Rhodoks's blog

Codeforces Round #810 Editorial
By Rhodoks, history, 2 years ago, In English

Sorry for the late editorial. May this editorial help you. If you have questions, feel free to ask.

1711A - Perfect Permutation

hint1.
solution
code

1711B - Party

hint1.
hint2.
solution
code

1710A - Color the Picture

hint1.
hint2.
solution
code

1710B - Rain

hint1.
hint2.
hint3.
solution
code

1710C - XOR Triangle

hint1.
hint2.
solution
code

1710D - Recover the Tree

Thank dario2994, the key part of the proof is from him.

hint1
hint2
hint3
solution
code

1710E - Two Arrays

hint1.
hint2.
hint3.
solution
code
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Comments Comments (111)
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2 years ago, # |
  Vote: I like it +536 Vote: I do not like it

Auto comment: topic has been updated by Rhodoks (previous revision, new revision, compare).

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2 years ago, # |
  Vote: I like it +529 Vote: I do not like it

Auto comment: topic has been updated by Rhodoks (previous revision, new revision, compare).

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2 years ago, # |
  Vote: I like it +14 Vote: I do not like it

Thanks !

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2 years ago, # |
  Vote: I like it +3 Vote: I do not like it

Thank you for the editorial!

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2 years ago, # |
  Vote: I like it +133 Vote: I do not like it

You don't deserve the downvotes man :(

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    2 years ago, # ^ |
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    Upvoting this Blog can decrease his negative contribution.

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      2 years ago, # ^ |
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      Let me be a snob and point out that this mathematically means it will become worse

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    2 years ago, # ^ |
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    The editorial is very well done, with hints/proofs of statements etc.

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2 years ago, # |
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Why do we need to sort array in 1710 A? It works fine without it

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    2 years ago, # ^ |
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    it is just a way to give a valid answer. There may be better ways and in this problem you in fact do not need to implement it.

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2 years ago, # |
← Rev. 2   Vote: I like it +154 Vote: I do not like it

Great Div.1 D. If there had been a normal E, it would have been the best contest this month (so far).

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2 years ago, # |
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Really liked the idea behind Div. 2 D, and the editorial explains it very well. Thanks!

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    2 years ago, # ^ |
      Vote: I like it +52 Vote: I do not like it

    Really? For me it reads like explaining nothing, just stating some formulars. I am lost in the second line.

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2 years ago, # |
  Vote: I like it +1 Vote: I do not like it

Div2 B,I spent a full hour and a half thinking about the y=0 case and did it wrong！The torial is like a drop of holy water on my rusty brain feel so good~

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2 years ago, # |
  Vote: I like it +16 Vote: I do not like it

Can someone explain in a little more detail Div 1 C?

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2 years ago, # |
  Vote: I like it +4 Vote: I do not like it

Is there an error in Div2 D's solution? I think it should be pi−|xi−xj|>=aj−m

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    2 years ago, # ^ |
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    you are right. Thank you very much.

    I have fixed it.

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2 years ago, # |
  Vote: I like it +475 Vote: I do not like it

Auto comment: topic has been updated by Rhodoks (previous revision, new revision, compare).

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2 years ago, # |
  Vote: I like it +5 Vote: I do not like it

I don't understand the idea of difference array for Div2D. Can anyone explain in an easier way?

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    2 years ago, # ^ |
    ← Rev. 4   Vote: I like it +28 Vote: I do not like it

    a[i] can also be calculated in this way.
    Firstly,we make the contributions two arithmetic sequences.
    for example x=6,p=5
    that's 1,2,3,4,5 for [2,6] and 4,3,2,1 for [6,10].

    then we divided the contribution of an arithmetic sequence into two parts: b+k*i
    for the above examples
    we add -1 for each position i [2,6],then increase each position i [2,6] by i.
    we add 11 for each position i [7,10],the decrease each position i [7,10] by i.

    now let's find two arrays for both the b and i.
    it's easy to calculate a 1D difference array.

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      2 years ago, # ^ |
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      Thank you so much for your reply. However, if you loop through each position i in [2,6] or [7,10], will that not make your runtime O(n^2)?

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        2 years ago, # ^ |
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        well,for a arithmetic sequence,we divided its contribution into two parts.
        the first part is add a number b to an interval [l,r], you can use difference array to maintain it.
        the second part is add i for position i within interval [l,r] ,You can calculate how many times a position i is added,then it becomes add 1 to an interval [l,r]

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          2 years ago, # ^ |
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          Oh, I think I see what you're saying now. Thank you!

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      2 years ago, # ^ |
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      But isn't the constraint on range of x and p too high. How are u supposed to calculated difference array like this without iterating over the complete range atleast once?

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2 years ago, # |
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Rhodoks orz

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2 years ago, # |
← Rev. 4   Vote: I like it +33 Vote: I do not like it

In Div 1C, there's an equivalent condition for the inequalities,

$$$(a \oplus b) + (a \oplus c) = ((a \oplus b) \oplus (a \oplus c)) + 2 ((a \oplus b) \land (a \oplus c)) = (b \oplus c) + 2 ((a \oplus b) \land (a \oplus c)),$$$

from the identity $$$x + y = (x \oplus y) + 2(x \land y)$$$.

So, the three satisfy the triangle inequality iff $$$a \oplus b$$$, $$$b \oplus c$$$ and $$$c \oplus a$$$ are all pairwise non-disjoint.

A digit DP solution from here is rather straightforward, 165666080.

Here's $$$\land$$$ and $$$\oplus$$$ refer to bitwise AND and XOR respectively.

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    2 years ago, # ^ |
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    Yeah I did this. The editorial soln has $$$2^9$$$ constant factor but this soln took only $$$2^6$$$.

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      2 years ago, # ^ |
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      Not really, you have atmost $$$8$$$ transitions from each state so it's more or less the same constant factor.

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    2 years ago, # ^ |
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    Can you please explain your dp states and transitions in brief?

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      2 years ago, # ^ |
      ← Rev. 6   Vote: I like it +9 Vote: I do not like it

      Rather than calling the numbers $$$a, b, c$$$ let's call them $$$x_0, x_1, x_2$$$.

      We iterate from the highest bit to the lowest, so call the highest bit $$$0$$$.

      We have the states $$$(\text{idx}, \text{tight}, \text{cond})$$$ where $$$\text{idx}$$$ is the current bit we are on, $$$\text{tight}$$$, and $$$\text{cond}$$$ are bitmasks of size $$$3$$$.

      • $$$\text{tight}_i$$$ is set if $$$x_i$$$ currently matches the input on the prefix $$$[0, \text{idx})$$$.

      • $$$\text{cond}_i$$$ is set if $$$(x_i \oplus x_{i + 1}) \land (x_i \oplus x_{i - 1})$$$ (the indices are modulo $$$3$$$) has a bit set among the $$$[0, \text{idx})$$$ bits.

      I believe the transitions should be clear if you have experience with digit DP.

      We finally require that $$$\text{cond}_0, \text{cond}_1, \text{cond}_2$$$ are set so the final answer is $$$\sum_i \text{dp}(\lvert n\rvert, i, 7)$$$.

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    2 years ago, # ^ |
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    what does "all pairwise non disjoint" mean?

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      2 years ago, # ^ |
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      By disjoint I mean that $$$x \land y = 0$$$, because if you view $$$x$$$ and $$$y$$$, as bitmasks/sets, $$$x \land y$$$ is the intersection of $$$x$$$ and $$$y$$$, and we say that sets $$$S$$$ and $$$T$$$ are disjoint if $$$S \cap T = \varnothing$$$.

      So $$$x, y, z$$$ being pairwise non-disjoint means that $$$x \land y$$$, $$$y \land z$$$ and $$$z \land x$$$ are all non-zero.

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2 years ago, # |
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As a member of botswana community, i upvoted this post !!!!!!!!!!!!

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2 years ago, # |
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Why does this solution give the wrong answer? http://pastebin.ubuntu.com/p/TrRV4gSKG5/

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2 years ago, # |
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For 1710B-Rain, why will the maximum position always be one of the centers? Can anybody help me to prove that?

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    2 years ago, # ^ |
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    A point $$$p$$$ which is between two center points, $$$L$$$ is all the points locates left to it and $$$R$$$ is all the points locates right to it. When we move $$$p$$$ from left to right, the rainfall from $$$L$$$ decrease(getting far away from it) or stay the same(some points is too far to give rainfall), the rainfall from $$$R$$$ increase or stay the same. In detail, the rainfall from $$$L$$$ decrease quickly first, then slowly(some points becomes too far) and the rainfall from $$$R$$$ increase slowly first, then fast(some points becomes close enough). That is the second derivative is always non-negative. So one of the center point must be the maximum point.

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      2 years ago, # ^ |
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      This makes so much sense. Thanks a lot :)

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2 years ago, # |
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Can someone help me understand the solution of div1B? I do not understand the implementation of the logic given in the editorial. What exactly is getIntersection() calculating?

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2 years ago, # |
← Rev. 4   Vote: I like it +16 Vote: I do not like it

I struggled a lot with understanding how we found the intersections in 1710B — Rain without doing exhausting casework, so I decided to explain the math in the code for anyone else who also had a hard time.

We will keep track of the region $$$S$$$ such $$$\forall (x, p)\in S$$$ then there is no flood, i.e $$$p - |x - j| \geq a_j - m$$$, for all invalid positions $$$j$$$.

Looking at the inequlity ( we can see that $$$(x,p)\in S \iff p\geq |x-h|+c$$$ (as mentioned in the editorial "One can check that the region of $$$(x,p)$$$ satisfying such an inequality is a quadrant rotated $$$45^\circ$$$ anticlockwise and translated.")

So we will keep track of a function $$$f(x) = |x-h| + c$$$ to keep track of the region. Initially the center of a region described by point $$$j$$$ is $$$(j, a_j - m)$$$ so $$$h = j$$$, $$$c = a_j - m$$$ (notice that point $$$(h,c)$$$ is the center of such a region). Now $$$(x,p)\in S \iff p \geq f(x)$$$

Now how can we interesect two regions $$$S_1,S_2$$$ described by $$$f_1, f_2$$$? We need to find the function $$$f_3 = \max(f_1, f_2)$$$. By visualizing (or doing a bit more complex case work) we can see that $$$S_3 = S_1 \cap S_2$$$ is also described by a similar function $$$f_3(x) = |x-h_3| + c_3$$$. How will we figure the parameters?

If $$$x\leq \min(h_1,h_2)$$$ then $$$f_3(x) = \max(f_1, f_2) = \max(h_1 + c_1 - x, h_2 + c_2 - x) = t_x - x$$$ $$$(I)$$$ ($$$t_x$$$ is defined like that in the code).

If $$$x\geq \max(h_1,h_2)$$$ then $$$f_3(x) = \max(f_1, f_2) = \max(c_1 - h_1 + x, c_2 - h_2 + x) = t_y + x$$$ $$$(II)$$$ (again $$$t_y$$$ is defined in the code).

As we mentioned $$$f_3(x) = |x-h_3| + c_3$$$ so if $$$x\leq h_3$$$ then $$$f_3(x) = c_3 + h_3 - x$$$ and because of $$$(I)$$$, particularly looking for $$$x\leq \min(h_1, h_2, h_3)$$$ it must hold that $$$h_3 + c_3 = t_x$$$. Doing the same for $$$x\geq h_3$$$ because of $$$(II)$$$ we see that $$$c_3 - h_3 = t_y$$$. Solving the linear system we see that $$$(h_3,c_3) = \left(\frac{t_x-t_y}{2}, \frac{t_x + t_y}{2}\right)$$$. Which are the parameters for region $$$S_3$$$.

In that way we can find the region $$$\displaystyle S = \bigcap_j S_j$$$ (going through all the invalid positions $$$j$$$) which was what we wanted to find.

There might be an easier way but it is the most intuitive way I could find to understand the solution.

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    2 years ago, # ^ |
    ← Rev. 2   Vote: I like it +14 Vote: I do not like it

    i think we can calculate intersection directly. suppose we have two region specified by $$$(a,b)$$$ and $$$(c,d)$$$ , and the intersection is $$$(x,y)$$$. then because the boundaries of regions are just line with slope 1 or -1, we can get

    $$$ \begin{align*} x + y = c + d \\ y - x = b - a \end{align*} $$$

    which gives

    $$$ \begin{align*} x = (a-b+c+d)/2 \\ y = (b-a+c+d)/2 \end{align*} $$$

    through my test, it seems the same as the one in tutorial code

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      2 years ago, # ^ |
      ← Rev. 7   Vote: I like it +8 Vote: I do not like it

      That may be the case for some situations but the formulas are not correct every time. That is why my goal in the post was to skip "exhausting casework".

      Consider two points which sit on the same horizontal line, specifically $$$(a, b)$$$ and $$$(c,d) = (a, d)$$$. Then the formulas give the intersection as $$$(x,y) = \left(\frac{2c+d-b}{2}, \frac{b+d}{2}\right)$$$. We can see how this is not correct since the we can easily see that the center point should be $$$(x,y) = (a,b)$$$. (suppose $$$b>d$$$)

      Note: you might argue that we never intersect points with the same $$$x$$$ value, so with some constraint the problem may be solved the way you described but it is not whole picture I wanted to give.

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        2 years ago, # ^ |
          Vote: I like it +8 Vote: I do not like it

        oh, well, you are right. the main reason is that when one region totally contain another region, the intersection assumed in my equation is actually not exist.

        so it can not achieve "check whether point (xi,pi) belongs to such region", but just calculate some "ideal" intersection ... QAQ

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          2 years ago, # ^ |
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          hello, after calculating the prefix and suffix sums and getting the total rainfall in all xi's, why can't we do an O(n^2) approach, where we go through each day and get the maximum after removing that day's rain and checking if it is <= m ? , i tried it but gave me WA in TC2 , can you tell me why my approach would be wrong?

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            2 years ago, # ^ |
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            if we put time complexity aside, your method is absolutely right.

            however this problem has no guarantee about input xi and pi, so there may be some cases that many pairs of same xi and pi, and your 'originalPos' only record one position. if there are more than one feasible solutions among them, you just change one position in answer string.

            one possible work around is use map<pair, queue> , you can refer to this

            meanwhile, i don't understand very well how you calculate 'left' and 'right' arrays , could you explain a little

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              2 years ago, # ^ |
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              okay i assumed xi's are always unique my bad, so i think there's an easier way to pre calculate the sums but for me the motivation was, for each xi (consider sorted pair<xi,ai> and the rainfall can extend only to the right of xi), it will be

              left[i]=a[i]+max(0,left[i-1]-valReduce[i]- cnt*abs(x[i]-x[i-1])),

              now what is cnt? cnt should ideally be the number of rains that have happened before x[i] (in terms of the number-line) but that does not happen as some rainfall might not extend till the current x[i] hence at every iteration for left prefix i check the max index to which i can extend my current rainfall to, and to the next index of that (redxnIndex) i increase cntReduce array so that when i arrive at that index i know i will need to reduce my cnt by this and also add some correction value to the above equation to valReduce array , similarly i do the same for right and then get the tot[i]=left[i]+right[i]-rainfall[i]

              so it will take O(N*log(N) + N) to calculate the total rainfall after N days

              also according to the constraints shouldn't O(N^2) run in 4 seconds? just curious

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                2 years ago, # ^ |
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                thanks for explanation.

                for time issues, firstly based on system test result we know it can't pass. then from my experience, if a problem is tend to be solved by a n^2 solution, the N is at most around 1e4.

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2 years ago, # |
  Vote: I like it +40 Vote: I do not like it

Let's upvote this editorial to show some support to Rhodoks.

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2 years ago, # |
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Can someone help me figure out what's wrong with my submission for C 165672596

long explanation follows

As you can probably tell, this problem frustrated me a lot in contest. I'm wondering if anyone has a testcase or logic flaw that breaks my solution.

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    2 years ago, # ^ |
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    Take a look at Ticket 15920 from CF Stress for a counter example.

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    2 years ago, # ^ |
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    Here is a test-case:

    5 5 4
15 10 10 10

    First color fills 3 rows, while the rest each fill 2 rows. The YES solution is simple: use 3 + 2 = 5. However, your approach calculates a sum of 3 + 2 + 2 + 2 = 9. Then the first iteration of the leftover loop checks if 3 can be dropped, concluding that yes, 3 can be dropped to reduce the sum to 6, and it also changes the 3 to 0. However, without the 3, it is impossible to reach a target value of 5, since all that's left are 2s.

    Even aside from this, I don't feel like your approach of calculating leftovers and adjusting the sums accordingly is guaranteed to reach a correct solution, especially since the loop doesn't revisit a color (just a plain for loop). Have you actually proved anything about this approach?

    Instead, consider this alternative observation: if we overshoot the target, then for each color that covered more than 2 rows, we can decrease how many rows it covers (don't implement this, just think). If you keep reducing the problem in this manner, what are the possible outcomes? Is there some state that can't be reduced in this manner, but still overshoots? Once you characterize this state, you should be able to figure out when such a state can lead to a YES and when it will lead to a NO.

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      2 years ago, # ^ |
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      You beat me to it.

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      2 years ago, # ^ |
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      yeah, thanks. It's obvious to me now that a constructive solution like mine would be harder to make work.

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    2 years ago, # ^ |
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    Say your list is [3,2,2,2] and you are trying to make 5. You algorithm will take 1 out of the 1st element, and report 'No'?

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      2 years ago, # ^ |
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      I like how we both thought of literally the exact same counterexample

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2 years ago, # |
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Is there a proof of Hint 2 from XOR Triangle Editorial ? a XOR b <= a + b

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    2 years ago, # ^ |
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    For each bit, XOR is the plus modulo $$$2$$$ which is less or equal to plus.

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2 years ago, # |
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In div 2 problem c my logic is same as given in this editorial but still it shows wrong answer on test 8 . Here's my solution https://codeforces.net/contest/1711/submission/165574557 Please help!!!

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    2 years ago, # ^ |
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    Oh, your ll is just long but not long long.

    So it will overflow.

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2 years ago, # |
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wow, the relation between div2D and region intersections is beyond my thought, interesting.

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2 years ago, # |
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To be honest, apart from Div. 1 E, this is a really great round. The questions are really well-thought.

Thank you for this round.

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2 years ago, # |
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Question not that related to this round: Was problem difficulty rating system different in the past? I found lot of problems that have very high difficulty rating for it's difficulty. Some of them are from like 10 years ago, but some of them like this one is from 3 years ago.

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2 years ago, # |
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i like the problems a lot ty <3

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2 years ago, # |
← Rev. 4   Vote: I like it +3 Vote: I do not like it

The editorial of Div2B (1711B) is a little misleading:

If $$$y \geq 1$$$, then only deleting one vertex with odd degree would have been enough.

If $$$y = 0$$$, then the parity of the edges at the end is determined only by the number of edges whose both endpoints are deleted. In particular, there must be at least two adjacent vertices deleted with even degree.

In particular, specifying the conditions of $$$y \geq 1$$$ and $$$y = 0$$$ (which are mutually exclusive conditions) suggests that these are two mutually exclusive cases to consider, i.e., if $$$y \geq 1$$$, we should only consider the odd degrees for removal, while the removal of adjacent vertices with even degree should only be considered when $$$y = 0$$$. But this is false, since even when $$$y \geq 1$$$, it is not enough to consider only the odd vertices, and the optimal solution may require the removal of two adjacent even vertices instead.

My suggestion would be to remove all mention of $$$x$$$ and $$$y$$$ (since their values are irrelevant for the algorithm design), and instead phrase these "cases" as options to consider (the I/O explanation in the problem also refers to them as options), with a brief note of why it's sufficient to only consider these two options: the other options not considered are the removal of a single even person only (which doesn't fix the total degree parity) or the removal of some superset of one of these two options (odd + some set, or even + even + some set).

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    2 years ago, # ^ |
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    well said

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    2 years ago, # ^ |
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    Well, what I want to express is that, if you have an answer:

    1. If $$$y \geq 1$$$, you can get a not worse (but may not be the best) answer by only deleting one odd degree vertex of the $$$y$$$ vertices. So you do not need to consider it except when $$$(x,y)=(0,1)$$$.

    2. If $$$y = 0$$$, you can get a not worse (but may not be the best) answer by only deleting two adjacent even-degree vertices. So you do not need to consider it except when $$$(x,y)=(2,0)$$$ and they are neighbours.

    So, If we want to find the best answer, we only need to consider such two cases as I said.

    And if you think what I express here is easier to understand, I will modify it.

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2 years ago, # |
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cough it's totally not because I suck at graphs that I ruined my ratings this round cough it shouldn't be violent cough

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2 years ago, # |
← Rev. 6   Vote: I like it +31 Vote: I do not like it

A different solution to Div1 D:

$$$Construct(L,R)$$$ is a function that finds the edges in the component of [L,R] ([L,R] is connected).

You can construct as follows:

Let $$$z$$$ be the largest number satisfies $$$z<R$$$ and $$$[L,z]$$$ is connected.

Let $$$x$$$ be the largest number satisfies $$$x\leq z$$$ and $$$[x,R]$$$ is connected.

There is a useful property :

  • There is no segment $$$[l',r'],(l'\in [L,z],r'\in (z,R))$$$ that is connected.

Then if you construct $$$[L,z]$$$ first , the only influence to $$$[L,R]$$$ are the segments $$$[y,R],(L\leq y\leq z)$$$

If $$$[z+1,R]$$$ is connected then the rest of construction is easy:

  1. $$$Construct[z+1,R]$$$
  2. Connect $$$R$$$ and $$$x$$$

If $$$[z+1,R]$$$ is not connected . Then $$$[z+1,R]$$$ can be split into some connected components $$$[l_1,r_1],[l_2,r_2]...[l_k,r_k],[l_{k+1},r_{k+1}]$$$.

The solution is also not hard:

  1. $$$Construct[l_i,r_i]$$$ for every $$$i$$$.

  2. Connect $$$r_1,r_2,r_3...r_{k-1}$$$ and $$$R$$$.

  3. Connect $$$r_k$$$ and $$$x$$$.

  4. Connect $$$R$$$ and $$$x$$$.

In fact the last two part can be implemented together.Submission: 165734051

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2 years ago, # |
← Rev. 2   Vote: I like it 0 Vote: I do not like it

I get stuck proving only stripe pattern make all cell to have at least 3 neighbors. (Div.2 C)

Is there a easy way to prove it?

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    2 years ago, # ^ |
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    In other words, if the picture is not a stripe, there must be a place which has $$$2$$$ neighbors with different colors.

    neighbors with different colors are blacken.

    1 1 1 1

    1 1 2 2

    2 2 2 2

    Node that the lenth / width of the stripes mush be bigger than $$$2$$$.

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      2 years ago, # ^ |
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      thank's for your reply.

      i proved this by different way!

      we can generalize "non stripe pattern" as a "(stripe pattern) — (hole)" when hole is not stripe.

      and there is a two case.

      1. holes appear in first column.

      2. no hole appears in first column.

      1st case ex)

      xxxxxx

      001000 <- 1 in this column is bad

      010000

      110011

      010001

      (x is color which is not 1 (possibly 0))

      then

      -> there exists always bad color 1 in column which color 1 is appear for the first time.

      2nd case ex)

      xxxxxx

      111111

      111111

      010000 <- some 0 in this column is bad

      110011

      010001

      then

      -> there exists always bad color 0 in column which color 0 is appear for the first time.

      and this model include all possible region -> proved!

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2 years ago, # |
← Rev. 2   Vote: I like it 0 Vote: I do not like it

oH mine gods, Sparky_Master_WCH1226 are the goodest in contributon rateing agan. thakns for you suppot sar.

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2 years ago, # |
← Rev. 3   Vote: I like it +21 Vote: I do not like it

D2D

Damn. This getIntersection magic is the one I wouldn've thought about.

You can do it simpler. Just maintain the minimum height current point should have to "cover" all the things on the left of it. It is a very simple dp.
Then calculate the same for right and check for each point whether it satisfies both left and right.

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2 years ago, # |
← Rev. 4   Vote: I like it 0 Vote: I do not like it

Hello. Can someone tell me please, why this solution for C is WA?

https://codeforces.net/contest/1711/submission/165895290

Thank you!

UPD: Got it: https://codeforces.net/contest/1711/submission/165896780

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2 years ago, # |
← Rev. 2   Vote: I like it +5 Vote: I do not like it

Getting MLE in Problem B (Party). My solution

How to resolve?

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2 years ago, # |
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For 1710A-Color the Picture, shouldn't $$$\sum_{i=1}^{n} \lfloor \frac{a_{i}}{m} \rfloor$$$ instead be $$$\sum_{i=1}^{k} \lfloor \frac{a_{i}}{m} \rfloor$$$? Also, I don't understand why if, when $$$n$$$ is even, $$$\sum_{i=1}^{k} \lfloor \frac{a_{i}}{m} \rfloor \geq n$$$ is sufficient. What if every $$$a_{i}$$$ was a single row? Don't you still check if you can make the sum using at least $$$2$$$ from each? It should be the sum of every $$$\lfloor \frac{a_{i}}{m} \rfloor$$$ that is at least $$$2$$$.

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    2 years ago, # ^ |
      Vote: I like it +41 Vote: I do not like it

    You are right, It is $$$k$$$ and I forget to mention skipping those $$$a_i < 2*m$$$.

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2 years ago, # |
← Rev. 2   Vote: I like it 0 Vote: I do not like it

I solved Div1B in incredibly ugly way using segment tree. It was good exercise though.

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    2 years ago, # ^ |
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    I also tried an approach with a seg tree but couldn't figure it out. If you have time, could you please explain how you did it?

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      2 years ago, # ^ |
      ← Rev. 2   Vote: I like it +9 Vote: I do not like it

      Let $$$T(x)$$$ be the total accumulation at position $$$x$$$. We can compute a bunch of intervals $$$[l_{i},r_{i})$$$ and pairs of integers $$$(a_{i},b_{i})$$$ such that for all $$$x\in[l_{i},r_{i})$$$, $$$T(x)=a_{i}x+b_{i}$$$ using prefix sum technique. If we let $$$f_{i}=\max_{x\in [l_{i},r_{i})} a_{i}x+b_{i}$$$ then it is obvious that we will get $$$f_{i}$$$ either at $$$x=l_{i}$$$ or $$$x=r_{i} - 1$$$. Now we want to consider the case when we can erase the $$$j$$$ th rain. After we erase the rain, if we let $$$W_{j}(v)$$$ be the total accumulation of the position $$$v$$$ if we erase $$$j$$$ th rains, then:

      $$$ W_{j}(v) = \begin{cases} T(v) & \text{if } v \lt x_{j} - p_{j} \\ T(v) - v - p_{j} + x_{j} & \text{if } x_{j} - p_{j} \le v \lt x_{j} \\ T(v) + v - p_{j} - x_{j} & \text{if } x_{j} \le v \lt x_{j} + p_{j} \\ T(v) & \text {if } x_{j} + p_{j} \le v \\ \end{cases} $$$

      Notice, that for every $$$i$$$, $$$\max_{v\in [l_{i},r_{i})} W_{j}(v)$$$ will also be achieved at either $$$v=l_{i}$$$ or $$$v=r_{i}-1$$$. Thus, all we have to do is compute the maximum of $$$W_{j}(v)$$$ for all $$$v\in [l_{i},r_{i})$$$ for each $$$i$$$ using segment tree and compare the maximum with $$$m$$$. (Forgive me for terrible explanation because it is quite difficult to explain in plain English which is not my mother tongue)

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2 years ago, # |
  Vote: I like it +24 Vote: I do not like it

I was having hard time figuring out the editorial soln for Div2D/Div1B. So this is what I came up with.

First we figure out all the $$$a_i$$$ values as stated in the editorial. We will only consider all the $$$x$$$ values for which $$$a_{x} > m$$$. Call them critical points.
Take a look at the $$$i$$$-th flood. We will check it's left and right side separately. For it to fix any point $$$j$$$ to it's left (that is $$$j < x_i$$$), this must be true: $$$a_j - (p_i - |x_i - j|) \leq m$$$ $$$\Rightarrow a_j - j - m \leq p_i - x_i$$$. Deletion of $$$i$$$-th flood must satisfy this for all critical points to it's left. Here we can keep prefix maximum for $$$a_j - j - m$$$ for all critical points and compare that with the R.H.S.

We can get similar equation for the right half. It's $$$a_j + j - m \leq p_i + x_i$$$. This can be checked by keeping suffix maximums for all critical points.

Overall the solution will be $$$\mathcal{O}(n\log{}n)$$$. One note for implementation, it's easier to check the two equations for all relevant points (even ones with $$$a_x <= m$$$) instead of just the critical ones. For points with $$$a_x <= m$$$ define their L.H.S of the two above equations to be $$$-\infty$$$ and it will be fine. Here is my submission.

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2 years ago, # |
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In div1C, can anyone explain how the transitions are taking place? What is m and how does it decide tmpmask1 and tmpmask2?

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    2 years ago, # ^ |
      Vote: I like it +20 Vote: I do not like it

    $$$m$$$ is the $$$i$$$-th bit of $$$a,b,c$$$. For example, when $$$m=(101)_2$$$ then $$$a_i=1,b_i=0,c_i=1$$$.

    First, you should check if $$$m$$$ is valid for the current state. If $$$cnt$$$ will let $$$a,b,c$$$ exceed the upper bound you should "continue".

    Then $$$tmpmask2$$$ is whether it will reach the upper bound ($$$0$$$ is yes while $$$1$$$ is no). $$$tmpmask1$$$ is whether it satisfies the conditions we mention in the editorial ($$$1$$$ is yes while $$$0$$$ is no).

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2 years ago, # |
  Vote: I like it +26 Vote: I do not like it

The editorial of 1E says:

If we consider the initial bipartite graph, it's easy to see that we only need to check whether $$$(1,1)$$$ is in all maximum matchings of the initial graph, becuase the maximum matching remains unchanged in the other $$$999$$$ parts.

(And it mistakes "because" for "becuase".)

I think it's not obvious (probably because I'm too foolish), so I try to prove it. We need to prove

  • If we want to get a maximum matching of the new graph, we can just copy a maximum matching of the initial graph $$$1000$$$ times.
  • If $$$(1,1)$$$ belongs to all the maximum matchings of the initial graph, and we want to get a maximum matching of the new graph with a copy of $$$(1,1)$$$ removed, we can just remove a copy of $$$(1,1)$$$ from the maximum matching of the new graph we get above.

The main idea of my proof is to prove that there exists no augmenting path in the two matchings we get above. To prove this, we assume that there exist an augmenting path, and we try to make it contains no more than one copy of any vertices. Then we show that this path is an augmenting path of the maximum matching of the initial graph or the initial graph without $$$(1,1)$$$.

Sorry for not writing the whole proof down here, but my poor English doesn't allow me to do that.

However, since the editorial said "it's easy to see", I believe there is a much cleverer and shorter proof. But I'm not able to come up with it. If anyone knows the proof, please teach me. Thanks.

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2 years ago, # |
  Vote: I like it +23 Vote: I do not like it

And the editorial says

For the maximum independent set without the initial cell, you just need to remove it when it is in $$$I$$$.

Note that if $$$da_{va}=da_{va-1}$$$ or $$$db_{vb}=db_{vb-1}$$$ ($$$(va,vb)$$$ is the starting cell), this strategy may not give the correct result for a certain $$$(i,j)$$$. I don't know whether it can always give the correct maximum independent set.

I'm too lazy to write the proof for why it's correct if $$$(va,vb)$$$ doesn't meet this condition, but it's not hard to prove.

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    2 years ago, # ^ |
      Vote: I like it +8 Vote: I do not like it

    Actually, I think you can get WA depending on how you implement it.

    Let $$$s(i,j), i \in [0,n], j \in [0,m],$$$ denote the size of the independent set if we pick the first $$$i$$$ rows and $$$j$$$ first columns white. Let $$$db[j]$$$ be the number of white cells in each column. Note that $$$db[j]$$$ decreases with $$$j$$$. Then, (when the initial position is considered black)

    $$$\delta(i,j) = s(i,j+1) - s(i,j) = \min(db[j], i) + \max(db[j], i) - n = db[j] + i - n.$$$

    That means that it is useful to increase $$$j$$$ if $$$\delta(i, j) > 0 \iff db[j] > n - i$$$. On the other hand, if $$$db[j] = n-i$$$, then we get the same size of independent set whether or not we increase $$$j$$$. Thus, you could implement it such that you increment $$$j$$$ whenever $$$db[j] >= n-i$$$ or whenever $$$db[j] > n-i$$$ and both would be correct when the initial position is considered black. However, when the initial position is not considered black, for some $$$j$$$ the condition becomes $$$db[j] > \text{or} >= n - i - 1$$$. Let us see that if you pick the strict option, $$$db[j] > n - i$$$, your code gives WA.

    Suppose $$$n = 4, m = 2$$$ and that the matrix is as follows, where X marks the initial position.

    WW
WW
BX
BB

    Then, $$$db[] = {2,2}$$$. For $$$i = 0$$$, $$$db[0] = 2$$$ is smaller than $$$n - 0 = 4$$$, so the optimal $$$j$$$ is $$$0$$$ and $$$s(0,0) = 3$$$ because the starting point is not considered black. The same happens when $$$i = 1$$$. When $$$i = 2$$$, though, $$$n - i = 2$$$ and $$$db[0] = 2$$$. Hence, if we increase $$$j$$$ we still have the same maximum independent set size. But, once we do so, the condition becomes $$$db[j] >= n - i -1$$$, meaning that it is strictly better to increase $$$j$$$. The optimal $$$j$$$ for $$$i = 2$$$ is $$$j = 2$$$. This corresponds to choosing all white cells as independent set, giving answer $$$4$$$. They key fact is that if we don't increase $$$j$$$ when the answer would not improve, we don't get to this position.

    It is true that in this example the same position would be obtained when $$$i = 3$$$, but I think that we can craft examples where this does not happen.

    To remember this easily, the mental heuristic would be to prefer maximum independent sets with more white vertices, since afterwards there is a black vertex that we are going to remove if it is in our independent set.

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2 years ago, # |
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Can anyone explain me Question B — Rain. I'm unable to understand the editorial.

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    2 years ago, # ^ |
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    Which part do you need to explain? Difference and prefix sum? Or the calculation of region?

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      2 years ago, # ^ |
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      Please explain calculation of region part?

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        2 years ago, # ^ |
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        First, the maximum of $$$a$$$ can always be achieved at the center of a certain rain. So if for the center position of any rain $$$j$$$, $$$a_j \leq m$$$ then we know it is valid.

        If $$$a_j > m$$$ then if erasing rain $$$i$$$ can make position $$$j$$$ valid, $$$p_i−|x_i−j| \geq a_j−m$$$ must hold. Let's see what such a region is like.

        $$$p_i−|x_i−j| \geq a_j−m \iff |x_i-j| \leq p_i+m-a_j$$$ where $$$m-a_j$$$ is known constant for a certain $$$j$$$.

        So point $$$(x_i,p_i)$$$ should be in such a region:

        $$$-p_i-C_1 \leq x_i-C_2 \leq p_i + C_1$$$

        Which is the translation of region $$$-p_i \leq x_i \leq p_i$$$, like:

        ..*********...
...*******....
....*****.....
.....***......
......*.......

        Multiple such regions' intersection still holds such structure, and it can be easily calculated. I think UUUnmei's picture shows it clearly.

        You can refer to my getIntersection function which uses a way of rotation (rotate 45° clockwise and take the maximum of $$$x,y$$$ then rotate them back) to calculate it.

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          2 years ago, # ^ |
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          So much thanks for this wonderful explanation.

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2 years ago, # |
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What topic should know first to understand the solving approach/ solution of problem D:Rain?

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    2 years ago, # ^ |
      Vote: I like it +8 Vote: I do not like it

    I think you need to understand difference and prefix sum. You can refer to this link.

    In this problem, you need a "second difference" and calculate it after discretization.

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2 years ago, # |
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Well, so finally Rhodoks' contribution turns positive again. That is a fair result at last anyway. Congratulations again, and also %%% CCOrz.

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2 years ago, # |
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Alternate solution to problem 1710B Rain which does not requires you to find intersection: The idea is similar to this problem. First calculate value of a at all points then calculate the equivalent points. Then take their range. For any valid point it must cover the entire range.

Solve function

AC Submission

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22 months ago, # |
← Rev. 2   Vote: I like it 0 Vote: I do not like it

In 1710C if we use a "magic" array t={0,1,2,4,4,2,1,0} (each element means that if the i-th bit of a,b,c is m, then t[m] is the "condition mask" of the condition m will satisfy), we could make it's code simpler than the official solution.

My submission:189673909

my code
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