Matrix-Ball Construction of affine Robinson-Schensted correspondence

In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between extended affine symmetric group and triples $(P, Q, \rho)$ where $P$ and $Q$ are tabloids and $\rho$ is a dominant weight. The weights $\rho$ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.