Abstract
Analysis of Algorithms 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 = Sigma(i) AiXi o 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 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] there exists t -> 1 + mint, 1 - t.
Highlights
We consider fixed points of the operatorK(μ) =D AiXi ◦ Bi + C. (1)i∈N on the set D of cadlag functions on the unit interval I = [0, 1]
The very first example of such fixed points appeared in the analysis of Find as the limit distribution of the Find algorithm [10]
The infimum is taken over all processes (X, Z) with one dimensional distributions μ, ν on some probability space. In this distance we show all finite dimensional distributions of Xn converge to those of Y and all finite dimensional distributions are continuous in the cylinder coordinates. (This does not imply weak convergence on D, since we do not prove tightness or equivalently the convergence of T -distributions with T ⊂ I dense and countable.)
Summary
Grubel and Rosler used a specific representation of the iterates Kn by some rvs with nice properties and showed the convergence of theses rvs in Skorodhod metric on D to a limit, the fixed point This approach works for the 2-version of Find, splitting into 2 sets, but not for the 3-version, splitting into 3. The median m-version was treated by Grubel [9] He provided pictures of the limiting distribution function and gave more details for the average running time, which becomes better compared to original Find. It seems obvious, that in average the algorithm will perform better, if we can choose a pivot element, which leaves for the round a smaller list (at least with very high probability). The algorithm PICK, developed by Blum, Floyd, Pratt, Rivest and Tarjan [21][8], shows an upper bound of 5.4305...n comparisons
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call