Counting preimages of TCP reordering patterns

TL;DR

This paper develops efficient algorithms to determine the existence and count the number of packet ID permutations that produce a given buffer sequence under TCP reordering models, enhancing analytical understanding of TCP traffic behavior.

Contribution

It introduces polynomial-time algorithms for counting and verifying permutations of packet sequences consistent with TCP reordering models, extending to sequences with repeated packets.

Findings

Existence of a linear time algorithm for permutation verification.

Polynomial time algorithm for counting permutations.

Extension of results to sequences with repeated packets.

Abstract

Packet reordering is an important property of network traffic that should be captured by analytical models of the Transmission Control Protocol (TCP). We study a combinatorial problem motivated by RESTORED, a TCP modeling methodology that incorporates information about packet dynamics. A significant component of this model is a many-to-one mapping B that transforms sequences of packet IDs into buffer sequences, in a manner that is compatible with TCP semantics. We show that the following hold: 1. There exists a linear time algorithm that, given a buffer sequence W of length n, decides whether there exists a permutation A of 1,2,..., n such that $A\in B^{-1}(W)$ (and constructs such a permutation, when it exists). 2. The problem of counting the number of permutations in $B^{-1}(W)$ has a polynomial time algorithm. We also show how to extend these results to sequences of IDs that contain repeated packets.