The Analysis of Find or Perpetuities on Cadlag Functions

In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of some stochastic fixed points equation of the form X D = ∑ i AiXi ◦Bi +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 some random time change. We show the existence of solutions via the Weighted Branching Process. Specifically we present m-median and adapted versions of Find. For an optimal adapted version the normalized asymptotic number of comparisons needed satisfies the degenerate fixed point equation