We will hold AtCoder Beginner Contest 296.

- Contest URL: https://atcoder.jp/contests/abc296
- Start Time: http://www.timeanddate.com/worldclock/fixedtime.html?iso=20230401T2100&p1=248
- Duration: 100 minutes
- Number of Tasks: 8
- Writer: kyopro_friends, math957963, physics0523, Nyaan, nok0
- Tester: leaf1415, MMNMM
- Rated range: ~ 1999 The point values will be 100-200-300-400-500-500-600-600.

We are looking forward to your participation!

Looking forward to participate and getting most of it

How to solve F?

Key observation $$$1$$$: Swapping $$$(i, j)$$$ in $$$A$$$ and $$$(i, k)$$$ in $$$B$$$ is equal to swapping $$$(i, k)$$$ in $$$B$$$ first followed by swapping $$$(i, j)$$$ in $$$B$$$. Therefore, we can always fix $$$A$$$ and operate on $$$B$$$ to make $$$B$$$ equals to $$$A$$$. Note that $$$(i k)(i j)$$$ could be merged into a rotation of three words, we call them $$$3$$$-rotations.

Key observation $$$2$$$: The answer is "Yes" iff $$$B$$$ is an even permutation of $$$A$$$. $$$B$$$ always swap even times, so if $$$B$$$ can reach $$$A$$$, $$$B$$$ is an even permutation. On the other hand, it is well-known that the alternative group $$$A_n$$$ (which consists of all even permutations) are generated by $$$3$$$-rotations $$$\{(123), (124), ..., (12n)\}$$$, therefore, if $$$B$$$ is an even permutation, then $$$B$$$ can convert into $$$A$$$.

Key observation $$$3$$$: If the content of $$$A$$$ and $$$B$$$ are different, it is impossible. Otherwise, if there are duplicate elements in $$$A$$$, it is always possible, as we can add a dummy swap between two elements with the same value to make an even permutation. Finally, we only need to consider the case where all elements are distinct. We can use the merge sort to calculate its inversion number in $$$O(nlogn)$$$.

Solve $$$A, B, C, E, F$$$ but WA on $$$D$$$ $$$8$$$ times. How $$$D$$$ please?

Upd: I am an idiot. I wrote something like $$$n \times n$$$ which easily overflows.

I looked up for a and b upto sqrt(m).

Can you give a hint for E?

I looked for cycle, got idea from the odd-even explanation of the last test case.

Build a functional graph which consists of edges $$$i \rightarrow a[i]$$$. Then T can win at $$$i$$$ iff $$$i$$$ is in a circle. To find all elements not in a circle we can utilize the toposort.

Can you share your submission? I thought the element which is in circles can be reached whatever the value of k will be. Atcoder editorial is quite complex to understand.

I think he wins iff i is in a circle.

Sorry, critical typo.

Spoilerwhat's going wrong in this 51 TestCases Passes and 5 Failed.I faced this 5 WA too. The solution is to check if n*n is overflow.

how to fix it did you use sqrt to compare and in result did you use LLONG_MAX?

Yes, my submission

can you tell what your ceil does?

n can be 1e12. n*n can become 1e24 which is too big.

change

to

worked thank you guys!!! didn't think of overflow during contest!! 😭

what was the reason for checking it only till root m?

In $$$G$$$ was killed by just only one case: when query point is equal to rightmost point in polygon.

How did you approach G?

This is mostly implementation based problem without any thinking. There is algorithm for checking if point lies in convex polygon.

First we do cyclic shift on polygon such that its leftmost point becomes minumum. Then for all other point $$$atan2(y[i] - y[0], x[i] - x[0]) \in [-\frac{\pi}{2}, \frac{\pi}{2}]$$$. Then we precalc for every $$$i$$$ this $$$atan2$$$. This is increasing array.

We have a query point. We have to check in which triangle of points with indices $$$0, i, i+1$$$ does in lie. Use lower_bound to do it in $$$log{n}$$$. Now we have to check, if point is in corresponding triangle and is on corresponding segment. To remove many ifs and epsilons and make life easier, we just apply these checks to all $$$[max(0, i - 3), min(n - 1, i + 3)]$$$.

Now we only have to check, if point is on segment, and if point is in triangle. This can be done in integers.

F is similar to 1585D - Ещё одна задача на сортировку / 1591D - Ещё одна задача на сортировку

Also 986B - Петя и перестановки

Can someone explain the proof of D in more detail?

Why does imposing a <= b does not restrict set of integers?

In D, i did binary search

My thought process was like from the array

`[1,2,3,...N]`

we will take l=1, r=N take mid as (l+r)/2 , then WLOG consider mid as "a" then select b from the array as ceil (M/a) , if my b <=N , then our ans will be a*b ans we will move to the left of search space , otherwise will move to the right of search spaceI am not able to think why this is not working i am not able to think of any corner case.

SpoilerAfter bunch of WA, i just guessed and traversed from 1 to 1e6 and for every "a" keep finding b and updating ans as a*b. I dont know why it works, but it works

There is a proof of why it works in editorial but it's confusing, if you understood the proof please explain it to me!

https://atcoder.jp/contests/abc296/submissions/40253903 can someone give a example where this fails.

3 2 3 2 The answer is 2

Thanks achvanov

Submission — one of the shortest solution to problem E using binary jumps

Can you explain the approach?

Step 1If $$$i$$$ is on a cycle, then we always win.

Step 2If $$$i$$$ is outside of the cycle, then our opponent can pick $$$k_i = N$$$ and he always wins.

Step 3Pair of intersected cycles don't exist in this graph.

Step 4So, the answer is the sum of lengths of cycles.

FinallyLet's do $$$2^{20}$$$ (or $$$2^{30}$$$, or $$$2^{60}$$$ — some value, which we can choose) steps for each vertex with using Binary Jumps technique. We will finish on the cycle (always) for each vertex. No one vertex on cycle will be missed because we are doing equal number of steps. Total number of distinct results (

`set.size()`

) is the total number of vertex on cycles — answer on the problem.amazing!! thanks a lot! the problem basically reduced to topo sorting. Awesome problem!

How can we always win if i is on a cycle

Take such vertex on the cycle which will equal $$$i$$$ after $$$k_i$$$ steps