We will hold AtCoder Beginner Contest 172.

- Contest URL: https://atcoder.jp/contests/abc172
- Start Time: http://www.timeanddate.com/worldclock/fixedtime.html?iso=20200627T2100&p1=248
- Duration: 100 minutes
- Number of Tasks: 6
- Writer: beet, evima, kyopro_friends, tozangezan, sheyasutaka, ynymxiaolongbao
- Rated range: ~ 1999

The point values will be 100-200-300-400-500-600.

We are looking forward to your participation!

Is there any way to view current solve-count of problems, without having to reload standings? Reloading standings take too much time with my internet connection.

It would be nice to be able to filter the ranking page by "show fav only", just like contest standing.

If by reloading standings you mean using the Refresh/Auto Refresh function (in the customize option in the standing page), then I think there is no other way. (Unlike CF, the problem list page of AtCoder does not show the current solve count.)

I hope the Auto Refresh solves your problem (as it likely updates while you are solving other problems), but it still somehow needs to download the whole standing.

The most interesting and hard ABC I see.

They made problems hard this time seeing last time many solved D and E.

D wasn't hard at all, it was just OEIS.

What is OEIS? I wasn't able to solve it.

OEIS is a website which has information about all sequences, This one was 1, 5, 11, 23

On-Line Encyclopedia of Integer Sequences Google this or simply OEIS. https://atcoder.jp/contests/abc172/submissions/14775013 D in O(nlogn)

it seems that you've successfully created an array div[n+1] , when n can be 1e7 .

i am getting

`segmentation fault(core dumped)`

when i try to create an array of 1e7+1 integers in`main()`

. i don't understand why! can you explain please ?There is something else wrong, array of size 1e7 is no problem.

Man, I used Sieve of Eratosthenes to calculate no of divisors of each number from 1 to n. Here's my AC Code

But I submitted it after the contest because when I run it on my laptop during the contest it didn't gave output to 3rd test case which was 10000000 because i thought it will give me TLE. Any possible reason for this?

You would need to provide a link to the code and an understandable question.

Please read my edited comment

Sorry, that link does not show the code.

I edited the link, don't know what has happened with me today. Sorry man.

Are you sure that this works:

`ll arr[n+1]={};`

Usually there is an 0 between the braces. For vector it works because it calls the default constructor, which initializes all to zero. But on array?

A local array (AKA an array created in a function scope) depends on the size of the stack. Default stack size is not enough to create such a huge size array.

You have 3 options:

Thanks for the suggestions. Also how to increase the stack size of function mentioned in 2nd option? Though I'll use global array now onwards. Just curious about 2nd option.

Assuming you are on Windows, passing

`-Wl,--stack=268435456`

to the compiler will give same stack size as Codeforces gives.On Linux, you can use

`ulimit`

.i got AC with this code, but it gave segmentation fault in my device. As far as I can remember i've done creating similar sized array before in my device. these type of unexpected behaviors are really dissatisfying :( hope i will get an explanation .

thank you very much for your reply :)

I cannot see anything wrong with that code, and its proofed by AC.

So maybe/likely there is something wrong on your device, some wired compiler options or the like?

What's a wired compiler option? I've a dell laptop i3(5th gen) 64-bit. Don't know apart from this. But yes the problem is with the machine may be. Some people on this thread had same problem like me. Can't do anything now. (-_-)

If you run the code on your laptop you first compile it to a program. For this you use a compiler, and you call this compiler somehow. All compilers suppert uncountable optional parameters to do certein things.

Perhaps it might be. :( thank you for you kind replies.

Or you could just use sieve of erasthothenes to keep count

Hey, I used sieve of eratosthenes but didn't get AC? How did you get it?

Matuagkeetarp

Here is Code which uses Sieve

Really liked the Problem C.

C was awesome

Man I tried it so many times.

Don't know where it went wrong.

same but i got it right at last 1 minute i was taking lower_bound instead of upper_bound

indeed!!

Try to solve it using prefix sum and binary search

Test cases are perfect. But your solution doesn't consider all the books that can give the better answer.

For example, Consider this test case-

6 4 25

5 15 1 1 1 1

4 15 8 8

Your answer- 3

Correct answer- 6.(As we can read all the books in first stack within given k)

I also couldn't solve it. It require a DP solution or some other solution explained by other coders. which I'm myself trying to understand :)

Can the books be sorted also ? Or is the order fixed?

No, they cannot be sorted. It says in the statement that you can only read the books from the top (The first element; this works more like a queue than a stack).

Not quite DP.

Calculate the prefix sums for the first array.

Then try to 'read books' from the second stack. For each number of books read from the second stack, use binary search to calculate how many books you can read from the first stack.

The answer is the maximum across the sum of books readable from both stacks.

Submission

How to solve E?

I solved it using inclusion-exclusion principle.Submission

I think F is quite similar to 1325D - Ehab the Xorcist. Knowing idea of this problem could be a massive help.

I didn’t find having solved that problem especially helpful. I would argue that the main challenge of F came from dealing with the condition of maximizing the first value without taking it above A[1].

For me the useful idea is, $$$a+b=(a\oplus b)+2$$$(a & b)$$$ $$$. I didn't find this during that contest and thought it's quite tricky. The idea of digit dp is quite simple in my opinion.

lol it doesn't require digit dp it was just x = a^b which is xoring boxes 2..n , y = a+b , a is A[0] , b is A[1] , then you just have to find the a&b which is w = (a+b-a^b)/2 , then the answer at first is w then you have to iterate from left to right on x and if you can add this bit to the answer without exceeding a then just add it. this is the solution briefly but you have to add the -1 cases inside

Yeah, I messed things up a bit. Thanks for telling. :)

I could infer we have to make xor of first two numbers equal to xor of the rest of the array. How to proceed further?

How to solve C? Question: https://atcoder.jp/contests/abc172/tasks/abc172_c My solution: https://atcoder.jp/contests/abc172/submissions/14773409

Make prefix sum of array a and b try taking upto ith element in a then find remaining k if ith books in A are being read , with this remaining k binary search in prefix sum of array B to find maximum possible number of books you can read, do the same by taking ith index of array B also.

Solved with partial sum and binary search

My code

hope it helps

My solutions to all the problems are outlined at this link.

can anyone tell the logic behind how to solve C?my all sample test cases were coming out right but still it was showing WA for many

apply binary searchlink to the solution solution

Can someone plz explain the solution of problem C.....??plzz

Problem C: It can be done using binary search. We will try to select (0, 1, 2, ... , n) elements from a[] and then the differnce that remains = k — prefix_sum[i]. Now, we will binary search the index in B[] in which prefix_sum1[j] is less than difference obtained above. The time complexity of this solution is O(n logm). My solution : https://atcoder.jp/contests/abc172/submissions/14766605

Using 2-pointer can solve it in linear time.

can you explain please? I also tried 2 pointers but failed.

How to solve E and F?

For E: Use the inclusion-exclusion principle. Solution

Is it an special algorithm ? or a technique which we use in math?

just Basic maths.

for E: I solved it using inclusion-exclusion principle Submission

F: From the nim-conclusion, the problem is equivalent to "given a, b, x, find k such that (a-k) xor (b+k) = x".

Given an interval [l, r] where l is an multiple of 2^20 and r is less than l + 2^20, we can check in O(1) if there can possibly be an answer k in said interval. Once we find the interval with smallest l, we directly enumerate all values to check if there actually exists one.

How do we check in O(1)? Notice that (a-l)>>21 and (a-r)>>21 must have a difference of at most 1, and likewise for (b+l)>>21 and (b+r)>>21. If there exists a pair (a, b), where a is one of [(a-l)>>21, (a-r)>>21] and b is one of [(b+l)>>21, (b+r)>>21], such that a^b=x>>21, then one exists.

your algorithm is interesting. however, i can't understand the last part:

if pair (aa,bb) exists and satisfies:

then there is an answer.

could you please explain it more in detail?

Hey! I'll try my best to explain my logic here:

Notice that for any i between l and r (l <= i <= r), then (a-i)>>21 must be either (a-l)>>21 or (a-r)>>21. Likewise, (b+i)>>21 must be either (b+l)>>21 or (b+r)>>21. Since in (a-i) xor (b+i) the first bits of (a-i) do not affect the other bits, we can regard them separately. Just taking the xor of the (almost) constant first bits, we can check if their could possibly exist an i in interval [l, r].

now I can prove some of your claims to help me understand:

and

proof:

suppose

`l=x*2^k`

,`r=(x+1)*2^k`

(which in your case k==20).likewise,

hence,

because

`0<=d1,d2<2^(k+1)`

,`-2^(k+1)<d2-d1<2^(k+1)`

. thussince

`(a-l)>>(k+1)`

and`(a-r)>>(k+1)`

are both integer, thus`(a-l)>>(k+1)-(a-r)>>(k+1)`

belongs to`[0,1]`

.Yep, as far as I can tell that's correct.

Once you find an interval [l, r] that can potentially contain an answer, you just iterate through the entire interval to check.

so your main strategy is:

am i right?

Yes, that is correct. This will give a total runtime of O(M sqrt(max(a))) where M is the number of higher bit matches. Although I don't know how to prove that M=O(1), it works very fast :) I'd be grateful if someone can hack my solution or prove that M is constant.

I tried implementing your idea here. It passes all of the tests, except one named

`after_contest_01.txt`

. I think this test case was specially constructed to fail this solution idea.The input array is: $$$2^{39}-2,\ 1,\ 2^{39}-3$$$.

There's no possible solution. But all of the intervals seem to appear as potential, thus timing out.

PS: I know this is necroposting. But many people train using Atcoder problems and view these discussions to learn new approaches. So it might be helpful to some.

How to solve F?

https://atcoder.jp/contests/abc172/submissions/14745041

Can anyone help me why this is wrong?

try test 3 3 5

a = [1, 2, 3]

b = [3, 1, 1]

answer = 3, just take all second array

E had a solution quite similar to Placing Rooks

what is wrong in my solution of problem Chttps://atcoder.jp/contests/abc172/submissions/14763812 getting WA on 10 test cases.Here greedy solution doesn't work as you might get a better solution picking the higher one, Consider the following case, N = 3, M = 3, K = 5, A = [1,2,3], B = [3,1,1], here you can pick the whole B array and answer will be 3

Ohh yes, now I get it, thank you so much

problem E can be solved by IEP. The closed form answer is:

What is IEP?

Inclusion–exclusion principle

does E has something to do with derangements?

Principle of Inclusion-Exclusion.

I am new in Atcoder. I did't see any English editorial for the problems. Is there any English tutorial available?

For International Readers: English editorial will be published in a few days.It's written in editorial

will take some time for the translation.

They generally come out a day later.

for problem E, assuming fixed A is [1,2,3...,n], ans = C(m,n) * n! * G(n, m)

which G(n, m) means given 1 ~ n for A, using 1 ~ m for B that satisfy the condition

then G(n, m) = (m — n) * G(n — 1, m — 1) + (n — 1) * ( G(n-1, m-1), G(n-2, m-2) )

G(0, m) = 1, G(1, m) = m — 1

when n == m, G(n, n) = D(n), which is derangements

Can you please explain how you got that recursive relation

or any link related to that derivation would be nice

Assuming A is [1,2,3,...,n]

When looking at what G(n, m) should be, let's imagine what the last number, x, in B should be.

There are (m - n) ways to select x greater than n, leading us to a position where our new size is n - 1 where we still have (m - n) available numbers greater than the size (we lose the number we selected, but we gain n as a choice) This gives us the (m - n) * G(n - 1, m - 1) part of the formula

There are (n - 1) ways to select x < n. From here, we have 2 options:

1) select the xth number in B to n.

This leave us with a new size to fill of n - 2, and there were still (m - n) numbers available that are not in A. This gives us the (n - 1) * G(n - 2, m - 2) part of the formula

2) select the xth number in B to be something other than n

Here we can imagine that x is our new 'last number', but instead of being unable to select x for it, it cannot select n for itself. Every other number works the same, so this is the same as G(n - 1,m - 1), giving us the (n - 1) * g(n - 1, m - 1) part of the formula

Is there a way to solve the problem without blindly using any memorized formulas? Some form of dp? What would be a useful definition of dp[i]?

Yes? It's just Principle of Inclusion-Exclusion.

Let $$$S_k$$$ be number of permutations with $$$>= k$$$ positions fixed (as given permutation).

$$$S_k =$$$ (no of ways to choose set of $$$k$$$ positions) * (no of ways to permute $$$(n - k)$$$ positions from $$$(m - k)$$$ elements) = $$${n \choose k} * {m - k \choose n - k} * (n - k)$$$.

Then, Derangements $$$= \Sigma_{i = 0}^{n} (-1) ^ i * S_i$$$

It looks to me that G(n,m) is easy to turn into a dp, since you always decrease m by the same amount you decrease n, so you can disregard m and just keep track of n in each state.

Submission

Thanks

https://atcoder.jp/contests/abc172/submissions/14765903

this is my submission in the contest

During the contest, I was able to derive this formula

`G(n, m) = (m — n) * G(n — 1, m — 1) + (n — 1) * (n - m + 1) * G(n - 2, m - 1)`

I am sure this recurrence is correct but this form didn't let the computation to be in O(N)

Thanks for sharing the formula @mickeyandkaka

I have a solution to the D problem with time complexity O(n).

$$$F(n) = \sum_{k=1}^n (\dfrac{k}{2} \times \left\lfloor{\dfrac{n}{k}}\right\rfloor\times \left\lfloor{1+\dfrac{n}{k}}\right\rfloor)$$$

Code (C++):Why does it work?

because OEIS

This

number 2 appears as the divisor in (n / 2) numbers from total n numbers

In 2,4,6, ... number 2 appears as a divisor exactly one time,

So our answer will increase by

2 * 1 + 4 * 1+ 6 * 1 + ...

2 *( 1 + 2 + 3 + .. + n / 2)

Do the same for 1 to n

SpoilerC was way harder than D for me , i dont know if its me or the problem.

I used 2 pointers instead of binary search in problem C. That was easier to implement in my opinion.

bro can you give me link to you solution using 2 pointers?

here's mine for using 2 pointer for C https://atcoder.jp/contests/abc172/submissions/14740634

Really liked problem E! Good contest!

is there any English editorial in atcoder? i am new at Atcoder

For International Readers: English editorial will be published in a few days.

but you can use google doc translater (its not accurate but you can understand the logic)

O(sqrt(n)) solution for D:

Here is my solution to C.

Can anyone tell me why the above code gets WA in 10 test cases. Isn't the problem solvable by 2 pointer kinda technique?

Try a case like :-

1 2 120 101 110 10

The correct answer should be 2 by reading the books from array B which take 110 and 10 minutes(collectively a value less than or equal to 120), but your solution gives 1(by picking 101 from array A instead).

Ahh, so that means greedy will not give the optimal solution right?! Any help on how to solve the problem? The English editorial is going to take some days to come :(

Since it is clear from the statement that we have to pick books exactly in the order as how they appear, the first intution can be to construct prefix sum arrays for both arrays A and B(say, arrays, prefA and prefB). Why? Because, having two prefix arrays (one for A and other for B) means that, we can find out the time to read books upto the ith book(included), in O(1) for both the arrays A and B, separately. Now, the only part remaining is to search efficiently, the maximum number of books we can read in <= K mins. We can do a binary search since, prefix arrays are always sorted. One approach could be to fix the number of books we can take from array A (0,1,2...N) one by one and if for any of these values, the time taken i.e., prefA[i] is <= K, we do a binary search on prefB to see how many books can be picked from array B, and keep updating the answer, accordingly. Hope it helps :)

why greedy fails for C? at each step pick up the minimum of the two.

UPD — Got it.

TestcaseAnswer — 4

UP2 — Thank you for so many testcases :)

Try this

The correct answer is 3, but greedy gives 2

https://codeforces.com/blog/entry/79405?#comment-651442, you can check this comment

Try this case5 3 70

50 2 3 4 5

49 48 47

Can any one tell why C fails using two pointer approach and any test case for the same.

This is a nice explanation of how inclusion-exclusion principle can be used to find the number of derangements. It can be extended to solve E.

Edit:Look for the proof under the formulae section.i wonder why this solution WA : https://ideone.com/VIzSXP

someone helps :((

consider the sequence on desk1 as $$$(1,2,3,4,8)$$$ and on deks2 as $$$(10,1,1,1,1,1,1,1,1)$$$ for $$$k=18$$$. Your $$$ans$$$ is $$$5$$$ but it's 9.

No English Editorial.

I am new to cpp, so not sure whats the problem. The code gives WA in one test case and RE in one. Rest all is accepted. Any help would be nice.

`ll B[n];`

wrong sizeyes, that was the problem tysm

C using upper boundCan somebody explain the solution for problem E, I am not able to understand it.

first of all watch this video then read this comment under this blogpost

thanks a lot, bro!!!!! Learned a lot from this question

https://atcoder.jp/contests/abc172/submissions/14775983 can anyone help in C I used sliding window and binary search but getting wrong answers on 4 tcs .UPD-I found the bug

Easy solution of D just by loop for time limit 3 secYou can go even simpler than thator in kotlin

Thank you for giving us more simpler way.

Hey can anyone help me out where I did wrong in problem C https://atcoder.jp/contests/abc172/submissions/14753662 Thanks in advance

You implemented a greey solution which does not work well. Consider first stack of books all 2, ie

`2 2 2 2 ...`

and the other one starting with 3, ie`3 1 1 1 1 1...`

.So you will read all 2-books, but never a 1-book.

Thank you sir

Does anybody know why my approach to prob D gives TLE? I ran the Sieve storing, for each number, its prime factors (instead of "crossing off" the number, I stored the prime it's a multiple of.) This should be NloglogN. Then, for each number, I compute the exponent of each of it's prime factors via prime factorization (I already know all of it's prime factors, so this should be NlogN, since we can divide the number at most logN times) in order to get the number of divisors.

Note that this approach works; I don't get any WA, only some TLEs.

Does anybody have any clue?

Thanks!

I solved D in exact same way. My run time was 2733 ms which is really tight. Actually they expect us to write an O(n) solution for this problem. BTW below is the link to my submission:-

https://atcoder.jp/contests/abc172/submissions/14789913

Does the problem can be solved in $$$O(N)$$$? I thought $$$O(N \log N)$$$ is the best.

Yes it can be solved in O(n). Just think about the contribution to sum of ith integer.

I did a similar approach, and I didn't have any issues

submission here

Can someone help? Why is this code giving WA?

Try this testcase:

3 4 8

4 10 11

5 1 1 1

Your output:1

Answer:4(Whole second array 5+1+1+1=8)

Two pointer would not work. I hope you got the mistake.

can somebody please explain me problem E with statement and how princple of inclusion exclusio works here

Here is my solution with some comments about what is going on (on a best effort basis).

SolutionThanks you, this is very helpful

Could someone help me with problem C? I got WA in 2 test case and i don't know why here's my submission : https://atcoder.jp/contests/abc172/submissions/15825396

i can not understand b the minor change. if any got plz tell what i have to do to solve it