71 lines
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3.4 KiB
Text
71 lines
No EOL
3.4 KiB
Text
---
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id: 2DRQ
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title: "2D Range Queries"
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author: Benjamin Qi
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prerequisites:
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- Platinum - Range Update Range Query
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description: "Extending Range Queries to 2D (and beyond)."
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---
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See [my implementations](https://github.com/bqi343/USACO/tree/master/Implementations/content/data-structures/2D%20Range%20Queries%20(15.2)).
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## Static Array Queries
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- [Multi-Dimensional RMQ (retrograd)](https://codeforces.com/blog/entry/53810)
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- GP of Serbia 2020 B
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## 2D BIT
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### Tutorials
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- [GFG 2D BIT](https://www.geeksforgeeks.org/two-dimensional-binary-indexed-tree-or-fenwick-tree/)
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- [TopCoder BIT](https://www.topcoder.com/community/competitive-programming/tutorials/binary-indexed-trees/)
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- Suppose that you want to update $N\approx 10^5$ points with both coordinates in the range $[1,N]$.
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- The 2D BITs mentioned above use $O(N^2)$ memory, which is too much.
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- Can be reduced to $O(N\log^2N)$ memory with unordered map, but this might also be too much (and too slow).
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- If you know the points to be updated beforehand then you can reduce to $O(N\log N)$ memory (and no maps).
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- [my 1D offline BIT](https://github.com/bqi343/USACO/blob/master/Implementations/content/data-structures/1D%20Range%20Queries%20(9.2)/BIToff.h)
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- [my 2D offline BIT](https://github.com/bqi343/USACO/blob/master/Implementations/content/data-structures/2D%20Range%20Queries%20(15.2)/BIT2DOff%20(15.2).h)
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<optional-content title="Range Update and Range Query in Higher Dimensions">
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You can extend the 1D BIT solution for range update range query to higher dimensions as well
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- [Paper](https://arxiv.org/pdf/1311.6093.pdf)
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- USACO Camp - "Cows Play Global Thermonuclear War" (2D case)
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</optional-content>
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### Problems
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- [CSES Forest Queries II](https://cses.fi/problemset/task/1739)
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- [thecodingwizard's implementation](https://github.com/thecodingwizard/competitive-programming/blob/master/cses/Forest%20Queries%20II.cpp)
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- [DMOJ Soriya's Programming Project](https://dmoj.ca/problem/dmopc19c7p5)
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- intended complexity is $O(N\log^2 N)$
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- compressed 2D BIT
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- rather difficult to pass within time and memory limits
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- Make sure to use `\n` instead of `endl`!
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- [thecodingwizard's implementation with Benq's 2d offline bit](https://github.com/thecodingwizard/competitive-programming/blob/master/DMOJ/Soriyas%20Programming%20Project.cpp)
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- or do divide & conquer with a 1D BIT
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- same as first problem [here](https://robert1003.github.io/2020/01/31/cdq-divide-and-conquer.html)
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- [thecodingwizard's (messy) implementation](https://github.com/thecodingwizard/competitive-programming/blob/master/DMOJ/Soriya%20Programming%20Project%201d%20BIT%20cdq%20dnc.cpp) based off of article above
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- [IOI 2007 Pairs](https://wcipeg.com/problem/ioi0722)
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- [DMOJ Crowded Cities](https://dmoj.ca/problem/bfs17p6)
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- [USACO Plat Friendcross](http://www.usaco.org/index.php?page=viewproblem2&cpid=722)
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- [USACO Plat Mowing](http://www.usaco.org/index.php?page=viewproblem2&cpid=601)
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## 2D Segment Tree
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Note: no lazy propagation in 2D.
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### Short Description
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- CSES 28.2 (Sparse Segment Tree), 28.4
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- Segment Tree (or BIT) nested inside segment tree
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- use 2D offline BIT instead whenever possible (faster, lower memory)
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### Problems
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- [POI Tetris 3D](https://szkopul.edu.pl/problemset/problem/OQjANSOOD_-c38gh8p6g3Gxp/site/?key=statement)
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- [IOI 2013 Game](http://wcipeg.com/problem/ioi1323)
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- good implementation? |