arXiv subjects

Léo Robert

Publications and source records attributed to Léo Robert.

3 recordsLinked to original sources

arXiv2026

Lozenge Tiling by Computing Distances

The Calisson puzzle is a tiling puzzle in which one must tile a triangular grid inside a hexagon with lozenges, under the constraint that certain prescribed edges remain tile boundaries and that adjacent lozenges along these edges have different orientations. We present the first polynomial-time algorithm for this problem, with cubic running time. This algorithm, called the advancing surface algorithm, can be executed in a simple and intuitive way, even by hand with a pencil and an eraser. Its apparent simplicity conceals a deeper algorithmic reinterpretation of the classical ideas of John Conway and William Thurston, revisited here from a theoretical computer science perspective. We introduce a graph-theoretic overlay based on directed cuts and systems of difference constraints that complements Thurston's theory of lozenge tilings and makes its algorithmic structure explicit. In Thurston's approach, lozenge tilings are lifted to monotone stepped surfaces in the three-dimensional cubic lattice and projected back to the plane using height functions, reducing tilability to the computation of heights. We show that selecting a monotone surface corresponds to selecting a directed cut in a periodic directed graph, while height functions arise as solutions of a system of difference constraints. In this formulation, a region is tilable if and only if the associated weighted directed graph contains no cycle of strictly negative weight. This additional graph layer shows that the Bellman-Ford algorithm suffices to decide feasibility and compute solutions. In particular, our framework allows one to decide whether the infinite triangular grid can be tiled while respecting a finite set of prescribed local constraints, even in the absence of boundary conditions.

cs.CG↗

arXiv2023

The Calissons Puzzle

In 2022, Olivier Longuet, a French mathematics teacher, created a game called the \textit{calissons puzzle}. Given a triangular grid in a hexagon and some given edges of the grid, the problem is to find a calisson tiling such that no input edge is overlapped and calissons adjacent to an input edge have different orientations. We extend the puzzle to regions $R$ that are not necessarily hexagonal. The first interesting property of this puzzle is that, unlike the usual calisson or domino problems, it is solved neither by a maximal matching algorithm, nor by Thurston's algorithm. This raises the question of its complexity. We prove that if the region $R$ is finite and simply connected, then the puzzle can be solved by an algorithm that we call the \textit{advancing surface algorithm} and whose complexity is $O(|\partial R|^3)$ where $\partial R|$ is the size of the boundary of the region $R$. In the case where the region is the entire infinite triangular grid, we prove that the existence of a solution can be solved with an algorithm of complexity $O(|X|^3)$ where $X$ is the set of input edges. To prove these theorems, we revisit William Thurston's results on the calisson tilability of a region $R$. The solutions involve equivalence between calisson tilings, stepped surfaces and certain DAG cuts that avoid passing through a set of edges that we call \textit{unbreakable}. It allows us to generalize Thurston's theorem characterizing tilable regions by rewriting it in terms of descending paths or absorbing cycles. Thurston's algorithm appears as a distance calculation algorithm following Dijkstra's paradigm. The introduction of a set $X$ of interior edges introduces negative weights that force a Bellman-Ford strategy to be preferred. These results extend Thurston's legacy by using computer science structures and algorithms.

cs.CG↗

arXiv2021

Optimal Threshold Padlock Systems

In 1968, Liu described the problem of securing documents in a shared secret project. In an example, at least six out of eleven participating scientists need to be present to open the lock securing the secret documents. Shamir proposed a mathematical solution to this physical problem in 1979, by designing an efficient $k$-out-of-$n$ secret sharing scheme based on Lagrange's interpolation. Liu and Shamir also claimed that the minimal solution using physical locks is clearly impractical and exponential in the number of participants. In this paper we relax some implicit assumptions in their claim and propose an optimal physical solution to the problem of Liu that uses physical padlocks, but the number of padlocks is not greater than the number of participants. Then, we show that no device can do better for $k$-out-of-$n$ threshold padlock systems as soon as $k\geq{\sqrt{2n}}$, which holds true in particular for Liu's example. More generally, we derive bounds required to implement any threshold system and prove a lower bound of $\mathcal{O}{\log(n)}$ padlocks for any threshold larger than $2$. For instance we propose an optimal scheme reaching that bound for $2$-out-of-$n$ threshold systems and requiring less than $2\log_2(n)$ padlocks. We also discuss more complex access structures, a wrapping technique, and other sublinear realizations like an algorithm to generate $3$-out-of-$n$ systems with $2.5\sqrt{n}$ padlocks. Finally we give an algorithm building $k$-out-of-$n$ threshold padlock systems with only $\mathcal{O}{\log(n)^{k-1}}$ padlocks. Apart from the physical world, our results also show that it is possible to implement secret sharing over small fields.

cs.CR↗