Circular Partition Algorithms

• Circular Partition Algorithms are advanced methods that combine combinatorial, geometric, and discretization strategies to partition circular structures into balanced subparts.
• They employ techniques such as convex versus nonconvex partitioning, cycle decomposition, and dummy padding to achieve optimal aspect ratios and ensure group balance.
• Applications include polygon decomposition, CNC toolpath planning, combinatorial design, and modeling of systems with circular symmetry.

Circular partition algorithms encompass a range of combinatorial, geometric, and algorithmic techniques for dividing finite sets or geometric objects—specifically those exhibiting circular symmetry—into substructures ("partitions") with prescribed balance, optimality, or approximation properties. Applications span polygon partitioning, discrete sampling in computational geometry, arc-wise grouping for combinatorics, and toolpath discretization in manufacturing.

1. Formal Problem Statements and Variants

Geometric Circular Partitioning

In computational geometry, a recurrent formulation is: Given a polygon (regular or otherwise), partition it into pieces with maximal "circularity," measured by the aspect ratio $\rho(P) = D_\mathrm{out}(P)/r_\mathrm{in}(P)$—the diameter of the smallest circumscribing circle to the radius of the largest inscribed disk of each piece $P$ [0304023]. The canonical objectives are:

• Convex partitioning: All pieces are convex subsets.
• Nonconvex partitioning: Nonconvex pieces permitted for improved circularity [0412095].

The goal is to construct a partition $\mathcal{P} = \{P_1, ..., P_N\}$ minimizing $\max_i \rho(P_i)$.

Combinatorial Circular Partitioning

Khormali et al. (Khormali et al., 4 Jan 2026) formalize the general "circular partition" problem:

• Given $n$ items arranged on a circle, partition into $r$ groups under a step size $k$ (modulo $n$).
• Two core algorithmic variants:
  • Removal (elimination): Josephus-type round-robin elimination. Items are removed one at a time with a fixed step and assigned to groups cyclically.
  • Non-removal (grouping): The indices visited by repeated step-$k$ modular advances decompose into cycles, which are then assigned as groups.

These models require that the resulting groups be as balanced as possible: group sizes differ by at most one.

Circle Discretization for Applications

In the context of CNC/computer-aided manufacturing, the problem reduces to discretizing a circle into $n$ points or segments given a user-specified dimensional tolerance $\epsilon$, such that the maximum deviation from the ideal arc does not exceed $\epsilon$ (Vercueil, 2013).

2. Mathematical and Algorithmic Foundations

Convex and Nonconvex Geometric Partitions

Let $Q \subseteq \mathbb{R}^2$ be a regular $k$-gon.

• Aspect Ratio: For convex pieces, $\rho(P)\geq 2$ with equality if and only if $P$ is a perfect disk.
• Optimization: For a given $Q$, the optimal aspect ratio $\rho^*(Q)$ is minimized over all partitions.

Special cases:

• Equilateral triangle ($k=3$): $\rho^*(T)=2$, only attainable with infinitely many pieces; explicit finite constructions achieve $\rho \leq 2+\epsilon$ in $O(1/\epsilon)$ pieces via horizontal strip partitions [0304023].
• Square ($k=4$): Several intricate convex schemes (e.g., corner-cut plus central octagon, $3\times 3$ grids) attain $\rho^*(S)\leq 2.3096$, close to—but not exactly at—the theoretical pigeonhole and packing lower bounds [0304023].
• Higher regular polygons ($k>5$): The unpartitioned polygon is already optimal: $\rho^*(P_k)=2/\cos(\pi/k)$ [0304023].

Nonconvex partitions allow for significantly improved circularity but at the cost of high piece counts (e.g., pentagon requiring $>20,000$ pieces) [0412095].

Combinatorial Circular Partition Algorithms

Given $n$ items, $r$ target groups, and step size $k$:

• Removal approach: Balanced partition without dummies possible if and only if $r\mid n$. Otherwise, the minimal dummy padding $s$ is added so $r\mid(n+s)$. Items are dealt round-robin into bins, and after process completion, dummy items are discarded and swaps performed as necessary for uniformity (Khormali et al., 4 Jan 2026).
• Non-removal approach: Partitioning exists without dummies if and only if $\gcd(n,k)=r$. Otherwise, dummies or extra dummy groups are added to reach a compatible size. Cycle-decomposition of the map $i\mapsto i+k\pmod{m}$ yields $g$ cycles of size $m/g$, with $g=\gcd(m,k)$. Recursively applying this or refining further yields the required number and size of groups (Khormali et al., 4 Jan 2026).

Algorithmic complexity is $O(n)$ in all cases, with $O(r)$ dummy/group overhead.

Discretization of the Circle

Given a radius $R$ and tolerance $\epsilon$:

• The number of discrete points is $$n = \left\lceil \pi / \arccos(1-\epsilon/R) \right\rceil$$.
• Each point $(x_i, y_i)$ is calculated as $x_i = X_0 + R\cos\theta_i$, $y_i = Y_0 + R\sin\theta_i$, where $\theta_i = i 2\pi / n$ for $i=0,\ldots,n-1$ (Vercueil, 2013).
• For small $\epsilon/R$, a stable approximation is $\Delta\approx2\sqrt{2\epsilon/R}$ for the angular step.

This allows fine control over the tradeoff between data size and geometric accuracy.

3. Structural Theorems and Existence Results

Key results from (Khormali et al., 4 Jan 2026):

• Direct removal partition: Possible iff $r|n$.
• Direct non-removal partition: Possible iff $\gcd(n,k)=r$.
• Universal existence with augmentation: Theorem 2.1 establishes existence of a balanced $r$-partition for all $n,r,k$ via dummy padding (either extra items or groups), requiring at most $O(r)$ augmentation steps. After the process, at most one dummy needs to be swapped per group to enforce the one-dummy-per-group constraint, after which dummies are dropped.

For geometric partitions, the tightest possible aspect ratio is shown to depend critically on both $k$ and the allowable piece complexity.

4. Representative Algorithmic Schemes

Geometric Partition Algorithms

Polygon	Convex Partition Pieces	Nonconvex Partition Pieces	Attainable $\rho^*$
Triangle ($k$=3)	$N$ ($N\to\infty$ for $\rho^*=2$)	$4$	$2$ (convex: inf. pieces; nonconvex: 4)
Square ($k$=4)	$9$ (grid); $5$ (central octagon)	$13$	$2.3096$ (convex); lower for nonconvex
Pentagon ($k$=5)	$1$ (already optimal)	$>20,000$	Close to $2/\cos(\pi/5)$
$k > 5$	$1$ (unpartitioned optimal)	General algorithm	$2/\cos(\pi/k)$

Combinatorial Circular Partition Algorithms

Approach	Direct Partition Condition	Dummy Padding?	Cycle Structure
Removal	$r|n$	Required if $r\nmid n$	Items removed one-by-one and assigned round-robin
Non-removal	$\gcd(n,k)=r$	Required if $\gcd(n,k)\neq r$	Cycles under $i\mapsto i+k\pmod{m}$ decomposition

5. Complexity and Implementation

For all combinatorial circular partition algorithms in (Khormali et al., 4 Jan 2026):

• Time complexity: $O(n)$ for both removal and non-removal variants, irrespective of dummy group/item padding.
• Space complexity: $O(n)$ for maintaining the circle; $O(r)$ for group buffers.
• Combinatorial enumeration: Total labelings: $r^n$; surjective (nonempty groups): $r!S(n,r)$; balanced labelings: ${r\choose s}\frac{n!}{(q+1)!^{s}q!^{r-s}}$, with $n=qr+s$.

Geometric partition algorithms typically require numerical root-finding or combinatorial enumeration to optimize cut-parameters, with overall complexity dominated by the number of pieces.

Circle discretization algorithms (Vercueil, 2013) require $O(1)$ time to compute $n$ and $O(n)$ to enumerate points.

6. Illustrative Examples

• Triangle (convex): For $\epsilon = 0.01$, an explicit horizontal strip partition yields $N \approx 182$ convex pieces, each with $\rho \leq 2.01$ [0304023].
• Square (convex): $3\times3$ grid or "four corner-cuts plus octagon" schemes yield max $\rho \approx 2.32$ [0304023].
• Removal approach (combinatorial, (Khormali et al., 4 Jan 2026)): $n=12$, $r=4$, $k=3$: Directly produces four size-3 groups. $n=10$, $r=3$, $k=4$: Requires adding 2 dummies for balanced assignment.
• Non-removal approach (Khormali et al., 4 Jan 2026): $n=15$, $r=5$, $k=5$: $\gcd(15,5)=5$—yields five cycles of length 3.
• Circle discretization (Vercueil, 2013): $R=10$, $\epsilon=0.1$: $n=23$ segments; $\epsilon=0.01$: $n=71$ segments.

7. Connections and Applications

Circular partition algorithms intersect with the following domains:

• Polygon decomposition: Seeking optimal geometric properties (e.g., tiling with bounded aspect ratios).
• Josephus-type problems: Generalizations used to model cyclic eliminations and resource assignment.
• Combinatorial design: Construction of equitable groupings with cyclic or block constraints.
• CNC/Manufacturing: Generation of efficient toolpaths and circular discretizations within specified tolerances (Vercueil, 2013).
• Formal languages/automata: Circular partitions underpin certain formulations of block patterns and run-length encodings in context-free grammars, enabling decidability proofs for constrained sentence-generation problems (Khormali et al., 4 Jan 2026).

Each domain utilizes bespoke structural theorems, combinatorial enumeration, and algorithmic strategies tailored to the principal constraints: optimality, exact balance, and structural regularity. Explicit pseudocode, complexity bounds, and worked examples across these contexts are detailed within [0304023], [0412095], (Vercueil, 2013), and (Khormali et al., 4 Jan 2026).