Faster sequential universal coding via block partitioning

Rissanen provided a sequential universal coding algorithm based on a block partitioning scheme, where the source model is estimated at the beginning of each block. This approach asymptotically approaches the entropy at the fastest possible rate of 1/2log(n) bits per unknown parameter. We show that the complexity of this algorithm is Ω(nlog(n)), which is comparable to existing sequential universal algorithms. We provide a sequential O(nlog(log(n))) algorithm by modifying Rissanen´s block partitioning scheme. The redundancy with our approach is greater than with Rissanen´s block partitioning scheme by a multiplicative factor 1+O(1/log(log(n))), hence it asymptotically approaches the entropy at the fastest possible rate.