usaco-guide/Gold_1DRQ.md

Gold - 1DRQ

Author: Benjamin Qi

Binary Indexed Tree

Assumes that you are familiar with prefix sum queries (CPH 9.1).

Introduction

Given an array, the task is to update the element at a single position in addition to querying the sum of a prefix.

Sample Problems:

• CSES Range Sum Queries II
• CSES Range XOR Queries
• SPOJ Inversion Counting

The easiest way to do all of these tasks is with a Binary Indexed Tree (or Fenwick Tree).

Tutorials:

• CPH (very good)
  • Section 9.2
• CSAcademy BIT (also very good)
• cp-algorithms Fenwick Tree
  • extends to range update range query, although this is not necessary for gold
• Topcoder BIT

My implementation can be found here, and can compute range sum queries for any number of dimensions.

Indexed Set

In the special case where all elements of the array are either zero or one (which is the case for several gold problems), users of C++ will find indexed set useful. Using this, we can solve "Inversion Counting" in just a few lines (with template).

#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
using namespace __gnu_pbds;
template <class T> using Tree = tree<T, null_type, less<T>, 
	rb_tree_tag, tree_order_statistics_node_update>; 
#define ook order_of_key
#define fbo find_by_order

int main() {
	setIO();
	int T; re(T);
	F0R(i,T) {
		int n; re(n);
		Tree<int> T; ll numInv = 0;
		F0R(j,n) { // T.ook(x+1) gives number of previous elements < (x+1)
			int x; re(x); numInv += j-T.ook(x+1); // so this gives # previous elements > x
			T.insert(x);
		}
		ps(numInv);
	}
}

Practice Problems

• USACO Gold
  • The first three problems are just small variations on inversion counting.
  • Haircut
  • Balanced Photo
  • Circle Cross
  • Sleepy Cow Sort
  • Out of Sorts (harder?)
• Other Problems:
  • Sorting Steps
    • I think this was a silver problem??
  • Cards

Segment Tree + Related (update later?)

Segment trees allow you to perform any associative operation over ranges, not just summation or XOR. Although not required for any USACO gold problems, it can still be helpful.

• Range Minimum Query??
  • Tutorial
    • Wikipedia
• Segment Tree
  • Tutorial
    • CPC.3
    • CSAcademy Tutorial
    • Codeforces Tutorial
  • USACO Springboards
    • can use segment tree with min query