Exterior Site Perimeter

• Exterior site perimeters are mathematically defined boundaries that encircle designated areas, fundamental in combinatorics, computational geometry, and security applications.
• Algorithmic frameworks, including Markov Decision Processes and Q-learning, efficiently identify and optimize perimeters for real-time monitoring and control.
• Strategic resource allocation leverages sensor placement, patrolling algorithms, and game-theoretic models to secure perimeters against adversarial incursions.

An exterior site perimeter is a mathematical and operational construct describing the boundary—typically as a closed curve or set of points—that encircles a designated area, physical site, or cluster. This concept is central in combinatorics (convex polyominoes, percolation clusters), computational geometry (polygonal domains), security and monitoring (sensor/radar placement, resource allocation, patrol/pursuit strategies), and sequential decision processes (boundary optimization in surveillance or traffic control). Theoretical treatments address not only how to represent or enumerate exterior perimeters but also how to discover, partition, defend, or optimally monitor them using algorithmic, probabilistic, or game-theoretic models.

1. Combinatorial and Probabilistic Formulations

The exterior site perimeter is rigorously defined within various discrete models:

• Convex Polyominoes: The outer-site perimeter $o(v)$ of a convex polyomino $v \subset \mathbb{Z}^2$ is the total number of outer-site cells—those outside $v$ but adjacent (via an edge) to the boundary of $v$. This statistic governs the boundary complexity of $v$ and drives exact and asymptotic enumeration via generating functions. The outer-site perimeter grows linearly with the (half-)perimeter, with expected value $E_n \sim \frac{25}{16}n$ among all convex polyominoes with half-perimeter $n$, and its conditional expectation relates linearly to the usual perimeter, $E[\text{perimeter}\,|\,o(v)=n] \sim 5^{1/4}n$ (Mansour et al., 2020).
• Random Planar Maps/Percolation: In critical site percolation on the Uniform Infinite Planar Triangulation (UIPT), the exterior site perimeter $|\partial \mathfrak{C}|$ is the length (number of vertices) of the cycle in $\mathfrak{C}$ separating the root cluster from the infinite exterior. The distribution follows $$\mathbb{P}\{|\partial \mathfrak{C}|=n\} \sim c\,n^{-4/3}$$ at criticality, a nontrivial exponent indicative of large cluster boundaries in two-dimensional phases (Ménard, 2022).

2. Algorithmic Perimeter Identification and Optimization

Exterior site perimeter identification can be cast as a sequential decision process over a graph or spatial grid underlying the site:

• Sequential Decision-Making Frameworks: The perimeter identification problem is formulated as a Markov Decision Process (MDP) $(S, A, T, R, H)$, where the agent iteratively adds/removes candidate points to improve the perimeter's fit (e.g., as a convex hull) to high-value site features (congestion, thermal zones). The optimal policy $\pi^*$ maximizes cumulative reward over a finite horizon, typically using Q-learning or policy-gradient methods for optimization. The state is the set $s$ of selected vertices, the action is add/remove, and the reward $R(s, s')$ encodes the marginal utility of perimeter modification—including penalties for covering uninformative or low-priority regions. Public data sources (Google Maps congestion, satellite NDVI, OpenStreetMap street networks) provide input for weight functions and candidate sets. Empirically, Q-learning achieves rapid convergence ($\sim 300$–$500$ episodes) to near-optimal perimeters under trade-off regimes determined by penalty $\lambda$ (Taitler, 2024).
• Practical Adaptations: For irregular non-convex perimeters, iterative partitioning and graph-based projection align the computed convex hulls with true site boundaries. Resolution of features ($w_p$), density of candidates, and real-time update requirements influence method selection.

3. Perimeter Guarding and Resource Allocation

Securing or monitoring an exterior perimeter often involves covering a closed curve $\Gamma$ with $k$ mobile or static sensors:

• Optimal Guarding with Range Sensors: Given $\Gamma \subset \mathbb{R}^2$, position $k$ sensors $c_1,\dots,c_k$ (with adjustable radii $r_i$) to cover $\Gamma$ such that $\max_i r_i$ is minimized. The problem is NP-hard to approximate better than factor $\alpha \sim 1.152$ even when each sensor covers at most two boundary arcs. If each sensor may cover only a single continuous arc, a fully polynomial time approximation scheme (FPTAS) is available based on discretization and sliding-window tiling. For the general case, a $(2+\epsilon)$-approximation via $k$-center clustering or ILP formulations can be used, with trade-offs in runtime and optimality (Feng et al., 2020).

Approach	Optimality Guarantee	Key Idea / Complexity
FPTAS (1 arc/sensor)	$1+\epsilon$ (nearly optimal)	Discretize, sliding window tiling, $O(N^2 \log N)$
General Approx	$(2+\epsilon)$	$k$-center farthest-first, $O(Nk), O(N^2 \log N)$
ILP (small N)	Exact (up to discretization $\epsilon$)	Binary search on $r$, constraints for coverage

Sensor deployment must further account for line-of-sight constraints and terrain adaptation.

• Patrolling Strategies: Optimal stochastic scheduling ensures (worst-case) detection of intrusions, independent of whether the patrollers are undercover or uniformed. The schedule is randomized to minimize detection variance and negate timing advantage for learning attackers. Detection probability is determined analytically as $$V(\lambda,T,p) = 1 - \bigl(r(1-p)^{m+1} + (1-r)(1-p)^{m}\bigr)$$, with $m, r$ determined by the expected number of patrol passes over the intrusion's duration (Lin, 2019).

4. Defense Games and Competitive Strategies

Exterior perimeter defense is often modeled as a pursuit-evasion or resource allocation game, with defenders and intruders following optimal strategies:

• Turret Defense with Service Time: Protecting a perimeter (e.g., circle or arc) with a limited-range turret against $N$ intruders gives rise to a rich scheduling problem. Each intruder defines a time-window for intercept, and the turret's defense reduces to a Travelling Repairperson Problem with Time Windows (TRP–TW), typically NP-hard. Online algorithms (e.g., Sweeping Turret, DPaC) have provable competitiveness bounds ($1$-competitive or $2$-competitive within parametric regimes), depending on turret angular speed, range, and service time $\Delta$ relative to intruder velocity and sector aperture (Bajaj et al., 2023).
• Aerial Perimeter Defense in Hemispherical Geometry: Differential games on curved perimeters (e.g., hemisphere base) with mobile defenders (UAVs) and intruders give rise to explicit feedback strategies—min/max time to the breach point—based on Hamilton–Jacobi–Isaacs theory. Implementation on real quadrotor platforms must account for actuation, curvature, and computation latency, and distributed assignments for scalable multi-defender systems (Lee et al., 2021).

5. Layered and Adaptive Perimeter Security

Robust exterior site defense often employs layered configurations and dynamic allocation:

• Layered Resource Allocation: The site is modeled as a layered network, with outer (exterior) and inner sensors, each assigned fractions of total detection budgets $Y$ and $X$, respectively. Detection rates $D_j(y_j)$ are piecewise-linear, and flows $F_j$ between layers are explicitly modeled. The objective—a mixed sum/product reflecting multi-stage detection—renders the optimization non-convex, solved efficiently via dynamic programming with provable discrete convergence. In adversarial settings, the max–min extension distributes resources to guarantee detection along every possible adversary-chosen path, eliminating soft spots in coverage (Asamov et al., 2022).

Defense Model	Optimized Quantity	Mathematical Solution
Base (sum-objective)	Expected total detection	Multi-layer DP, Bellman rec.
Adaptive adversary	Minimal detection along any path	Max–min DP with min in merges

Layered strategies adapt well to changes in perimeters, varying flows, and partial knowledge of adversary behavior.

6. Generating Function and Asymptotic Analysis

Exact and asymptotic enumeration of exterior site perimeters is achieved via combinatorial generating functions:

• Polyominoes: The number $a_n$ of convex polyominoes with outer-site perimeter $n$ satisfies $$a_n \sim \frac{3(\sqrt5-1)}{20 \sqrt{\pi n} 5^{1/4}} \bigl( \tfrac{3+\sqrt5}{2} \bigr)^n$$ as $n \to \infty$. The full generating function $G(t) = \sum_{n \ge 4} a_n t^n$ has explicit algebraic form; similarly, bivariate GFs control further statistics (Mansour et al., 2020).
• Percolation Clusters: In the UIPT, the perimeter generating function and its singular expansion are linked to “gasket decomposition” and analytic combinatorics; the perimeter law's coefficient sequence has tail decay $n^{-4/3}$ at criticality (Ménard, 2022).

These analytic approaches provide foundational statistics for benchmarking and validation of empirical detection, reconstruction, and security protocols on exterior site perimeters.

7. Operational and Modeling Considerations

Robust exterior site perimeter analysis requires alignment between mathematical models and real-world constraints:

• Feature map resolution ($w_p$ granularity) and candidate point density are critical for capturing geometric and functional details in identification tasks.
• Sequential decision frameworks accommodate real-time updates and data-driven prioritization, enabling adaptation to changing site topologies or operational objectives.
• Defensive resourcing, whether in continuous spatial domains or discrete checkpoint networks, leverages formal optimization and learning guarantees to balance operational cost, detection probability, and adversarial adaptivity.

The unifying theme is the integration of combinatorial, algorithmic, stochastic, and strategic tools for the precise analysis and optimization of exterior site perimeters across domains of monitoring, security, and spatial modeling.