A greedy maximal sweepline algorithm for a Jordan curve
We give a greedy sweepline algorithm for a Jordan curve and prove that it is maximal in the sense of [1]. Our proof uses Kőnig's lemma.
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arXiv subjects
Apurva Mudgal
Publications and source records attributed to Apurva Mudgal.
A proof of Jordan curve theorem based on the sweepline algorithm for trapezoidal decomposition of a polygon
We prove the Jordan curve theorem by generalizing the sweepline algorithm for trapezoidal decomposition of a polygon. Our proof uses Zorn's lemma (or, equivalently the axiom of choice). Though several proofs have been given for the Jordan curve theorem by various authors, ours is the {\bf first algorithmic proof} of Jordan curve theorem using computational geometry. Further, with some preparation, the proof can be taught as part of an undergraduate discrete mathematics course, where till now the proof of this theorem was considered inaccessible.
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Hardness and Approximation Schemes for Discrete Packing and Domination
We present polynomial-time approximation schemes based on local search} technique for both geometric (discrete) independent set (\mdis) and geometric (discrete) dominating set (\mdds) problems, where the objects are arbitrary radii disks and arbitrary side length axis-parallel squares. Further, we show that the \mdds~problem is \apx-hard for various shapes in the plane. Finally, we prove that both \mdis~and \mdds~problems are \np-hard for unit disks intersecting a horizontal line and axis-parallel unit squares intersecting a straight line with slope $-1$.
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Convexity of near-optimal orthogonal-pair-free sets on the unit sphere
A subset $S$ of the unit sphere $\mathbb{S}^2$ is called orthogonal-pair-free if and only if there do not exist two distinct points $u, v \in S$ at distance $\fracπ{2}$ from each other. Witsenhausen \cite{witsenhausen} asked the following question: {\it What is the least upper bound $α_3$ on the Lesbegue measure of any measurable orthogonal-pair-free subset of $\mathbb{S}^2$?} We prove the following result in this paper: Let $\mathcal{A}$ be the collection of all orthogonal-pair-free sets $S$ such that $S$ consists of a finite number of mutually disjoint convex sets. Then, $α_3 = \limsup_{S \in \mathcal{A}} μ(S)$. Thus, if the double cap conjecture \cite{kalai1} is not true, there is a set in $\mathcal{A}$ with measure strictly greater than the measure of the double cap.
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