On geometric shape construction via growth operations

Nada Almalki, Othon Michail

Abstract: We study algorithmic growth processes under a geometric setting. Each process begins with an initial shape of nodes SI=S0<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">I</mi></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub></math> and, in every time step t≥1<math><mi is="true">t</mi><mo is="true">≥</mo><mn is="true">1</mn></math>, by applying (in parallel) one or more growth operations of a specific type to the current shape, St−1<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">t</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></msub></math>, generates the next, St<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">t</mi></mrow></msub></math>, always satisfying |St|>|St−1|<math><mo stretchy="false" is="true">|</mo><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">t</mi></mrow></msub><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mo stretchy="false" is="true">|</mo><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">t</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></msub><mo stretchy="false" is="true">|</mo></math>. We define three types of growth operations and explore the algorithmic and structural properties of their resulting processes. Our goal is to characterize the classes of shapes that can be constructed in O(log⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> or polylog n time steps, n being the size of the final shape SF<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">F</mi></mrow></msub></math>. Moreover, we want to determine whether a given shape SF<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">F</mi></mrow></msub></math> can be constructed from a given initial shape SI<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">I</mi></mrow></msub></math> using a finite sequence of growth operations of a given type, called a constructor of SF<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">F</mi></mrow></msub></math>. We give exact and partial characterizations of classes of shapes that can be constructed in polylog n time steps, polynomial-time centralized algorithms for deciding reachability between pairs of input shapes (SI,SF)<math><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">I</mi></mrow></msub><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">F</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> and for generating constructors when SF<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">F</mi></mrow></msub></math> can be constructed from SI<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">I</mi></mrow></msub></math>, as well as some negative results.