Identifying Almost Sorted Permutations from TCP Buffer Dynamics

• Main Author: G. Istrate
• Format: Article
• Language: English
• Published: Alexandru Ioan Cuza University of Iasi 2015-06-01
• Series: Scientific Annals of Computer Science
• Online Access: http://www.info.uaic.ro/bin/download/Annals/XXV1/XXV1_5.pdf

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 The result is no longer valid if we replace the upper bound 3 by 4.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