Linear time universal coding and time reversal of tree sources via FSM closure

Tree models are efficient parametrizations of finite-memory processes, offering potentially significant model cost savings. The information theory literature has focused mostly on redundancy aspects of the universal estimation and coding of these models. In this paper, we investigate representations and supporting data structures for finite-memory processes, as well as the major impact these structures have on the universal algorithms in which they are used. We first generalize the class of tree models, and then define and investigate the properties of the finite-state machine (FSM) closure of a tree, which is the smallest FSM that generates all the processes generated by the tree. The interaction between FSM closures, generalized context trees (GCTs), and classical data structures such as compact suffix trees brings together the information-theoretic and the computational aspects, leading to the first algorithm for linear time encoding/decoding of a lossless twice-universal code in the class of three models. The implemented code is a two-pass version of Context. The corresponding optimal context selection rule and context transitions use tools similar to those employed in efficient implementation of the popular Burrows-Wheeler transform (BWT), yielding similar computational complexities. We also present a reversible transform that displays the same "context deinterleaving" feature as the BWT but is naturally based on an optimal context tree. FSM closures are also applied to an investigation of the effect of time reversal on tree models, motivated in part by the following question: When compressing a data sequence using a universal scheme in the class of tree models, can it make a difference whether we read the sequence from left to right or from right to left? Given a tree model of a process, we show constructively that the number of states in the tree model corresponding to the reversed process might be, in the extreme case, quadratic in the number of states of the original tree. This result answers the above motivating question in the affirmative.