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#### Identifying almost-sorted permutations from TCP buffer dynamics

Autori: Gabriel Istrate

Editorial: Scientific Annals of Computer Science (special issue dedicated to professor Sergiu Rudeanu's 80th birthday), XXV (1), p.133-154, 2015.

Rezumat:

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 The result is no longer valid if we replace the upper bound 3 by 4.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.

Cuvinte cheie: SUS, permutations