Concepts13
Segment Tree with Range Affine Transformation
A segment tree with lazy propagation can support range updates of the form x → a·x + b (affine transformations) and range-sum queries in O(log n) per operation.
Top Tree
Top trees are dynamic tree data structures that represent a forest as a hierarchy of clusters, allowing O(log n) amortized time for link, cut, path queries/updates, and many subtree queries.
Link-Cut Tree
A Link-Cut Tree (LCT) maintains a dynamic forest and supports link, cut, and path queries in O(log n) amortized time.
HLD - Path Queries and Updates
Heavy-Light Decomposition (HLD) breaks a tree into a small number of vertical chains so any path (u,v) becomes O(log n) contiguous segments in an array.
Implicit Treap
An implicit treap is a randomized balanced binary tree that treats array positions as keys without storing them explicitly.
Treap
A treap is a binary search tree on keys combined with a heap on random priorities, which keeps the tree balanced in expectation.
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.
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.