Abstract. A common complaint about adaptive prefix coding is that it is much slower than static prefix coding. Karpinski and Nekrich recently took an important step towards resolving this: they gave an adaptive Shannon coding algorithm that encodes each character in O(1) amortized time and decodes it in O(log H) amortized time, where H is the empirical entropy of the input string s. For comparison, Gagie’s adaptive Shannon coder and both Knuth’s and Vitter’s adaptive Huffman coders all use Θ(H) amortized time for each character. In this paper we give an adaptive Shannon coder that both encodes and decodes each character in O(1) worst-case time. As with both previous adaptive Shannon coders, we store s in at most (H + 1)|s | + o(|s|) bits. We also show that this encoding length is worst-case optimal up to the lower order term. 1

We give the first sorting algorithm with bounds in terms of higher-order entropies: let S be a sequence of length m containing n distinct elements and let H # (S) be the #th-order empirical entropy of S, log n # O(m); our algorithm sorts S using (H # (S) + O(1))m comparisons.

We present four new asymmetric communication protocols, with which a server with high bandwidth can help clients with low bandwidth send it messages. Three of our protocols are the first to use only a single round of communication for each message. Unlike previous authors, we do not assume the server knows the messages' distribution.

In this paper, we demonstrate that we can effectively use results from the field of adaptive self-organizing data structures in enhancing compression schemes. Unlike adaptive lists, which have already been used in compression, to the best of our knowledge, adaptive self-organizing trees have not been used in this regard. To achieve this, we introduce a new data structure, the Partitioning Binary Search Tree (PBST) which, although based on the well-known Binary Search Tree (BST), also appropriately partitions the data elements into mutually exclusive sets. When used in conjunction with Fano encoding, the PBST leads to the so-called Fano Binary Search Tree (FBST), which, indeed, incorporates the required Fano coding (nearly-equal-probability) property into the BST. We demonstrate how both the PBST and FBST can be maintained adaptively and in a self-organizing manner. The updating procedure that converts a PBST into an FBST, and the corresponding new tree-based operators, namely the Shift-To-Left (STL) and the Shift-To-Right (STR) operators, are explicitly presented. The encoding and decoding procedures that also update the FBST have been implemented and rigorously tested. Our empirical results on files of the well-known benchmark, the Canterbury corpus, show that the adaptive Fano coding using FBSTs, the Huffman, and the greedy adaptive Fano coding achieve similar compression ratios. However, in terms of encoding/decoding speed, the new scheme is much faster than the latter two in the encoding phase, and they achieve approximately the same speed in the decoding phase. We believe that the same philosophy, namely that of using an adaptive self-organizing BST to maintain the frequencies, can also be utilized for other data encoding mechanisms, even as the Fenwick scheme has been used in arithmetic coding. 1

In this paper we study the adaptive prefix coding problem in cases where the size of the input alphabet is large. We present an online prefix coding algorithm that uses O(σ 1/λ+ǫ) bits of space for any constants ε> 0, λ> 1, and encodes the string of symbols in O(log log σ) time per symbol in the worst case, where σ is the size of the alphabet. The upper bound on the encoding length is λnH(s)+(λln 2+2+ǫ)n+O(σ 1/λ log 2 σ) bits. 1

Abstract. A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves ’ depths, it minimizes the maximum of any leaf’s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, O(nlog n)-time algorithm for building them. Drmota and Szpankowski (2002) gave another O(nlog n)-time algorithm, which checks the Kraft Inequality in each step of a binary search. In this paper we show how Drmota and Szpankowski’s algorithm can be made to run in linear time on a word RAM with Ω(log n)-bit words. We also discuss how our solution applies to problems in data compression, group testing and circuit design. 1