Optimal Prefix Free Codes with Partial Sorting

Abstract

We describe an algorithm computing an optimal prefix free code for n unsorted positive weights in time within O(n(1 + lg α)) ⊆ O(n lg n), where the alternation α ∈ [1..n − 1] measures the amount of sorting required by the computation. This asymptotical complexity is within a constant factor of the optimal in the algebraic decision tree computational model, in the worst case over all instances of size n and alternation α. Such results refine the state of the art complexity of Θ(n lg n) in the worst case over instances of size n in the same computational model, a landmark in compression and coding since 1952, by the mere combination of van Leeuwen's algorithm to compute optimal prefix free codes from sorted weights (known since 1976), with Deferred Data Structures to partially sort a multiset depending on the queries on it (known since 1988).

Editorial: Special Issue on Data Compression Algorithms and Their Applications

“…After several rounds of revisions and reviewing, seven of the submitted articles were accepted for inclusion in the Special Issue. These seven articles present new results for a wide variety of data compression topics: prefix free codes [1], finding patterns using lossy compression [2], compression in PDE solvers [3], compression and embedding of trees [4], compaction of Church numerals [5], compressed sensing [6], time-universal data compression [7]. Combined these results represent some of the current trends in the field.…”

State of the Art and Future Trends in Data Reduction for High-Performance Computing

“…At first, the data has to be scanned and the frequencies must be calculated. The nodes are either inserted into the binary tree or combined depending on the frequencies [20]. While Huffman coding splits input data into components that are encoded separately, arithmetic coding encodes the entire input into a number.…”