More Applications of Segment Tree

Walking on a Segment Tree

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

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

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

Solution - Hotel Queries

Problems

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.

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.

1using T = AR<mi,2>;
2T comb(const T& a, const T& b) { return {a[0]*b[0],a[1]*b[0]+b[1]}; }
3
4template<class T> struct BIT {
5    const T ID = {1,0};
6    int SZ = 1; V<T> x, bit[2];
7    void init(int N) {
8        while (SZ <= N) SZ *= 2;
9        x = V<T>(SZ+1,ID); F0R(i,2) bit[i] = x;
10        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.

Problems