Notes 1: ABC 133F

Revision en12, by NeoYL, 2023-12-12 03:42:52

This is my personal note and might be some kind of user editorial/learning material for some people. This is the first episode of this "note" series. I will write notes on problems, which are completely solved by myself without looking at the editorial, that are both interesting and educational. If you want to motivate me to write a continuation (aka note 2), an upvote would be well appreciated!

Try to solve the task independently before continuing the blog.

First of all, we can observe that queries can be done offline and the only thing that matters for each query is the color.

Lets do each color one by one, we can observe that distance between two nodes in the query = original distance $-$ sum of edge weight of color $x$ + number of edges that have color $x$ in the path * new weight length.

Now let us try to find what can be done. We can see that the number of edges with color $x$ going out of subtree $u$ will $+1$ if the starting point is from subtree $u$. So we can just $+1$ to all nodes inside subtree $u$. Then we can see that number of edges from node $x$ to $y$. Similarly, we can do $+W$ for weight sum.

Distance in original graph can be found in $O(N log N)$ using binary lifting and a dfs function. Simultaneously, since we need to add +1 to each node one by one, we will need to do $O(N^2)$ to do all necessary operations.

This reduces the problem to $O(N ^ 2)$, lets optimize it!

We can try finding a better way to do the $+1$ and $+W$ operations. Well, if you have learnt dfn, you should be able to solve the problem from here. If we represent $tin_{ i }$ be the time we entered subtree $i$ and $tout_{ i }$ we left subtree $i$. Then we can use this to observe that we need to do $+val$ in a range and find a value at a point. What do we need to have if we want to do these operations fast? Yes, segment tree!

However, we cannot just declare a new segment tree each time. We need to reuse the segment tree. So, we'll have to do $-1$ and $-W$ operations to revert the segment tree, or do something like range set.

This allows an $O(N log N)$ solution

AC code here

#### History

Revisions

Rev. Lang. By When Δ Comment
en30 NeoYL 2023-12-31 12:37:32 7
en29 NeoYL 2023-12-14 19:34:13 18
en28 NeoYL 2023-12-14 19:32:15 348
en27 NeoYL 2023-12-14 19:26:45 17 Tiny change: '\n<spoiler> \ncpp' -> '\n<spoiler summary="Code"> \ncpp'
en26 NeoYL 2023-12-14 19:26:18 5298
en25 NeoYL 2023-12-14 17:59:14 5 Tiny change: 'timization">\nWe can' -> 'timization used">\nWe can'
en24 NeoYL 2023-12-14 13:29:32 12 Tiny change: ' summary="Half solution"' -> ' summary="Incomplete solution"'
en23 NeoYL 2023-12-14 08:58:57 215
en22 NeoYL 2023-12-14 08:57:58 141 Tiny change: ' summary="$O(N^2)$ s' -> ' summary=" $O(N^2)$ s'
en21 NeoYL 2023-12-13 13:27:55 15
en20 NeoYL 2023-12-12 13:33:13 4
en19 NeoYL 2023-12-12 09:13:23 2 Tiny change: 'ed to add +1 to each n' -> 'ed to add $+1$ to each n'
en18 NeoYL 2023-12-12 05:48:22 2 Tiny change: 'round 2400 ish proble' -> 'round 2400-ish proble'
en17 NeoYL 2023-12-12 04:44:37 8
en16 NeoYL 2023-12-12 03:55:24 86
en15 NeoYL 2023-12-12 03:47:05 36 Tiny change: 'n problems, which ar' -> 'n problems (normally around 2300 ish problems), which ar'
en14 NeoYL 2023-12-12 03:46:04 16
en13 NeoYL 2023-12-12 03:45:29 20 Tiny change: 'note 2), an upvote would be ' -> 'note 2), a significant upvote from you would be '
en12 NeoYL 2023-12-12 03:42:52 160
en11 NeoYL 2023-12-12 03:38:43 22
en10 NeoYL 2023-12-12 03:37:41 324
en9 NeoYL 2023-12-11 19:35:45 4 Tiny change: 'at a point $p$. What do ' -> 'at a point. What do '
en8 NeoYL 2023-12-11 19:34:44 19 Tiny change: 'e able to continue from here' -> 'e able to solve the problem from here'
en7 NeoYL 2023-12-11 19:27:44 64
en6 NeoYL 2023-12-11 19:26:56 40 Tiny change: 'e set.\n\n[AC co' -> 'e set.\n\nThis allows an $O(N log N)$ solution\n\n[AC co'
en5 NeoYL 2023-12-11 19:26:07 20 Tiny change: 'n the path.\n\nNow' -> 'n the path * new weight length.\n\nNow'
en4 NeoYL 2023-12-11 19:25:11 262
en3 NeoYL 2023-12-11 19:23:21 8421 (published)
en2 NeoYL 2023-12-11 19:07:37 7364
en1 NeoYL 2023-12-11 19:00:00 1500 Initial revision (saved to drafts)