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Tutorial of Codeforces Round #520 (Div. 2)

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Codeforces (c) Copyright 2010-2022 Mike Mirzayanov

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As for D, there is a simple O(N) solution. First of all, let's forget about a and b less than 0 We can observe that score is increased on each integer as many times, as this integer appears a divisor of some other integer. This happens floor(n / i) — 1 times for number i. Then we can see, that we have a mirror situation with a and b less than 0. And with a < 0, b > 0 and a > 0, b < 0 works the same. That's why answer is sum of i * (floor(n / i) — 1) for i from 2 to n multiplied by 4.

Here is my solution 45732442

Why every "i * (n / i — 1)" can be choosed?

eg: 6->2, -6->2, 6->-2, -6->-2 in all the above case 3 is the divisor of the integer 6 so suppose n=18 then 3 appears as divisor for 6,9,12,15,18 so the score becomes 3(|x|:divisor) * 5(floor(n/i)-1) * 4(for a,b +ve 0r -ve cases).

thats what i think how it is please correct me if wrong.

Thank you!给你花花！

Thanks for suggesting this trick.Learnt something new.

A can be done by an O(n) algorithm by just traversing once along the array

yes i have done the same :)

In problem D.

Could someone elaborate more on the proof that all the nodes are in the same component?

Set of numbers that can transform into a number x is always even (if a can transform to x, -a can either), so the graph built by connections that based on two numbers which is transformable to each other or not will have vertices with even degree, so the graph is Eulerian, which means you can travel through all the vertices without coming back to previous vertices. And so all the vertices must be in a same component.

I guess.

Suppose you transform

aintoband |a| < |b| thenx=b/aright? It's clear thata,bandxare in the same component. You can see that |b| ≤nand |a| ≥ 2 so |b/a| ≤n/ 2, which means you are able to transformxinto 2xas well, that makesx, 2 and 2xbelong to the same component. So for every node that has at least an edge attached to it, we know that node is in the same component as 2.how to solve this problem? Can someone provide a simpler version of this problem or can explain this problem? Thanks in advance :)

O(sqrt(N)) code for D: https://codeforces.com/contest/1062/submission/45742622

Cool solution, that uses the same idea as my, but faster

fascinating solution :)

can you please explain it? especially the line below:

https://codeforces.com/contest/1062/submission/45724128 It's this solution but optimized. The answer is: for each i, for each multiple of i, sum j / i as can be seen from this code. N / i is the number of multiples and you can take that sum in O(1). Now, just group the i's by the value of N / i.

Can someone share their code for problem E with same solution as the editorial?

My implementation is almost the same as the editorial.

One difference is that my range queries return the two smallest and two largest dfs-order labels in [l, r]. We will either delete the smallest and use the LCA of the second smallest and the largest, or we will delete the largest and use the LCA of the smallest and second largest.

The editorial instead suggests to make additional queries on [

l,u) and (u,r] to simulate deletion of nodeu.You can find my solution here. It's the same approach as the author's, provided a quick explanation to the thought process commented above the code.

If you have any questions, please let me know :)

Thank you very much! Your clean code helped me learn and implement lca and solve this problem.

You're welcome! Glad i could help :)

Can someone share their code for problem D with same solution as the editorial?

following.

Codefores is the best!!!

Then why to cheat?

Am i the only one who solved D in less than 10 lines?

Me too, and it was a O(n) solution.

Hereis my O(n) solution.I an not able to understand how to solve that problem. Please can you help me with this. I know the theory behind Euler's path and graphs but how to think this problem can be solved using graph? Any help would be appreciated.

There is an O(n) complexity algorithm for A. Since we need to check for the longest set of increasing difference-by-1 consecutive elements in the array, we don't need to use all pairs i,j. It can be done by for i in size of array, if a[i]==a[i-1]+1 and a[i]==a[i+1]-1 then we increase the delable_currentsize_var by 1; else (means we find the barrier of the increasing set) we update max_delable_size = max(max_delable_size, delable_current_size) and reset the delable_currentsize_var to 0.

https://codeforces.com/contest/1062/submission/45727163

An alternative solution to problem F:

First, find any topological order of the DAG.

Observation 1: A city is important if and only if all the cities before it can reach it and all the cities after it can be reached from it.

Observation 2: In a DAG, vertex v can be reached from any other vertex if and only if v is the only vertex to have no outedges.

From 1 and 2, we can find which cities are important(by adding the vertices in the topological order and maintain a set of the cities with an outdegree equal to zero). This is O((m + n) log n) and with some tricks it can be done in O(m + n).

Now, let's focus on semi-important vertices:

Observation 3: Assume that v is a vertex without outedges in a DAG. Then v can't be reached from exactly 1 city if and only if:

So through restoring the vertices with exactly one outedge and how many times each vertex is pointed to by these vertices, we can easily deal with semi-important vertices in O(m + n).

Submission: 45770465

Correct me if i'm wrong. The topological ordering is not so important with two node that can't reach each other, right?

So with test case 3,

5 4 1 2 3 2 2 4 4 5

if i change the topo order before we apply solve(tsq, G), particularly node 3 and node 1 ( swap(tsq[3], tsq[4] ) ), your code will give answer 4, instead of 5, which is WA.

Swapping node 3 and node 1 should be swap(tsq[0], tsq[1]).

Nice solution. I have found a similar one, but mine is a little bit more complex.

What is the intuition behind C's solution? Someone explain please.

Solve the problem by hand for some cases, and you should realize that each time the numbers keep doubling. If the whole array is ones, then a number at step i will contribute (2^(i — 1)).

However that doesn't hold for zeros, so you assume everything is ones, then subtract the excess.

By doing 2^(length-1) — 2^(number of zeros), it gives wrong answer(due to some negative values in testcase number 3),but by doing 2^(number of zeros)*(2^(number of ones)-1),it gives correct answer,why?

it is like when you transfer from binary to decimal and you should take attention that you should take all ones first so if the number is 101001 you can say the number is 111000 * take care that if the number is 111111 the decimal will be 2^6 — 1 and if if the number is 111000 the number is 2^6 — 2^3 so the solution is 2^(all segment) — 2^(number of zero's) hope that is clear for you

Can you help me? What is wrong in this code 45786145 ? I changed all things but even is getting the same result.

45787128

Thank you very much!!!

roadtrue Why are we multiplying 2^(no_of_ones)-1 and 2^(no_of_zeroes) ?

when you transform from binary to decimal if the one's are at the left and the zero's are at the right like 111000 you can transform it quickly by make the number equal 1000000 — 111 and 1000000 = 2^(all segment) -> 111 = 2^(no_of_zeroes)

I got that part. But why are we multiplying these two values?

There is a dynamic programming approach that independs of pots of 2.

Let

nbe the number of 1's in the interval. Now you can have two processes: 1) the total deliciousness of thenpieces will be the deliciousness of then-th piece plus the total deliciousness ofn- 1 pieces. 2) the deliciousness of then-th piece will be the total deliciousness ofn- 1 pieces plus 1.Let

xbe the total deliciousness for thenpieces, andmbe the number of 0's in the interval. Then the answer for these guys will bex* total deliciousness ofmpieces (since, instead of starting with 1, these guys are starting withx)if anyone solved problem D by editorial approach, please provide the code.

is the tutorial for E has a error? it should be out[u]=max(out[v1],out[v2],...),or my understanding of in[] is wrong?

"a child of u" includes all the direct and undirect childs of u. Apparently,

in_{u}=max(in_{v1},in_{v2}, ...,in_{vk}) (vis a child ofu, both direct and undirect) andin_{u}=max(out_{v1},out_{v2}, ...,out_{vk}) (vis a DIRECT child ofu) are basically the same.Thanks!

In second sentence in E we should have the reverse implication i.e

impliesin_{u}≤in_{v}≤out_{u}is an ancestor ofu, instead ofvis an ancestor ofuimpliesvwhich is written, since this is the direction used later in the proof.in_{u}≤in_{v}≤out_{u}You're right. It should have written as

uis an ancestor ofvif and only ifin_{u}≤in_{v}≤out_{u}can anyone explain problem B again please ?

That was an easy problem bro. Check this http://codeforces.com/contest/1062/submission/45742566. I have added some comments to make things clear :)

thank you bro <3

Can someone explain me why in the 7 pretest of A problem answer is 2 but not 4?

Okay, I just read the task one more time ;D

The statement says "the

maximum_ number of consecutive elements" not "thetotalnumber of consecutive elements".Can someone explains why the complexity of the final step in tutorial F is O(N+M),please? (......After that we pop those nodes from the stack (or whatever), mark them as not visited and continue to iterate(迭代) to si−1.) (......To calculate in we reverse the directs of the edges and do the same. Because each node is visited 1 time by nodes on (P) and at most 2 times by candidate nodes ) in such a way why some nodes won't be visited more than 1 time?

Assume we have: 1->2->3->4 1->5->3 2->6->4 6-> a set of nodes S1 5-> a set of nodes S2

Let go backward from 4 to 1. First, at 4, because R_6 points to 4 so we start a search at 6, each node in S1 is iterated one time. After that, we reach 3, because R_5 points to 3 so we also start a search at 5 and each node in S2 is iterated one time. There maybe a chance that nodes in S1 are revisited (there is an arc from 5 to 6). Okay so let's deal with the worst case here: nodes in S1 are visited the second time. Then we reach 2, we start a search that goes from 2 to 5 and to all nodes in S1. So that's the third time we visit nodes in S1. The important thing is, this time we mark all nodes in S1 as visited and we will never visit them again. Therefore, each node is visited at most three times, one time by the nodes lie on the longest path and two times by the candidate nodes.

thank you bro ;D

Does anyone wrote the code for Problem D in editorial approach? please comment link.

In problem A, I'm getting an error on Test 7.

9 1 4 5 6 7 100 101 102 103

My answer is 4. However, the Jury's Answer is 2. Why is it 2? I can delete 5,6,101,102 for a total of four, right?

Here is my solution to problem D in less than 20 lines of code 47263480. It is a simple math problem and can be solved without any knowledge of graph. Complexity is O(N)

In A, for the test case7 :

9

1 4 5 6 7 100 101 102 103

The answer according to me should be 4 as we can remove elements 5, 6, 101 and 102 from the given array. But the jury's answer is given as 2. Please refer.

could someone explain problem b ?

Can Anybody explain why I am getting WA on Test Case 15 ?

My Code

Great Test Cases for Problem A. Congratulations!

video tutorial for D. Fun with Integers: https://www.youtube.com/watch?v=1TTorx9q1i0

Can anyone explain the derivation of formula for problem C ?

Thanks