L-Shaped Neighborhood Structure

• L-shaped neighborhood structures are spatial patterns defined by extending northward and eastward from a reference point, yielding asymmetric local interactions.
• They underpin models in cellular automata and urban tessellation, influencing computational complexity and phase transitions through directed dependencies.
• Their applications in geometric graph theory, statistical physics, and floor-planning enable efficient local encoding and optimization of spatial and computational systems.

An L-shaped neighborhood structure is a spatial concept characterized by the selection of sites forming an "L"—typically subsets extending northward and eastward from a central reference site—central to discrete models in statistical physics, cellular automata, graph theory, urban modeling, floor-planning, and optimization. The L-shaped configuration imposes unique local interaction constraints and often results in boundary, tiling, or computational properties distinct from those engendered by more isotropic (e.g., square or Moore) neighborhoods. Below, the principal mathematical definitions, theoretical frameworks, and applied consequences across diverse disciplines are outlined.

1. Mathematical Definition and Core Properties

Formally, an L-shaped neighborhood, denoted $N(i, j)$ about lattice coordinate $(i, j)$, is defined in the CA literature by two sets $S_n$ (north offsets) and $S_e$ (east offsets): $$N(i, j) = \{ (i, j+k) : k \in S_n \} \cup \{ (i+k, j) : k \in S_e \}$$ The classical "Toom" L-shaped neighborhood employs $S_n = S_e = \{1\}$, corresponding to a minimal L comprising immediate north and east neighbors. This asymmetry breaks rotational invariance, conferring directed dependencies—a property with deep computational implications (Concha-Vega et al., 19 Sep 2025).

L-shaped neighborhoods are also employed as unit building-blocks in tiling, tessellation, and in neighborhood context models for spatial Markov processes, where they constitute a combinatorially small yet expressive subset of possible local configurations (Magalhaes et al., 29 Jan 2024). In geometric graph theory, "L-shape" refers to the geometric union of a horizontal and vertical segment at a common endpoint, which, by their intersection, encode combinatorial neighborhood relations (Felsner et al., 2014).

2. L-Shaped Neighborhoods in Cellular Automata and Complexity

L-shaped neighborhoods underpin freezing majority cellular automata (FMCA), where local update rules exhibit irreversible behaviour: states in $\{-1, +1\}$ evolve by the majority among north/east L-shaped neighbors, with $+1$ states freezing. The FMCA update rule is: $$F(x)_u = \begin{cases} +1 & \text{if } x_u = +1 \text{ or } \sum_{v \in N(u)} x_v > 0\ -1 & \text{otherwise} \end{cases}$$ For minimal L-shaped neighborhoods (single north and east), the Prediction problem—deciding a cell's state at time $t$—is in $\mathsf{NC}$, admitting polylogarithmic parallel algorithms via cycle detection in the directed dependency graph $G_x$. In contrast, when $|S_n|, |S_e| \geq 2$, the problem is $\mathsf{P}$-complete: logical circuit simulations embedded in larger neighborhoods induce inherent sequentiality, disabling efficient parallelization (Concha-Vega et al., 19 Sep 2025). The following table summarizes complexity results for FMCA:

Neighborhood size	Complexity class	Key property
$|S_n| = |S_e| = 1$ (minimal L)	$\mathsf{NC}$	Efficient parallel computability
$|S_n|, |S_e| \geq 2$	$\mathsf{P}$-complete	Inherently sequential behavior

These results underscore how increasing the "reach" of the L-shaped neighborhood enhances local computational expressiveness at the cost of parallelizability, reflecting a phase transition in algorithmic tractability.

3. L-Shaped Structures in Urban Segregation and Tessellation

In spatial segregation models, specifically extensions of the Schelling model, L-shaped neighborhood patterns emerge as energetically favorable local motifs. When agents' utility is maximized at a mixed local composition (e.g., under the triangle utility $U_T = 4 - |4 - \#(\text{like neighbors})|$), the asymptotic lattice configuration manifests as a tessellated mosaic. Here, interlocking L-shaped blocks behave as minimal "satisfaction clusters", each agent in an L-pattern typically having the optimal number of like-type neighbors (four in a Moore neighborhood) (Singh et al., 2010):

• L-shaped clusters efficiently tile the urban lattice, forming the backbone of mixed and integrated macroscopic urban configurations.
• The prevalence and regularity of L-shaped structures are highly sensitive to happiness thresholds (e.g., moving from $T=3$ to $T=4$), group ratios, and the precise form of the utility function.

In grid-like urban models with economic constraints (e.g., internal boundaries or differential housing cost fields), localized L-shaped ghettos can emerge when geometric barriers interact with agents' relocation and affordability criteria. These clusters often exhibit power-law size distributions and their shape—sometimes distinctly L-shaped—reflects the interplay of boundary constraints and economic heterogeneity (Ortega et al., 2022).

4. L-Shaped Neighborhoods in Geometric Graph Representations

L-shaped intersection representations provide a means to encode adjacency in classes of planar and related graphs. An L-shape in the plane is the union of a horizontal and a vertical segment with a shared endpoint, and four rotations are admissible. In an $\{\llcorner\}$-graph, each vertex is assigned such an L-shape, and adjacency corresponds to intersection.

• Planar $3$-trees, line graphs of planar graphs, and full subdivisions of planar graphs can all be represented as $\{\llcorner\}$-graphs (Felsner et al., 2014).
• The use of "private regions" and "bottomless rectangles" during inductive construction ensures that local neighborhood structure is captured precisely via geometric intersection, supporting efficient encoding of local adjacency.
• These representations have implications for algorithmic optimization: they can provide lower-complexity models for problems such as Maximum Clique or chromatic number, relating geometric neighborhood shapes to combinatorial complexity.

5. L-Shaped Domains in Statistical Physics and Dynamical Systems

The "L-shaped domain" paradigm appears directly in several statistical mechanics models:

• Six-Vertex Model and Arctic Curves: In studies of the six-vertex model on L-shaped domains (i.e., a square with a corner removed), the system undergoes a third-order phase transition as the size of the L-shaped cut increases and interacts with the bulk "frozen" region, as marked by the Arctic ellipse. The explicit form of the free energy per site shows that for small L-shaped cuts the bulk remains unchanged, but crossing the Arctic boundary introduces a singularity in the third derivative of the free energy (Colomo et al., 2015).
• Translation Flows and L-Shaped Billiards: For translation surfaces of genus two, particularly those arising from unfolding billiard paths on L-shaped polygons, the return time to an L-shaped neighborhood is governed by a Diophantine-type exponent, quantifying arithmetic complexity. The limsup exponent of the hitting time varies between linear and quadratic in the Diophantine exponent, dictated by the number-theoretic properties of the flow direction (Kim et al., 2017).

Thus, L-shaped neighborhood structures influence both local dependencies (phase transitions, recurrence rates) and global limit shapes in these systems.

6. L-Shaped Structures in Algorithmic Layout and Floor-Planning

L-shaped neighborhood constraints play a crucial role in VLSI floor-planning and orthogonal tree drawing:

• Floor-Planning: An L-shaped floor-plan is defined as an orthogonal layout with a single intrinsic concave (inward) corner. The existence of a non-trivial L-shaped floor-plan for a properly triangulated planar graph (PTPG) is characterized by the presence of a boundary triplet (a, b, c) with prescribed adjacency restrictions, ensuring that the L-concavity is forced by the combinatorial structure and cannot be "rectangularized" without altering connectivity. A constructive $O(n^2)$ algorithm is provided for such layouts (Raveena et al., 2022).
• Orthogonal Tree Drawing: In graph drawing, L-shaped edges (having one bend) are used to construct planar embeddings of trees on point sets. Recursive partitioning and bounding arguments yield improved upper bounds on universal point set size for various tree classes (e.g., $O(n^{1.22})$ for binary trees), but strict L-shaped constraints lead to the existence of non-embeddable trees for some $(n, P)$ pairs (Biedl et al., 2017, Mütze et al., 2018). The necessity for L-shaped neighborhoods therefore imposes nontrivial combinatorial and geometric restrictions.

7. L-Shaped Neighborhoods in Probabilistic and Markov Models

In spatial statistics, probabilistic context neighborhood (PCN) models employ fixed frame-like geometries around each site, akin to L-shaped structures in the sense that only subsets of the full local environment are considered informative. The salient features include:

• A fixed geometry (commonly an annular or ring-shaped "frame") is applied to all sites; the order (number of frames stacked) that is needed varies depending on observed values.
• The dependency structure is organized as a tree where contexts correspond to specific local configurations, which may include L-shaped sub-patterns if only partial frames are exploited (e.g., considering only north and east neighbors).
• In practical applications such as fire occurrence in spatial grids, most conditional dependencies are captured by first-order contexts, occasionally necessitating an extension to larger (potentially L-shaped) neighborhoods (Magalhaes et al., 29 Jan 2024).

The PCN model demonstrates that L-shaped or frame-restricted neighborhood structures can be statistically efficient and computationally tractable for modeling spatial dependencies with variable orders.

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Summary Table: L-Shaped Neighborhoods in Key Contexts

Field	L-Shaped Role	Technical Consequence
Cellular Automata (Concha-Vega et al., 19 Sep 2025)	Update dependency & influence	Parallelism (NC) vs. P-completeness
Urban/Tessellation Models (Singh et al., 2010, Ortega et al., 2022)	Minimal energy clusters, ghetto shape	Regular tilings, sensitive to utility/threshold choice
Geometric Graph Theory (Felsner et al., 2014)	Adjacency via intersection	Efficient encodings, links to VPG-/SEG-graph hierarchies
Statistical Physics (Colomo et al., 2015)	Boundary domain, free energy	Third-order phase transitions, Arctic ellipse
Floor-Planning & Tree Drawing (Raveena et al., 2022Biedl et al., 2017, Mütze et al., 2018)	Topological constraint, embedding	Construction algorithms, forbidden configurations
Markov/PCN Models (Magalhaes et al., 29 Jan 2024)	Fixed (frame) geometry, context tree	Parsimonious, interpretable, variable order dependence

Conclusion

L-shaped neighborhood structures constitute a mathematically and algorithmically rich class of local interaction patterns with ramifications for computational complexity, spatial patterning, geometric representation, and statistical modeling. Their (often asymmetric) shape sharply influences dynamical evolution, computational feasibility, and physical or urban spatial structure; as such, their paper illuminates the profound effects that local geometry can exert on global system behavior.