Published in Volume XXV, Issue 1, 2015, pages 133-154, doi: 10.7561/SACS.2015.1.133

**Authors:** G. Istrate

### Abstract

Associate to each sequence A of integers (intending to model packet IDs in a TCP/IP stream) a sequence of positive integers of the same length M(A). The i’th entry of M(A) is the size (at time i) of the smallest buffer needed to hold out-of-order packets, where space is accounted for unreceived packets as well. Call two sequences A, B equivalent (written A≡FB B) if M(A) = M(B).

For a sequence of integers A define SUS(A) to be the shuffled-up-sequences reordering measure defined as the smallest possible number of classes in a partition of the original sequence into increasing subsequences. We prove the following result: any two permutations A, B of the same length with SUS(A), SUS(B) ≤ 3 such that A ≡FB B are identical. The result is no longer valid if we replace the upper bound 3 by 4.

We also consider a similar problem for permutations with repeats. In this case the uniqueness of the preimage is no longer true, but we obtain a characterization of all the preimages of a given sequence, which in particular allows us to count them in polynomial time.

The results were motivated by explaining the behavior and engineering RESTORED, a receiver-oriented model of traffic we introduced and experimentally validated in earlier work.

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### Bibtex

@article{sacscuza:istrate2015iaspftbd, title={Identifying Almost Sorted Permutations from TCP Buffer Dynamics}, author={G. Istrate}, journal={Scientific Annals of Computer Science}, volume={25}, number={1}, organization={``A.I. Cuza'' University, Iasi, Romania}, year={2015}, pages={133--154}, doi={10.7561/SACS.2015.1.133}, publisher={``A.I. Cuza'' University Press} }