Landau's Theorem Revisited Again

TL;DR

This paper presents a new proof of Landau's theorem on tournament score sequences, introduces an efficient $O(n^2)$ construction algorithm, and compares it with existing methods.

Contribution

It offers a novel proof of Landau's conditions and develops an efficient algorithm for constructing tournaments with specified score sequences.

Findings

Provides an $O(n^2)$ algorithm for tournament construction

Introduces a new proof of Landau's sufficiency conditions

Compares different algorithms for sequence ordering

Abstract

We give a new proof of the sufficiency of Landau's conditions for a non-decreasing sequence of integers to be the score sequence of a tournament. The proof involves jumping down a total order on sequences satisfying Landau's conditions and provides a $O(n^2)$ algorithm that can be used to construct a tournament whose score sequence is any in the total order. We also compare this algorithm with to other algorithms that jump along this total order, one jumping down and one jumping up.