usaco-guide/content/6_Plat/2DRQ.md

id	title	author	prerequisites
2DRQ	2D Range Queries	Benjamin Qi	Platinum - 1D Range Queries

Extending Range Queries to 2D (and beyond).

See my implementations.

Static Array Queries

• Multi-Dimensional RMQ (retrograd)
• GP of Serbia 2020 B

2D BIT

Tutorials

• GFG 2D BIT
• TopCoder BIT
• Suppose that you want to update N\approx 10^5 points with both coordinates in the range [1,N].
  • The 2D BITs mentioned above use O(N^2) memory, which is too much.
  • Can be reduced to O(N\log^2N) memory with unordered map, but this might also be too much (and too slow).
  • If you know the points to be updated beforehand then you can reduce to O(N\log N) memory (and no maps).
    • my 1D offline BIT
    • my 2D offline BIT

You can extend the 1D BIT solution for range update + range query (mentioned in 1DRQ) to higher dimensions as well (USACO Camp - "Cows Play Global Thermonuclear War").

• Paper

Problems

• CSES Forest Queries II
  • thecodingwizard's implementation
• DMOJ Soriya's Programming Project
  • intended complexity is O(N\log^2 N)
  • compressed 2D BIT
    • rather difficult to pass within time and memory limits
      • Make sure to use \n instead of endl!
    • thecodingwizard's implementation with Benq's 2d offline bit
  • or do divide & conquer with a 1D BIT
    • same as first problem here
    • thecodingwizard's (messy) implementation based off of article above
• IOI 2007 Pairs
• DMOJ Crowded Cities
• USACO Plat Friendcross
• USACO Plat Mowing

2D Segment Tree

Note: no lazy propagation in 2D.

Short Description

• CSES 28.2, 28.4
• Segment Tree (or BIT) nested inside segment tree
• use 2D offline BIT instead whenever possible (faster, lower memory)

Problems

• POI Tetris 3D
• IOI 2013 Game