Concepts318
Category
Treap
A treap is a binary search tree on keys combined with a heap on random priorities, which keeps the tree balanced in expectation.
Merge Sort Tree
A Merge Sort Tree is a segment tree where every node stores the sorted list of values in its segment.
Sqrt Decomposition
Sqrt decomposition partitions an array into about \(\sqrt{n}\) blocks, each of size about \(\sqrt{n}\), to speed up range queries and updates.
Iterative Segment Tree
An iterative segment tree stores all leaves in tree[n..2n-1] and internal nodes in tree[1..n-1], enabling O(\log n) point updates and range queries without recursion.
Segment Tree Beats
Segment Tree Beats is a segment tree variant that supports range chmin/chmax (clamping) together with queries like range sum, min, and max in amortized logarithmic time.
Binary Trie for XOR
A binary trie (also called a bitwise trie) stores numbers by their binary bits, branching on 0/1 at each level.
Segment Tree - Handling Multiple Lazy Operations
When a segment tree supports multiple range updates, you must define how lazy tags compose, because the order of operations matters and composition is not commutative.
Segment Tree with Lazy Propagation
A segment tree with lazy propagation supports fast range updates and range queries in O(\log n) time.
Dynamic Segment Tree
A dynamic segment tree stores values over a huge coordinate range by creating nodes only when an operation touches their interval.
Trie (Prefix Tree)
A trie (prefix tree) stores strings or bit-sequences so that common prefixes share nodes, making operations depend on the key length L rather than the set size.
Segment Tree Basics
A segment tree is a complete binary tree that stores information about array intervals to answer range queries and support point updates in O(log n).
2D Fenwick Tree
A 2D Fenwick Tree (Binary Indexed Tree) supports point updates and rectangle sum queries in O(log n × log m) time.