Optimal Convex Hull Formation on a Grid by Asynchronous Robots With Lights

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We consider the distributed setting of <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="sharma-ieq1-3158202.gif"/></alternatives></inline-formula> autonomous mobile robots that operate in Look-Compute-Move cycles and communicate with other robots using a constant number of colored lights (the <i>robots with lights</i> model). We assume obstructed visibility where collinear robots do not see each other. In addition, we consider a grid-based terrain embedded in the 2-dimensional euclidean plane. The <small>Convex Hull Formation</small> problem is to relocate the <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="sharma-ieq2-3158202.gif"/></alternatives></inline-formula> robots (starting at arbitrary, but distinct, initial positions) so that each robot is positioned on a vertex of a convex hull. In this article, we provide a framework for solving <small>Convex Hull Formation</small>. We then provide four asynchronous algorithms under this framework. Key measures of the algorithms&#x2019; performance include the time taken and the space occupied. The presented algorithms are randomized and their time bounds hold with high probability. The first <inline-formula><tex-math notation="LaTeX">$O(\max \lbrace n^{2},D\rbrace)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq3-3158202.gif"/></alternatives></inline-formula>-time, <inline-formula><tex-math notation="LaTeX">$O({n^{2}})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq4-3158202.gif"/></alternatives></inline-formula>-perimeter, and <inline-formula><tex-math notation="LaTeX">$O({n^{3}})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq5-3158202.gif"/></alternatives></inline-formula>-area algorithm serves to introduce key ideas, where <inline-formula><tex-math notation="LaTeX">$D$</tex-math><alternatives><mml:math><mml:mi>D</mml:mi></mml:math><inline-graphic xlink:href="sharma-ieq6-3158202.gif"/></alternatives></inline-formula> is the diameter of the initial configuration. The subsequent algorithms, differing in computational requirements, run in <inline-formula><tex-math notation="LaTeX">$O(\max \lbrace n^{\frac{3}{2}},D\rbrace)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq7-3158202.gif"/></alternatives></inline-formula> time with a perimeter of <inline-formula><tex-math notation="LaTeX">$O(n^{\frac{3}{2}})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq8-3158202.gif"/></alternatives></inline-formula> and area of <inline-formula><tex-math notation="LaTeX">$O(n^{3})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq9-3158202.gif"/></alternatives></inline-formula>. We also prove lower bounds of <inline-formula><tex-math notation="LaTeX">$\Omega (n^{\frac{3}{2}})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>&#x03A9;</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq10-3158202.gif"/></alternatives></inline-formula> for time and perimeter and <inline-formula><tex-math notation="LaTeX">$\Omega (n^{3})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>&#x03A9;</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq11-3158202.gif"/></alternatives></inline-formula> for area, for any <small>Convex Hull Formation</small> algorithm; i.e., our <inline-formula><tex-math notation="LaTeX">$O(\max \lbrace n^{\frac{3}{2}},D\rbrace)-$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mrow><mml:mo>&#x007B;</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mo>&#x007D;</mml:mo></mml:mrow><mml:mo>)</mml:mo><mml:mo>-</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="sharma-ieq12-3158202.gif"/></alternatives></inline-formula>time algorithm is optimal in time, perimeter, and area.