Evolutionary Algorithms for Cluster Geometry

The paper presents a genetic algorithm for maximizing the Isoperimetric Quotient of poly hedral clusters This algorithm helped extending the de nition of Isoperimetric Quotient to clusters with nonplanar faces Detailed description of the genetic algorithm is followed by the performance analysis Some other observations of genetic algorithm s behavior are also presented Introduction Evolutionary algorithms are becoming widely used in di erent elds of technology and science They are inspired by modeling natural selection which gives them certain properties not common in classic algorithmic approach One of these properties is the fact that they can adapt to the problem they are solving and this can help discovering some unpredicted points of view of that problem This paper describes how one can use evolutionary algorithm to determine the position of cluster s vertices in D space The idea is to de ne a tness function which describes the acceptability of particular arrangement of points and then search for the optimum of this function This approach was already used by some graph drawing algorithms for exam ple NiceGraph algorithm that minimizes the strain in the bonds among vertices using simulated annealing In the experiment described in this paper the so called Polyhedral Isoperimetric Quotient was used as tness function and genetic algorithm was used to nd its maximum Polyhedral Isoperimetric Quotient is basically a measure for how spherical a given poly hedron is It is a dimensionless quantity which ranges from for at planar objects to which is obtained only for sphere A detailed de nition is given later De nition of Polyhedral Isoperimetric Quotient Polya introduced Isoperimetric Quotient IQ as a measure of how spherical a given polyhedron M is IQ is de ned as a normalized ratio of the square of polyhedron s volume and the cube of its surface