rhymehatch's blog

Problem C — Codeforces Round #645 (Div. 2)
By rhymehatch, history, 23 months ago, In English

https://codeforces.com/problemset/problem/1358/C

I cannot think simply enough.

First I amused myself with the number of possible paths:

$$$\frac{(x_2-x_1+y_2-y_1)!}{(x_2-x_1)!(y_2-y_1)!}$$$

I realized it is useless due to the inputs going to $$$10^9$$$.

Then I focused on figuring out how to calculate value for given coordinate $$$(i,j)$$$.

It took me a while to figure that out:

$$$ \bigg(\sum\limits_{k=1}^{i+j-1} k \bigg) - j + 1 = \frac{(i+j-1)(i+j)}{2}-j+1 $$$

Then I did not know what to do, so I started calculating all of the possible sums. Then I realized that there is a minimal sum and a maximal sum. Minimal is obtained by going through row $$$x_1$$$ for all $$$y_1$$$ to $$$y_2$$$. Then down the column $$$y_2$$$ for all $$$x_1+1$$$ to $$$x_2$$$. Maximal by first going down by column $$$y_1$$$ and then going right through row $$$x_2$$$.

Given the huge number of possible paths ($$$\approx {10}^{1000000000}$$$) I assumed that the difference between maximal and minimal sum will contain all numbers. So it is max-min+1 .

Then I just quit trying to work out the formula with a pencil and wrote the whole sum calculation in WolframAlpha:

Simplify[
  Sum[Sum[i,{i, 1, x2+j-1}] -j +1,  {j,y1,y2}] 
+ Sum[Sum[i,{i, 1, j+y1-1}] -y1+1, {j,x1,x2-1}]
- Sum[Sum[i,{i, 1, x1+j-1}] -j +1,  {j,y1,y2}] 
- Sum[Sum[i,{i, 1, j+y2-1}] -y2+1, {j,x1+1,x2}]
]

The output was $$$(y_1-y_2)(x_1-x_2)$$$. I added 1 and tada.

Insane waste of time. I still have no idea how to think simply enough to not go through all of these steps.

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Comments Comments (3)
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23 months ago, # |
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just consider the minimum sum and the maximum sum of the paths. It is easy to see that there is a path of the sum of any number between those two.

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

    Yes, thanks, I realized that, but I do not know how to avoid the symbolic math stuff.

    How can I realize $$$(y_1-y_2)(x_1-x_2)+1$$$ without explicitly formulating expressions for maximal and minimal sum?

    The thinking steps that lead me to solution are too complicated. It took me an hour to solve something that has a 1 line solution and that does not even care about the formula of a particular cell or sums of sums.

    I am sure people who solved this at the contest in very short time think very simply and have not seen the problem before.

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      23 months ago, # ^ |
      ← Rev. 3   Vote: I like it 0 Vote: I do not like it

      When you change the subpath of (down -> right) into (right -> down), it is decrementing the sum by 1. Then how many times do you have to make this step, in order to make the maximum sum path (down -> ... -> down -> right -> ... -> right) into the minumum sum path (right -> ... -> right -> down -> ... -> down)? They are the same questions.

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