The Li-Chao Tree: Algorithm Specification and Analysis
This paper provides a formal specification and analysis of a data structure that is already widely adopted in competitive programming, addressing a gap in peer-reviewed literature for this community.
This paper formally specifies the Li-Chao tree, an efficient data structure for dynamic lower envelope maintenance widely used in competitive programming. It provides complete algorithmic specifications, correctness proofs, theoretical complexity analysis, and empirical performance characterization.
The Li-Chao tree (LICT) was first introduced in lecture as an efficient data structure for dynamic lower envelope maintenance. In the years since, it has achieved widespread adoption within the competitive programming community, yet no formal specification has appeared in the peer-reviewed literature. This paper provides the definitive formalization of the Li-Chao tree, serving as both the official specification and an expansion of the original lecture. We present complete algorithmic specifications, establish formal correctness proofs, analyze theoretical complexity, and provide empirical performance characterization. The LICT offers distinct advantages in implementation simplicity, numerical stability, and extensibility to advanced variants such as persistence and line segments.