] — a[k+1]), on rearranging : (a[n] — a[1]) — ((a[i+1] — a[i]) + (a[j+1] — a[j]) +.....+ (a[k+1] — a[k])). So, we will store the difference of a[i+1]-a[i] for each 1<=i<n , sort them in decreasing order and take first k-1 differences and finally subtract from (a[n]-a[1]).

Olerinskiy

lol

Can someone explain how to solve E problem. Why do we need a segment tree. And how do we find the value of the minimum for the whole set of the dolls?

aijey

← Rev. 3 →

If there is no $$$j$$$, that $$$matryoshka_j.in \ge matryoshka_i.out$$$, then $$$d_i = (matryoshka_i.in, 1)$$$

else

$$$d_i.first = min[pos,n] - (matryoshka_i.out - matryoshka_i.in)$$$, where $$$pos$$$ is the first $$$j$$$, that $$$matryoshka_j.in \ge matryoshka_i.out$$$ (It means that we can put matryoshka $$$i$$$ inside matryoshka $$$j$$$)

$$$d_i.second = numberOfMinimums[pos,n]$$$

And $$$d$$$ — is our segment tree, so we need it to calculate minimum and number of minimums on suffix!

You can check my solution, I think it's clear enought.

stringray

yep , your solution is self descriptive , but i have seen this https://codeforces.net/contest/1197/submission/57527419 which doesn't even use segtree , like no heavy DS stuff .

I haven't mastered segment trees yet (did an object-pointer based one for practice but probably need to learn how to do it with bitwise-math and array indices for better efficiency) but I managed to solve this one using only a priority queue; code here if interested:

https://codeforces.net/contest/1197/submission/57589994

o_v_shake

What does len mean in the editorial of D? And how does that expression len * m come?

I suppose len is the number of segments that are fully covered. So $$$best_i$$$ is the answer if only segments of length divisible by m are considered.

MASTERASSASSIN

I come with something for problem D but I can't develop it to a solution if anyone solves it by the help of the following please tell me:

we deal with the array as the blocks, we try each possible one let's call it j from 1 to m, and for every j we try to start from 0 to j — 1 and move by j step to partition the array.

it does nothing but I think it could be developed in some way.

Very nice explanation of problem E. Thank you so much)

-osdajigu-

I was discussing problem E with my friend idgaf and after he solved it we read the editorial. And we noticed this part ( It's because not big enough subsets are not optimal in terms of minimality of extra space.)

However, we are not sure if the part that adedalic says is true, please correct us if we're wrong.

Take this example with 2 dolls {4, 1} {6, 5}

The minimum extra space of big enough is 2, but if you take the first doll only, the extra space is 1

adedalic

Let's extend your example as {2,1} {4,2} {6,5}

$$$d[3] = (5, 1)$$$ — it should be obvious. When we try to calculate $$$d[2]$$$ we can see, that the second matryoshka can be nested inside the third one — so "we must put it inside". Then $$$d[2] = (5-2=3, 1)$$$.

The first doll can be nested both in the second and third dolls, but putting it inside the third doll will lead us to not big enough subset. But! it also makes the extra space not optimal since $$$d[2].first < d[3].first$$$. Then $$$d[1] = (3-1=2, 1)$$$ and it's correct.

Ohh, I understand, I was a little confused, but now is totally clear. Thanks.

__Apocalypse__

Can someone explain the intution behind D and as to how are we connecting the maximum sum subarray problem to this problem in greater depth. Also I'm not able to understand the intuition behind the idea of introducing len (as done in the editorial) Please help.

Ramprosad

This is an alternate Solution 57537548 of Problem 1197C - Array Splitting

bittudalal

In problem E, Why cann't we add subset {4,7} as one of good subsets in Test Case 1.

Because it is not big enough. We can put $$$7$$$ inside other doll

ok,got it.

sebi420p

can someone give me some further explanations about problem c? more specific: why should we "add up the k−1 minimal ones to the answer"?

In the solution given for D why do we fix some elements of best_i with best_i + sum(i-len,i-1) — k ?

I think this is a typo: https://codeforces.net/blog/entry/68615?#comment-530337

durmanko

Can someone explain in more details solution of C, with proofs? Thanks

meooow

+27

I don't see why we need a segment tree for E since all queries are suffix queries and all updates are at the position we are currently at. We can just maintain a suffix minimum as we go along.

Solution with editorial's approach minus segment tree: 57644319

FireChromeExplorer

So, I actually got hacked on problem A because of exceeded time limit. How do you know if your solution needs to be O(n) or O(nlogn) given the time constraint? Would you just assume 2 seconds means my solution needs to be O(n)?

Sorry for too many questions, I'm just new to CP.

AnandOza

The rule of thumb I use is to plug the maximum input size into the big O, and look for something $$$\le 10^8$$$.

So for $$$n=10^3$$$ for example, $$$O(n^2)$$$ is fine ($$$10^6 < 10^8$$$), but for $$$n=10^5$$$, it'd be too slow ($$$10^{10}>10^8$$$).

In your solution you used quicksort, which is usually $$$O(n \log n)$$$, fast enough, but you were hacked by an input that causes it to become $$$O(n^2)$$$, which is too slow.

vp-asp

Could somebody explain me, if we want to calc T(i, j), we must know colors of r1, r2, r3. But I can't find any mention of it in the tutorial.

lazywitt

Logic behind soulution of C?

aviroop123

I think problem F can be solved simply by reducing the number of states. The number of states that are possible to reach is actually way lower than 64, it's 25. Then I don't think there's any need for optimization, just brute 25^3*1000*log2(10^9) will work. (Correct me if I'm wrong).

In the tutorial for D, last line, you wrote: $$$best_i + sum(i - len, i - 1) - k$$$. But in the code posted below it seems that you are calculating $$$best_i + sum(i + 1, i + len)$$$. I think that is a typo in your tutorial.

birchtree

Can E be solved by Graph Theory? Just saw the "Shortest Path" tag.

JKLover

There's an edge from i to j with length $$$in_i-out_j$$$ ,if and only if $$$in_i\ge out_j$$$ holds. Sort the matryoshkas by $$$out_i$$$ ,then we can use segment tree to build the graph. To solve the problem more easily,we use two more nodes,S and T. For all node i with indeg=0,add an edge from S to i with length 0. For all node i with outdeg=0,add an edge from i to T with length $$$in_i$$$ . The answer is the number of shortest paths from S to T. Since the graph is a DAG,the solution above works in $$$O(n\log n)$$$ time.

I got an WA on test 8 two days ago.My approach is similar to yours. 57926487

just found my mistake. Thanks anyway.

AC code: 58050060

Rezwan.Arefin01

+10

Here is an easier $$$O(n \log m)$$$ solution for D:

Let $$$p$$$ be the prefix sum of $$$a$$$. Then maximum cost of subarray ending at $$$i$$$ is $$$ \max_{j < i} \left\lbrace p_i - p_j - k\left \lceil \frac{i - j}{m} \right\rceil\right\rbrace$$$

Main observation is --

$$$\left\lceil \frac{i - j}{m} \right\rceil = \begin{cases}\left\lfloor \frac{i}{m} \right\rfloor - \left\lfloor \frac{j}{m} \right\rfloor + 1, & \text{if }(j \text{ mod } m) < (i \text{ mod } m) \\ \left\lfloor \frac{i}{m} \right\rfloor - \left\lfloor \frac{j}{m} \right\rfloor, & \text{if } (j \text{ mod } m) \geq (i \text{ mod } m) \end{cases}$$$

So, our formula becomes --

$$$\begin{cases}p_i - k\left\lfloor \frac{i}{m} \right\rfloor + \left(k\left\lfloor \frac{j}{m} \right\rfloor - p_j\right) - k, & \text{if }(j \text{ mod } m) < (i \text{ mod } m) \\ p_i - k\left\lfloor \frac{i}{m} \right\rfloor + \left(k\left\lfloor \frac{j}{m} \right\rfloor - p_j\right), & \text{if }(j \text{ mod } m) \geq (i \text{ mod } m) \end{cases}$$$

To evaluate the formula quickly, we can keep the maximum of $$$\left\lbrace k\left\lfloor \frac{i}{m} \right\rfloor - p_i\right\rbrace$$$ at index $$$(i\text{ mod } m)$$$ of a segment tree. Which enables us to get the prefix/suffix maximum in $$$O(\log m)$$$ resulting in total complexity $$$O(n \log m)$$$.

GQfighting

the second example of B: 3 1 2. why it is NO？we can move it like this.

-------------------------1------------------1---------------1-------------------------------2
3-1-2 ——> 3-null-2 ——> null-3-2 ——> null-3-2 ——> 1-3-2 ——> 1-3-null ——> finished and yes

ly_61

Please notice the second condition among the three in the problem statement:

2.pillar i contains exactly one disk

According to this, you won't be able to perform your third (also fourth) move.

ankyy_0103

In problem B,what happens when there are two discs of same radius , doesn't it affect the answer? My both solutions taking the case of equal radii discs and ignoring that case are accepted. I'm little confuse here ,help me out.

HynDuf

"The second line contains n integers a1, a2, ..., ai (1≤ai≤n), where ai is the radius of the disk initially placed on the i-th pillar. All numbers ai are distinct." — As the problem statement says.

Verdusk

I've devised a non-dp-based solution for D, taking advantage that the m is only up to 10.

Mark the array elements in 0, m, 2m, ... . Treat these elements as having k less cost than their original. Then calculate the max sum of the array but you may only start from a marked element.

Start with the leftmost marked element and traverse right, adding to the sum as you do so, and every time you visit a marked element, if it's better to start from that element instead, do so (if sum < cost of current marked element, already reduced by k from original, then set sum to that cost instead)

Then do the next rotation, that is, instead of marking 0, m, 2m, ..., you mark 1, m+1, 2m+1, ... . Doing m rotation ensures that all the elements got marked in one of the rotations, so you cover the cases of starting from every element.

The whole point of this is to take advantage of the fact that, if you mark every m element, the multiple of k that you have to "pay" is exactly as many as the marked elements included in your subarray, as long as you start your subarray in one of the marked elements.

Takes O(n*m) but very little memory.

zephxr

Can Anyone Explain Problem C.I am confused that how we come up with the solution. Thanks in Advance.

AQZZ

Idea is simple, we need to pick the starting point of the k subarrays. Of course the first subarray must have starting point 0. The other k - 1 starting points must be distinct indices in the range 1 .. n - 1 but there is no other restriction on them.

When you pick a starting point i, it adds a[i - 1] - a[i] to the total cost because a[i - 1] becomes an end point and a[i] becomes a starting point.

Thus we can just make a vector 'starts' with starts[i] = a[i - 1] - a[i], sort it, and take the sum of the first k - 1 elements. Then add a[n - 1] - a[0] to the sum to account for the fact that a[0] is a start point and a[n - 1] is an end point.

My code: 82349539

Thanks for the beautiful explaination AQZZ,

because a[i — 1] becomes a max and a[i] becomes a min.

What does it mean?

I edited it. I mean a[i — 1] becomes an end point (since it is the max of a subarray) and a[i] becomes a start point (since it is the min of a subarray).

Shouldn't it be that:

a[i] - end point

a[i-1]-starting point

If not , then whats the reason? AQZZ

In my approach we are picking the starting points of the subarray. If we make index i a starting point then a[i - 1] becomes the endpoint of a subarray and a[i] becomes the starting point of a subarray.

Let k = 2 and a = {1, 2, 5, 6, 7}. Then if we pick i = 2 to start a subarray it means the subarrays will be {1, 2} and {5, 6, 7}.

Starting point and minimum element of a subarray are exactly the same because the array is sorted. Same with end point and maximum.

lakshay05

can anyone tell me mistake in my code 84062159

snehal007

4 years ago, # |

I found Um_nik's submssion for E really neat and simple. Segment Tree is an overkill as we are only looking for suffix queries.

AjaySabarish

Simple explanation for D, the reason why kadane algorithm doesn't work is because of the length factor, let us assume the current length of the segment is len. And the factor we have to subtract( ceil(len/m)) is penalty.

Then let j be len%m, if j is 1, we are starting a new subsegment then we have to subtract penalty, else penalty was already subtracted before, so we don't need to care about it and we can proceed to add the current value. The approach is based on the fact that ceil(x/m) is 1 for all x from 1 to m. Let dp[i][j] denote the answer if segment ends at i and len%m is j.

So let us all iterate for all possible combinations i(1 to m) and j(0 to m-1), our answer is basically just the max of all.

Don't forget the special case of m = 1, where we have to add penalty for every element added.

Submission : 90408059