## Identifying Almost Sorted Permutations from TCP Buffer Dynamics

Published in Volume XXV, Issue 1, 2015, p. 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 The result is no longer valid if we replace the upper bound 3 by 4.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.

**Keywords**: algorithms, packet reordering, shuffled up sequences.

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