usaco-guide/content/6_Plat/2_Plat_1DRQ.md

id	title	author	prerequisites
1DRQ	1D Range Queries	Benjamin Qi	Gold - Binary Indexed Trees

General range queries for associative operations, including segment tree.

Static Range Queries

Given a static array A[1],A[2],\ldots,A[N], you want to answer queries in the form A[l]\ominus A[l+1]\ominus \cdots \ominus A[r] in O(1) time each with O(N\log N) time preprocessing, where \ominus denotes any associative operation. In the case when \ominus denotes min, preprocessing can be done in O(1) time.

Tutorial

• Range Minimum Query
  • Use for O(1) LCA
  • CPH 9.1
  • cp-algorithms RMQ

Divide & Conquer

Divide & conquer can refer to many different techniques. In this case, we use it to answer Q queries offline in O((N+Q)\log N) time.

Suppose that all queries satisfiy L\le l\le r\le R (initially, L=1 and R=N). Letting M=\left\lfloor \frac{L+R}{2}\right\rfloor, we can compute

lef[l]=A[l]\ominus A[l+1]\ominus \cdots \ominus A[M]

for all L\le l\le M and

rig[r]=A[M+1]\ominus A[M+2] \ominus \cdots\ominus A[r]

for each M< r\le R. Then the answer for all queries satisfying l\le M<r is simply lef[l]\ominus rig[r] due to the associativity condition. After that, we recurse on all query intervals completely contained within [L,M] and [M+1,R] independently.

Actually, this can be adjusted to answer queries online in O(1) time each. See my implementation here.

See my implementation here.

Problems

• DMOJ Continued Fractions
• USACO Plat - NonDec

Segment Tree

Author: Benjamin Qi

This data structure allows you to do point update and range query in O(\log N) time each for any associative operation. In particular, note that lazy updates allow you to range updates as well.

Tutorials

• CPH 9.3, 28.1 (Segment Trees Revisited)
• Codeforces Tutorial
• CSAcademy Tutorial
• cp-algorithms
• Slides from CPC.3

Problems

• Normal SegTree
  • USACO Old Gold Seating (check ...)
  • USACO Old Gold Optimal Milking (check ...)
  • USACO Old Gold Marathon (check ...)
  • USACO Plat Balancing (check ...)
  • USACO Plat Nocross
  • USACO Gold Springboards
    • can use segment tree with min query in place of the map mentioned in analysis
  • POI Cards
  • CSES Area of Rectangles
    • use segment tree that keeps track of minimum and # of minimums
• Lazy Updates
  • USACO Old Gold The Lazy Cow (check ...)
  • USACO Plat Counting Haybales

BIT Revisited

Binary Indexed Trees can support range increments in addition to range sum queries.

• GFG Range Update Point Query
• GFG Range Update Range Query
• My Implementation

Example problem:

• DMOJ Range Update Range Query