The Incompressibility Method

Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source. It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner. This is impossible for classical probability theory. Kolmogorov complexity is known variously asàlgorithmic information', `algorithmic entropy', `Kolmogorov-Chaitin complexity', `descriptional complexity', `shortest program length', `algorithmic randomness', and others. Using it, we developed a new mathematical proof technique, now known as thèincompressibility method'. The incompressibility method is a basic general technique such as thèpigeon hole' argument, `the counting method' or thèprobabilistic method'. The new method has been quite successful and we present recent examples. The rst example concerns a \static" problem in combinatorial geometry. From among ? n 3 triangles with vertices chosen from among n points in the unit square, U, let T be the one with the smallest area, and let A be the area of T. Heil-bronn's triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c=n 3 < n < C=n 3 for all large enough n, where n is the expectation of A. Moreover, c=n 3 < A < C=n 3 for almost all A, that is, almost all A are close to the expectation value so that we determine the area of the smallest triangle for an arrangement in \general position." Our second example concerns a \dynamic" problem in average-case running time of algorithms. The question of a nontrivial general lower bound (or upper bound) on the average-case complexity of Shellsort has been open for about forty years. We obtain the rst such lower bound.