Sparse Segment Trees

In problems where the query range is at most something like $10^6$, a normal segment tree suffices. However, as soon as we move to bigger ranges ($10^{12}$ in some cases), a normal segment tree results in MLE.

Luckily, we can still use a segment tree to solve these types of problems.

Resources

The main idea is that we don't have to store all the nodes at all times - we create nodes only when needed. In a normal segment tree, an update only affects $\mathcal{O}(\log N)$ nodes, so in a sparse segment tree, we only store $\mathcal{O}(Q \log N)$ nodes!

We can implement this efficiently using pointers to a node's children - just like a trie! (Then again, a segment tree is basically a fancier binary trie.)

Solution

We need to support two operations on a range:

• Count the number of red-ripe trees in a range (range sum)
• Set all trees in a range to red-ripe (range paint)

We can use a segment tree with lazy propagation to solve this, but the query range is up to $10^9$, so we have to use a sparse segment tree.

Luckily, lazy propagation still works here.

Copy
#include <bits/stdc++.h>
#pragma GCC optimize("O3")
#define FOR(i, x, y) for (int i = x; i < y; i++)
#define MOD 1000000007
typedef long long ll;
using namespace std;

struct Node {
	int sum, lazy, tl, tr, l, r;
	Node() : sum(0), lazy(0), l(-1), r(-1) {}

Problems

Some of these problems are solvable with a normal segment tree if you use coordinate compression. Sparse segment trees are rarely ever required to solve a particular problem, but they can make things a lot more convenient.