The Li-Chao Tree: Algorithm Specification and Analysis

TL;DR

This paper formally defines and analyzes the Li-Chao tree data structure, providing a comprehensive specification, correctness proofs, complexity analysis, and empirical performance insights, enhancing its understanding and application.

Contribution

It offers the first formal specification and analysis of the Li-Chao tree, expanding its theoretical foundation and practical usability.

Findings

Complete algorithmic specification provided

Formal correctness proofs established

Empirical performance characterization included

Abstract

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.