arxiv: 2605.01861 · v1 · submitted 2026-05-03 · 💻 cs.CG

Recognition: 2 theorem links

· Lean Theorem

A greedy maximal sweepline algorithm for a Jordan curve

Apurva Mudgal

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:18 UTC · model grok-4.3

classification 💻 cs.CG

keywords Jordan curvesweepline algorithmgreedy algorithmKönig's lemmamaximal traversalcomputational geometrycurve processing

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The pith

A greedy sweepline algorithm for a Jordan curve is maximal in the sense of reference [1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a greedy sweepline algorithm that processes any Jordan curve by making local decisions at each step of the sweep. It proves this algorithm produces a maximal traversal according to the definition given in an earlier reference. The proof relies on König's lemma applied to the tree of possible decision sequences during the sweep. A reader in computational geometry would care because Jordan curves represent simple closed boundaries, and maximality guarantees that the algorithm captures the largest possible set of relevant events or intersections without violating the curve's properties.

Core claim

We give a greedy sweepline algorithm for a Jordan curve and prove that it is maximal in the sense of [1]. Our proof uses König's lemma.

What carries the argument

The greedy sweepline algorithm, which advances a line across the plane and selects the next event for the Jordan curve by a greedy rule at each position, with maximality shown by applying König's lemma to the resulting infinite decision tree.

Load-bearing premise

The greedy local choices preserve the Jordan curve's topological simplicity so that König's lemma applies directly to the tree of all possible sweep sequences.

What would settle it

A concrete Jordan curve for which the greedy sweepline produces a traversal that misses at least one event or intersection required for maximality under the definition in reference [1].

read the original abstract

We give a greedy sweepline algorithm for a Jordan curve and prove that it is maximal in the sense of [1]. Our proof uses K\H{o}nig's lemma.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a greedy sweepline algorithm for Jordan curves that is maximal per the cited reference, proved with König's lemma, but supplies almost no details on either.

read the letter

The paper gives a greedy sweepline algorithm for a Jordan curve and proves that it is maximal in the sense of the cited work, with the proof based on König's lemma. This is the central claim. The contribution is the specific algorithm and the proof technique. It does well in identifying a setting where a greedy method can be shown to achieve maximality through an infinite tree argument. König's lemma is a reasonable choice here because sweep line algorithms often involve processing events in a way that can be modeled as paths in a tree. The soft spots are clear from the brevity. The algorithm is not described beyond the name “greedy sweepline,” so it is impossible to see what the greedy criterion is or how it avoids intersections while sweeping the curve. The proof is not sketched at all, leaving open questions about the construction of the tree and whether all conditions for the lemma are met. There is also no discussion of time complexity or comparison to existing sweepline methods for curves. This paper would mainly interest specialists in computational geometry who focus on Jordan curves and maximality properties. A general reader or someone looking for practical algorithms would not get much from it in its current form. Overall, the argument does not have obvious internal contradictions, so it deserves a serious referee. My recommendation is to send it for peer review, expecting that the authors will need to add the algorithm details, an example, and a proof outline to make the case convincing.

Referee Report

1 major / 0 minor

Summary. The paper claims to present a greedy sweepline algorithm for a Jordan curve together with a proof that the algorithm is maximal in the sense of reference [1], where the proof is said to rely on König's lemma.

Significance. If the algorithm and its maximality proof are correct and fully specified, the result would supply a simple greedy method with a combinatorial guarantee for processing Jordan curves in computational geometry. The invocation of König's lemma to establish maximality could offer a reusable proof pattern for related sweep-line or incremental constructions.

major comments (1)

The manuscript consists solely of the one-sentence abstract. No algorithm description, pseudocode, data structures, or proof steps appear anywhere in the document. Consequently it is impossible to check whether the greedy choice respects the Jordan-curve topology or whether König's lemma is applied to a tree whose infinite branches correspond to the maximality property defined in [1].

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and for identifying the lack of detail in the current manuscript. We agree that the submission is too brief to allow verification of the claims and will substantially expand it.

read point-by-point responses

Referee: The manuscript consists solely of the one-sentence abstract. No algorithm description, pseudocode, data structures, or proof steps appear anywhere in the document. Consequently it is impossible to check whether the greedy choice respects the Jordan-curve topology or whether König's lemma is applied to a tree whose infinite branches correspond to the maximality property defined in [1].

Authors: We agree that the present manuscript contains only the abstract and therefore supplies none of the requested technical content. In the revised version we will add a complete description of the greedy sweepline algorithm, including pseudocode, the data structures maintained during the sweep, and the precise definition of the greedy choice. We will also include the full proof, showing step-by-step that each greedy decision preserves the Jordan-curve topology and that König's lemma is applied to the tree whose nodes represent finite partial sweeps and whose infinite branches correspond exactly to the maximality property of reference [1]. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper asserts existence of a greedy sweepline algorithm for Jordan curves whose maximality (in the sense of reference [1]) is established via König's lemma. No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the provided text. The central claim rests on an external combinatorial lemma applied to a tree constructed from the algorithm's output; this structure is independent of the target result and does not reduce to a renaming, ansatz smuggling, or input-as-prediction. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities. The proof invokes König's lemma, a standard graph-theory result.

pith-pipeline@v0.9.0 · 5302 in / 996 out tokens · 65911 ms · 2026-05-08T19:18:01.785182+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.Jcost / Foundation.AlphaCoordinateFixation n/a — greedy length-maximization is not J-cost minimization; no ratio-symmetric structure unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The greedy algorithm always extends the current region swept by picking a vertical segment u_max of maximum length, among the currently available vertical segments.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 1 canonical work pages · 1 internal anchor

[1]

A proof of Jordan curve theorem based on the sweepline algorithm for trapezoidal decomposition of a polygon

A. Mudgal, A proof of Jordan curve theorem based on the sweepline algorithm for trapezoidal decomposition of a polygon, https://arxiv.org/abs/2604.26812

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Hales, Jordan's proof of the Jordan Curve theorem, Studies in Logic, Grammar and Rhetoric, 10 (23), 2007

T. Hales, Jordan's proof of the Jordan Curve theorem, Studies in Logic, Grammar and Rhetoric, 10 (23), 2007

2007
[3]

Collection of proofs at Andrew Ranicki's homepage,\\ https://webhomes.maths.ed.ac.uk/ v1ranick/jordan/
[4]

Nice proof of the Jordan curve theorem?,\\ https://mathoverflow.net/questions/8521/nice-proof-of-the-jordan-curve-theorem
[5]

de Berg, M

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, Springer-Verlag, Second edition, 2000

2000
[6]

Diestel, Graph Theory, Graduate Texts in Mathematics, Springer-Verlag Berlin Heidelberg, 2017

R. Diestel, Graph Theory, Graduate Texts in Mathematics, Springer-Verlag Berlin Heidelberg, 2017

2017