# Sparse Segment Trees

Author: Andi Qu

Querying big ranges.

### Prerequisites

Focus Problem – try your best to solve this problem before continuing!

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.)

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Resources | ||||
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Benq |

### Pro Tip

Sparse Segment Tree is more commonly referred to as "Dynamic Segment Tree."

## 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.

C++

#include <bits/stdc++.h>#pragma GCC optimize("O3")#define FOR(i, x, y) for (int i = x; i < y; i++)#define MOD 1000000007typedef long long ll;using namespace std;struct Node {int sum, lazy, tl, tr, l, r;Node() : sum(0), lazy(0), l(-1), r(-1) {}

### Optional

It's possible to reduce the memory of a sparse segment tree to $\mathcal{O}(N)$, as described here.

### 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.

Status | Source | Problem Name | Difficulty | Tags | |
---|---|---|---|---|---|

IOI | Normal | ||||

Balkan OI | Normal |

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