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# More Applications of Segment Tree

Authors: Benjamin Qi, Andi Qu

### Prerequisites

Walking on a Segment Tree, Non-Commutative Combiner Functions

Resources
CF

both of these topics

cp-algo

Includes these two applications and more.

## Walking on a Segment Tree

Focus Problem – read through this problem before continuing!

You want to support queries of the following form on an array $a_1,\ldots,a_N$ (along with point updates).

Find the first $i$ such that $a_i\ge x$.

Of course, you can do this in $\mathcal{O}(\log^2N)$ time with a max segment tree and binary searching on the first $i$ such that $\max(a_1,\ldots,a_i)\ge x$. But try to do this in $\mathcal{O}(\log N)$ time.

Solution

### Problems

StatusSourceProblem NameDifficultyTags
Old GoldNormal
PlatNormal

## Non-Commutative Combiner Functions

Previously, we only considered commutative operations like + and max. However, segment trees allow you to answer range queries for any associative operation.

Focus Problem – read through this problem before continuing!

Focus Problem – read through this problem before continuing!

### Solution - Point Set Range Composite

The segment tree from the prerequisite module should suffice. You can also use two BITs as described here, although it's more complicated.

using T = AR<mi,2>;T comb(const T& a, const T& b) { return {a[0]*b[0],a[1]*b[0]+b[1]}; }
template<class T> struct BIT {	const T ID = {1,0};	int SZ = 1; V<T> x, bit[2];	void init(int N) {		while (SZ <= N) SZ *= 2;		x = V<T>(SZ+1,ID); F0R(i,2) bit[i] = x;		FOR(i,1,N+1) re(x[i]);

### Solution - Subarray Sum Queries

Hint: In each node of the segment tree, you'll need to store four pieces of information.

Solution

### Problems

StatusSourceProblem NameDifficultyTags
CSESEasy
Old GoldEasy
Old GoldNormal
POINormal
COCIHard
Show Tags
PlatHard
Show TagsGreedy, PURQ
Balkan OIHard

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