A Unified Adaptive Recoding Framework for Batched Network Coding

Abstract

Batched network coding is a variation of random linear network coding which has low computational and storage costs. In order to adapt random fluctuations in the number of erasures in individual batches, it is not optimal to recode and transmit the same number of packets for all batches. Different distributed optimization models, which are called adaptive recoding schemes, were formulated for this purpose.The key component of these optimization problems is the expected value of the rank distribution of a batch at the next network node, which is also known as the expected rank. In this paper, we put forth a unified adaptive recoding framework. We show that the expected rank functions are concave when the packet loss pattern is a stationary stochastic process regardless of the field size, which covers but not limited to independent packet loss and burst packet loss. Under this concavity assumption, we show that there always exists a solution which not only can minimize the randomness on the number of recoded packets but also can tolerate rank distribution errors due to inaccurate measurements or limited precision of the machine. To obtain such an optimal solution, we propose tuning schemes that can turn any feasible solution into a desired optimal solution.

A Protocol Design Paradigm for Batched Sparse Codes

“…The above two models of adaptive recoding schemes were unified in Reference [ 82 ] when the expected rank functions are concave with respect to t . The unified model is similar to the one in Reference [ 44 ] but it becomes a concave optimization problem by performing linear interpolations on the discrete expected rank functions.…”

“…Regarding the optimal solution, Reference [ 82 ] showed that there is a robust solution which can tolerate inaccuracy on the input rank distribution. Also, there is a robust solution where the number of recoded packets is almost deterministic , that is, there is at most one rank such that the batches of this rank have a choice to transmit one more recoded packet.…”

“…This solution is called a preferred solution . The optimal solution given by a solver may not be a preferred solution, so Reference [ 82 ] provided some tuning schemes which can tune any feasible (not necessary optimal) solution into a preferred solution.…”

“…Practical implementations in References [ 41 , 44 ] suggest that the above limit can give a close approximation when the recoding field size is . In this case, we have It was shown in Reference [ 82 ] that when RLNC is used to generate all the recoded packets and the channel model follows a stationary stochastic process, then the expected rank function is monotonic increasing and concave with respect to t regardless of the recoding field size. Note that if itself is a sequence of binary random variables and f is the identity function, then we have for all t , that is, the stationarity of is inherited by .…”

“…Denote all those values of given a by the mapping . According to Reference [ 82 ], the plot can also be interpreted as a plot of Lagrange multiplier verse for the batches of rank r . That is, we can evaluate the mapping easily from Figure A3 .…”