### The Analysis of Find and Versions of it

*Diether Knof, Uwe Roesler*

#### Abstract

In the running time analysis of the algorithm Find and versions of it
appear as limiting distributions solutions of stochastic fixed points
equation of the form X D= ∑

_{i}A_{i}X_{i}ˆB_{i}+C on the space D of cadlag functions. The distribution of the D-valued process X is invariant by some random linear affine transformation of space and random time change. We show the existence of solutions in some generality via the Weighted Branching Process. Finite exponential moments are connected to stochastic fixed point of supremum type X D= sup_{i}(A_{i}X_{i}+C_{i}) on the positive reals. Specifically we present a running time analysis of m-median and adapted versions of Find. The finite dimensional distributions converge in L^{1}and are continuous in the cylinder coordinates. We present the optimal adapted version in the sense of low asymptotic average number of comparisons. The limit distribution of the optimal adapted version of Find is a point measure on the function [0,1]∋t↦1 + min{t, 1-t}.Full Text: PDF PostScript