Rényi to Rényi ¿ Source Coding under Siege
2006 IEEE International Symposium on Information Theory
A novel lossless source coding paradigm applies to problems of unreliable lossless channels with low bitrate, in which a vital message needs to be transmitted prior to termination of communications. This paradigm can be applied to Alfréd Rényi's secondhand account of an ancient siege in which a spy was sent to scout the enemy but was captured. After escaping, the spy returned to his base in no condition to speak and unable to write. His commander asked him questions that he could answer by
... ng or shaking his head, and the fortress was defended with this information. Rényi told this story with reference to traditional lossless source coding, in which the objective is minimization of expected codeword length. The goal of maximizing probability of survival in the siege scenario is distinct from yet related to this traditional objective. Rather than finding a code minimizing expected codeword length P n i=1 p(i)l(i), this variant involves maximizing P n i=1 p(i)θ l(i) for a known θ ∈ (0, 1). When there are no restrictions on codewords, this problem can be solved using a known generalization of Huffman coding. The optimal solution has coding bounds which are functions of Rényi entropy; in addition to known bounds, new bounds are derived here. The alphabetically constrained version of this problem has applications in search trees and diagnostic testing. A novel dynamic programming algorithm -based upon the oldest known algorithm for the traditional alphabetic problem -optimizes this problem in O(n 3 ) time and O(n 2 ) space, whereas two novel approximation algorithms can find a suboptimal solution faster: one in linear time, the other in O(n log n). Coding bounds for the alphabetic version of this problem are also presented.