Average case analysis for batched disk scheduling and increasing subsequences

We consider the problem of estimating the tour length and finding approximation algorithms for the asymmetric traveling salesman problem arising from the disk scheduling problem. Given N requests, we show that if the seek function has positive derivative at 0 the tour length is concentrated in probability around the value C_f,pN^1/2 for an explicit constant C_f,p dependent on the seek function and the distribution of requests. For linear seek function we provide even tighter bounds and provide an O(Nlog(N)) time algorithm for finding the optimal tour. The proof uses several results on the size and location of maximal increasing subsequences. To handle more general seek functions we introduce a more general concept of increasing subsequences. we provide order of magnitude estimates on the tour length for a wide class of seek functions with vanishing derivative at 0. For general seek functions we use some geometric information on the location of maximal generalized increasing subsequences obtained via Talagrand's isoperimetric inequalities to produce a probabilistic 1 + ε approximation algorithm. These results complement the results on guaranteed approximation algorithms for this problem presented in [2].