Tinkoff Internship Warmup Round 2018 and Codeforces Round #475 will start at 17 April 2018 17:05,

Thanks to Vladislav winger Isenbaev, Alex AlexFetisov Fetisov, Ildar 300iq Gainullin, Egor Egor.Lifar Lifar, Philip grphil Gribov and Gleb Glebodin Lobanov for testing my problems.

Special thanks to Grigory vintage_Vlad_Makeev Reznikov and Nikolay KAN Kalinin for coordinating throughout the round and Mike MikeMirzayanov Mirzayanov for remarkable systems Codeforces and polygon.

Round is rated for both divisions. There are 5 problems, and 2 hours to solve them. The points distribution will be updated later.

Good luck and Have fun!

**UPD** The points distribution for both divisions: 500 — 1000 — 1500 — 2000 — 2500

**UPD** Congratulations to the winners!

Div 1:

Div 2:

Wish high ratings everyone

Please stop all these comments about

`Wish high rating`

and`is it rated`

and`short problem statements`

.Every one wishes that. So it is not important to write these comments. Keep comment section clear for more important things.

If everyone was high rated no one would be high rated :o

Why isn't there an English version of the blog?

fixed

Is it rated?

For all

`is it rated`

spammersno u

I don't know, but there's a more important question: is it rated?

No one replied your question. I think it is unrated.

Same for you, no one replied to you comment, so I think it is rated.

I got worried when the registration was only privet :D

2 previous Tinkoff rounds

## Is It Rated?

.

Announce:Round is rated for both divisions

thanks, finally it's 5 problems

Champions in

Chinaso ..Today there is a good chance for some to achieve advanced positions ..

The points distribution will be updated later.This sentence is wrong, correct one is :

The points distribution will not be updated before contest.This was amazing guys ! Thank you for scoring :)

Why I can't register for the contest now???

Hii Stefdasca :)

Hi

please fix registration I want to participate voidmax

What was Div2 B?

if C > B, it is better to leave message unread as long as possible. So read the message on time T

C = B, time doesn't matter

C < B, it is better to read message as fast as possible. So read the message on t_i

why this right ?

If you read the message

xseconds after receiving it, you will receiveA+ (C-B) *xmoney.Any Idea of pretest 7 for Div2C?

a = b ?

Have that..

I did ..Multipied by (n+1)/k..still wa on 7 :(

(

b/a)^{k}≡ 1 (mod 10^{9}+ 9)Explain I didn't get it..

I got pretest passed with this I expect to pass full test.

Your problem is that the amount of extra terms is no fixed

Please!! What's the hack data for B?

Is it just me or B was much more difficult than usual?

#easy

Hope it is

acceptedbrotherToo much maths for Div2

Any idea of pretest 9 in DIV2C and pretest 7 in DIV2D?

Dude that was so frustrating

pretest 9 in DIV2C is (a + b) % mod == 0 and k % 2 == 0, so (a^-1 b) ^ k = 1, but blocks of size k have different signs.

OMG... Only geniuses could have thought of such test case.

Pretest 7 in Div2 D is one where root isn't 1, but something else. I only found that bug a few minutes before the end of contest.

It is a tree, I do not see where this change the answer.

In my dfs() I had "if(v != par[u]) dfs(v);", so unless I start the dfs at the real root of the tree, I don't get AC. Fixing this gave me AC.

When (

a^{ - 1}b)^{k}= 1, so each block of sizekcontributes the same.Pretest 9 DIV2C/DIV1A is case when a = b Example: 2 2 2 3 +-+ Answer should be 4

I guess it's not a = b, because my mistake was in case a = -b.

can u elaborate why a = -b needs to be handled?

http://codeforces.com/blog/entry/58959?#comment-425608 This thing have not the same answer as a = b, but need to be handled because (a^-1 b) ^ k = 1 for even k.

I mean everybody solved this problem with sum of geometric progression, it's formula is like

`p * (q^n - 1) / (q - 1)`

Pretest 9 is when q = 1, so geometric progression consists of the same members.

I have that,but still WA on 7..

No that's not the Pretest 9. My code outputs 4 for the above input, but failed on pretest 9.

Try when (A^K)%MOD == (B^K)%MOD.

Yeah got it!

Except a=b what are the other possibilities?

Just try asking the problemsetter.

a = p — b

http://codeforces.com/blog/entry/58959?#comment-425616

No, it's not. I get 4 and have WA on that test

For WA7 on B.

how to solve problem D in div2 ? so sad of wr7

x2

x3

5 0 1 2 2 4

My initial code failed this test and it was getting WA in test 7.

I think that the answer is to remove the deepest node with even degree, because none of its children have even degree so they can't be removed unless this node is removed first.

I think you can remove any such node that all the remaining trees would have odd sizes. This is easily done using BFS or a set(), since each removal only changes the "parity" of adjacent nodes.

oh..the contest is over...maybe it is too hard for me.....

In Div2 D isn't it always possible to destroy the tree by removing the leaf vertices and then proceeding upwards in this fashion?

The leaf nodes will always have a degree zero which is even right ?

They will have degree 1

I think leaves have degree 1 :)

Thanks. It seems there is an overloaded definition that's why I was confused

https://stackoverflow.com/a/33987475

I found it somewhat funny that the problem statement included a definition for tree, but didn't include a definition for degree.

Can you tell me why registeration was closed?? voidmax i waited for you a whole hour and you didn't response??

Can anyone explain how can this O(n) solution 37407108 for Div2A pass when I tried to hack it with test case n = 10^9 ? voidmax

Because C++ has optimizer. This programm works in O(1).

Oh. This trivial information costed me 50 points penalty.

Please explain how compiler identifies it can be optimized and how it does it? Thanks

Compiler optimized it to n/2

Compiler optimization?

Regarding "Destroying a tree", it matters the order in which the vertices are selected. For example, consider this test case:

If you start from vertex 1, then it's impossible to destroy the tree, but if you start with vertex 2, then the order "2 4 3 1 5" is valid.

Unfortunately, I couldn't make it up to find out a strategy that always brings the correct result and maybe that's why I failed pretest 7.

Kudos to voidmax for the interesting problemset!

Root the tree from a random vertex. It's always optimal to destroy a vertex with largest depth (and even degree of course).

Can you explain why?

This is probably not a proof, it is just my thinking during the contestSuppose that you have two vertex with even degree (one is parent of other).

If you remove the parent, the child will be affected, you will need to remove some child to have even degree again... It's kinda hard to check this from parent to child (root to leaf).

Try to go in the opposite way: from leaf to root.

You have a leaf (obviously odd degree), how can you remove it? You need to remove it's parent... If the parent is odd, then you need to keep checking the parent until you reach a vertex with even degree. Now you remove it.

Then you have just disconnected the tree, the new tree should be solvable, otherwise

"NO"Hi Yash, did you figure out a proof. I really want to know how this works but I can't prove it. The comment linked by the author below also doesn't help :/

PROBLEM D GREEDY PROOF

OK I figured it out for anyone in the future who falls in trouble:

wake me up when systests start

Systests have started for Div1 and Div2!

Div 2 as well.

What is pretest 15 in Div1 C?

Why this o(n) sollution passes system testing? 37405016

Because C++ has optimizer. It easily optimizes this sort of for, so programm works in O(1).

Please, just run systests faster and let us go to sleep, we have rehershal tommorow :/

You're my heroes :*

What is correct solution to B anyway? I know that:

There will definitely be a vertex with even degree breaking the tree into odd components. We can first remove all even sized subtrees of the root. Once all even sized subtrees have been visited, print current vertex and then visit odd sized subtrees. In this method, we are considering either odd sized trees with no parent edge, or even sized trees with a parent edge, and you can prove using induction that this works :

What is this meaning of parity calculation in your submission ?

That's what he asked to be proved.

You can remove the vertices greedily by its heights: remove the ones with greater height first.

If you are removing a vertex

V, its subtree has no vertexX≠Vwith odd degree, otherwise it would be removed beforeV. The children ofVwill now have even degrees, since just one edge is removed from them, and you can apply the same idea to them, since their subtrees have no nodes with odd degree beside them.So, all components that were on

V's subtree are good to go; you just need to prove that the upper component is good to go too. That's easy: since the number of edges inV's subtree is even, then the parity of removed edges of the upper component did not change, which implies that there is another even-degree vertex to be removed.Thanks for the explaination. Feel sad for not figuring this out in the contest...

Could you explain why answer is No when tree has even number of vertices?

Even number of vertices have odd number edges .And you always delete some even number of edges

What a simple idea it was. Deleting a vertex will separate trees, that thought confused me. thanks.

Could anyone please drop a couple of hints on Div2C? (Div1A)

How to approach this kind of problems?

Update:Oh, I got it now.

Thank you, aangairbender.

You can calculate (we will call it

T) inO(k*log(k)) time complexity. It will be the sum of one period.Two consecutive periods are times different — we can call that value

Z.The number of periods will be (

n+ 1) /k=Q.Therefore, the final answer should be: , or .

All of these can be calculated with modulo addition/multiplication/inverse.

Update:Just see the update of your comment. Nevermind, hope you can find something useful in mine ;)Thank you!

I indeed find your comment useful.

Specifically, it did not come to my mind during the contest that consecutive periods differ by (

a_{k}/b_{k}) times.Let

Sum[0,k) be the summation of firstKterms of that series.Now you can observe that

Sum[k, 2 *k) = (b^{k}/a^{k}) *Sum[0,k)Similarly, You can divide the whole series in (

n+ 1) /kterms which is a G.P with common ratio (b^{k}/a^{k}) and then use the formula for the sum of geometric progression (don't forget the case when common ratio of this G.P is 1).It is

b^{k}/a^{k}Fixed, Thanks.

Can anyone tell me if this solution for Div2.D is correct? I came upon it too late in the contest and didn't have time to implement it. So, here it goes:

Lets start at the leaves, it is obvious that their parent will have to be destroyed before them.

Now let's go 1 level up on the tree, if a parent of a leaf has even degree, then for sure its parent will have to be after before it, otherwise it wouldn't be possible to destroy it.

If it has an odd degree, then it will have to be destroyed before it. Now, lets substract the number of sons that will have to be clicked before their parent from the parent degree and continue doing the same procedure, going up the tree.

Lets make a directed graph, that has edges (u,v) for every u and v such that they are adjacent in the tree and u has to be clicked before v.

Do a topological sort on this graph and start clicking vertices in order, if at some step we have to click a vertex with odd degree, there is no solution. Otherwise, simply clicking in order will produce a solution.

Actually I will kill myself. I used f***ing

std:map< string >and I didn't realize it is |S|·log.P.S. I read comment it is always optimal to destroy vertex with the largest depth, why it is not good to destroy vertex with largest id ?

Because ids are arbitrary. You can relabel the nodes in any way that you want. (as long as you output them with their original labels)

Also don't kill yourself.

I am probably the only person who tried to solve Problem B using Dynamic Programming. I am sooooo stupid.

"Oh, they changed the mod from 1e9+7, that must mean it's not prime."

Why would you do this to us

1e9+9 is prime number

So, what's the problem with 10

^{9}+ 7? It's confusing...http://codeforces.com/blog/entry/58959?#comment-425616

With 1e9 + 9 I can generate more “smart” tests.

Yes, I know now, but I didn't check during contest. Instead I concluded "they must've changed it for a reason, not to be prime". Whoops

When you get wrong answer on E after system testing...

help please with this submission of DIV 2 C: http://codeforces.com/contest/964/submission/37413366 upd: solved, thanks voidmax

Can anyone explain, why it is always optimal to remove nodes from deeper level in div2 D?

http://codeforces.com/blog/entry/58959?mobile=false#comment-425677

1) Evil 2) Very Evil 3) Not putting any pretest cases where C-B > 1 for B :3

LOL I was hacked perhaps by that exact case :D

Well, thinking about it, being hacked seems to be fortunate :)

Haha, yes

Can someone help me？My code of div1.A is similar to others accepted submissions but why I got Wrong Answer on test9？

http://codeforces.com/blog/entry/58959?#comment-425616

Try a = b or a = p — b

Thanks.I got it. But i still don't know why when a=p-b will make (a^-1 b)^k=1.Is it a special condition when mod is 1e9+9？

I can generate the test, then k devide (p — 1).

Thank you.

1e9+6=(5e8+3)*2 (two prime numbers)

I normally have something to complain about the rounds (even though I never comment) but this round's problems were great.

Thanks :3

Now failed systest on A too... Why not allow solution...

It seemed that I still have a lot of thing to do if I want a slot in ACM-ICPC WF

Revision history of the comment is me irl

If I had to guess, I'd say it's not a problem of time complexity, but of correctness.

In your submission, you write

`int t =k*n/2`

, but the multiplication can exceed the limits of the int type, causing undefined behavior. What I'm guessing is that at some point this`t`

becomes negative and since you passed it as an exponent to your power function, it will cause an infinite loop.Just a hunch though, I'm not at my laptop right now so I can't test :)

twill never exceednin the problem statement.Why? Which part of the problem statement leads you to believe that?

I called the function in this manner:

So t will never exceed N/2 (let N be the 'n' in the problem statement so it won't be ambigious).

I got accepted after improving the complexity to , and I didn't change the Calc function in my code.

37415485

You can calculate first block in complexity

O(K+log(A+B). First calculating , so you can calculate all degrees up to .What's wrong with my solution ofr div2 C? http://codeforces.com/contest/964/submission/37415514

you forget modulo in cout<<sum*((n+1)/k);

Oh my God... Thank you

One more question, when does (

b/a)^{k}= 1 holds ifa! =bI don't get itEditorial?

The feeling of becoming Expert.

I think the time limit for Div1 A/ Div2 C should have been 2 seconds considering order of 10^6 modulo operations.

But is that true, that with given constraints (

k< = 10^{5})(b/a)^{k}= 1 implies (b/a) = 1?nope, a = 1 / (p — b) and k = 2, so (b/a) = -1 and (b/a)**k = 1

Thanks, got it

Jesus! I can't believe my bogus solutions for B and D passed System Tests !!!!!!

My solution for B: Maintain a set with all nodes with even degree ordered by depth first (distance to root). At each step, remove the deepest node (farthest from the root) and update degrees accordingly, removing or adding to the set as necessary.

My solution for D: Perform Suffix Array on

S. Then for each query (t,k), all occurences oftinSwill form a contiguous segment in the Suffix Array. Let the suffix array beSA, and suppose that the subarray for a given query in it is (l,r), then we need to find a setXofkindicesi_{1},i_{2}, ...,i_{k}from the range (l,r) such thatmax(SA[i]) -min(SA[i]) is minimized (). I did that by pushing allSA[i] withl≤i≤rto a vector, then sorting the vector and checking all values ofv_{i}-v_{i - k + 1}for alli≥ {k- 1}. I was certain this approach would get TLE in a very early test case. I can't believe such a thing passed systests....... but I'm really happy it did! :)Your solution for D works in . Let's consider all strings of the same length. The total number of their occurrences in

sisO(n), because at each position there can be at most one string. The number of defferent lengths is . So the number of all occurences is . You can also get rid of additional if you use Aho-Corasick instead of suffix array.Thanks a lot for the insight!

I will certainly look into that algorithm :)

i can prove your solution for B and it's true :))

so tests are okay ...

Yeah, I have no doubt tests are okay. It's just that I didn't prove it myself, I just thought "I guess this approach will work" and decided to blindly submit.

The same for D, I didn't put too much thought into how many different lengths and matches could occur and didnt' do the math. I just coded the first thing that came to my head and submitted.

That's why I didn't expect either of them to pass. Seeing my rating increase and ranking 50+ was a very pleasant surprise :)

My solution for B div1:

Always process the child nodes first before their parent (like a topological sorting where the directed edges go from the nodes to their parent). Initially start with the leaf nodes in the queue of the nodes to be processed. When processing a node, if its degree is even, delete the whole subtree rooted at it, and decrease the degree of its parent by 1. And after processing the node, look at its parent, if this parent has its children all processed then push this parent to the queue. At the end, you succeed if you have deleted the whole tree.

This idea is based on the fact that the information you start with is that the leaf nodes can't be removed unless their parents are removed. And if you processed the whole subtree under some parent and this parent still can't be removed you have to remove its parent, and so on. If you reached the root and couldn't remove it, so it is impossible to remove all nodes.

I see many people have provided greedy solutions for Div1B/Div2D. Here's my DP on tree solution (which ACed) for those interested:

First, observe that for any vertex u, having parent v, you can destroy the subtree of u either entirely before destroying vertex v or entirely after destroying vertex v. So, let's classify our vertices into two types: 'before' vertex — a vertex whose subtree is destroyed before destroying it's parent and 'after' vertex — a vertex whose subtree is destroyed after destroying it's parent.

Now, identifying vertices is easy. First, note that all leaves are 'after' vertices. They can only be deleted after their parent is deleted. Now, observe that a vertex is a 'after' vertex if and only if it has even number of children who are 'after' vertices, otherwise it is a 'before' vertex.

So, all vertices can be classified into 'after' and 'before' with a single DFS. The answer to our question is YES if the root is an 'after' vertex otherwise it is NO. Printing the solution is also easy, for printing the solution of the subtree of vertex u, print the solutions of all it's children who are 'before' vertices, then print u, then print the solutions of all 'after' vertices.

Here's my code: Destruction of a Tree

How to solve div1 E?

Please, wait for editorial, i'm preparing it now.

Still waiting for the editorial...

On div 1 Problem D test 22. The n is 80445 by assert, and the kind of lengths of mi is at least 325 by assert too. So Sum of length of all strings in input is at least 133420. That's illegal to the tips of the subject.

There really is a problem but I don't think that would be. The problem is that the problem mentioned "Sum of length of all strings in input doesn't exceed 10^5", the total length of query is less than or equal to 10^5, but including the string s, the total length is greater than 10^5. But I don't think it caused any trouble.

WHY got TLE in Div1 D using a Aho-Corasick automaton to brute force maintain the occurrences of each query strings?

I thought the time complexity is O(n sqrt(n))

my code here:37415123

So when can we see the tutorial of the contest? I really need it. thx!

Thanks for the round voidmax, great problems! (except div1 D)

Thanks :3

Since there's no editorial, here are some solutions:

B: No clever tricks, just tree DP — is it possible to delete the subtree of

vif the parent ofvis deleted beforev/ afterv? If we determine which sons ofvshould be deleted beforevfor both of these cases, the order of deleted vertices can be computed recursively (to delete the subtree ofv, delete subtrees of sons deleted beforev, deletev, delete subtrees of remaining sons). For the tree DP, ifvhas a son that can't be deleted in either case, the answer is no for both; if it has a son that can be deleted in both cases, the answer is yes for both, and otherwise, we know exactly which sons should be deleted before/afterv.C: If the answer is non-zero, the problem can be formulated this way: we have a complete edge-weighted bipartite graph (vertices are widths and heights of rectangles, edges are rectangles, weights are

c_{i}); count the number of ways to assign integer weights to the vertices in such a way that the weight of each edge is the product of weights of its endpoints. This can be solved independently for each prime factor — we're adding exponents instead of multiplying integers, everything is determined uniquely if we fix one exponent, and since these exponents are small, we can try all possibilities for one of them and check by bruteforce if it actually leads to a valid assignment. We only need to deal with prime factors of w.l.o.g.c_{1}; afterwards, we're left with a graph that hasc_{1}= 1, and need to check if the only possible assignment in this graph is valid. Time complexity: .D: If we know all positions where a query string occurs in

s, the answer for this query can be easily computed in linear time with no additional overhead. Actually, there are sufficiently few of these positions, so we can find them all. For a query string with length greater than sqrt (let's say 100), we can find them by KMP with overheadO(|s|); there are only sqrt of these strings. All query strings with length at most sqrt can be stuffed into a trie. Now, for each suffix ofs, we can find a path in the trie corresponding to this suffix (with length at most sqrt, since that's the depth of the trie), and all query strings that end in vertices on this path occur on the position corresponding to this suffix. Time complexity: . You can get MLE or TLE, watch out for your trie implementation, try to put smaller strings in it or use a hashmap instead.Problem B has a greedy approach too -> here

That's the approach with clever tricks.

Solving D with only Aho-Corasick is also possible. Just put all query strings in the trie and compute the suffix links. Then iterate over the text characterwise and record all occurences.

Aho-Corasick is solid overkill compared to KMP and tries. Yes, this is a deterministic memory-efficient solution, but I like to avoid Aho-Corasick whenever possible.

You seem to not understand Aho-Corasick if you think that KMP and trie is simpler. Because Aho-Corasick is literally KMP and trie. It even can be written as combination of those.

Ok, wise guy, answer this: in what order do you compute backlinks and where in KMP or trie do you need to think about it?

In order of length increasing, same for kmp.

In KMP, it's also in DFS order, or in the order in which vertices were added. So you didn't answer the second part of my question: where in KMP do you need to think about which order (out of at least 3 possibilities) it should be?

You don't need to think of it in both Kmp and Aho-Corasick if you have proper images of how it works in your head. Both algo proven by induction over length, so you use order of lengthes in your implementation. That's just ridiculuous that you think that it is hard to choose between dfs and bfs in such a case.

Can any one please help me about my solution of Problem B I use a O(n) algorithm but I get TLE.. http://codeforces.com/contest/963/submission/37431208

ok I know why I'm wrong

When will you write the editorial for the contest?

Why is my solution for Div.2 C computing 1 as the answer for TC 7? LINK

Thank you in advance.

While addition, subtraction and multiplication are straightforward in modular arithmetic

( (a op b) % mod = ((a % mod) op (b % mod)) % mod, where op is one of +,-,* )

the same cannot be said about division:

(a / b) % mod != ((a % mod) / (b % mod)) % mod. For instance, take (8/2) % 5.

To do division in modular arithmetic, you multiply the numerator with the modular inverse of the denominator. A modular inverse for number a under the modulo n is such a number b that . This number b exists if and only if

gcd(a,n) = 1. By Euler's theorem, we can conclude thatb=a^{φ (n) - 1}. You can read more about it online:Wikipedia

1e9 + 9 is the prime number so you can use Fermat's little theorem.

That is correct and is often what you use for solving tasks. I was just trying to offer a more general answer.

a·b= 1b=a^{p - 2}Thank you Bogdanisar and voidmax

Out of curiosity, how will the following statement result in an overflow:

(a / b) % m

There's no overflow. The fact that division doesn't work in modular arithmetic doesn't have anything to do with a programming limitation such as integer variables having max values and overflowing. It just doesn't work mathematically.

hey, if someone could explain please why is the total weight of split of 8 equal 5? At 964A ... I mean, i found 4 of its splits but i can't find the fifth ... Thank you!

2,2,2,2

2,2,2,1,1

2,2,1,1,1,1

2,1,1,1,1,1,1

1,1,1,1,1,1,1,1

Thank you!